3.12.5 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=346 \[ -\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 e (A e+4 B d)-4 b c d (3 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5 \sqrt {d}}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (7 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-3 c x (c d-b e) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )\right )}{4 b^4 c \left (b x+c x^2\right ) (c d-b e)}+\frac {3 \left (b^2 c e (5 A e+8 B d)-4 b c^2 d (5 A e+2 B d)+16 A c^3 d^2+b^3 (-B) e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c} \sqrt {c d-b e}} \]

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Rubi [A]  time = 0.87, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {818, 822, 826, 1166, 208} \begin {gather*} \frac {3 \left (b^2 c e (5 A e+8 B d)-4 b c^2 d (5 A e+2 B d)+16 A c^3 d^2+b^3 (-B) e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c} \sqrt {c d-b e}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (b^2 e (A e+4 B d)-4 b c d (3 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5 \sqrt {d}}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (7 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-3 c x (c d-b e) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )\right )}{4 b^4 c \left (b x+c x^2\right ) (c d-b e)}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(3/2))/(b*x + c*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(2*b^2*c*(b*x + c*x^2)^2) - (Sqrt[d + e
*x]*(b*(c*d - b*e)*(6*b*B*c*d - 12*A*c^2*d - 2*b^2*B*e + 7*A*b*c*e) - 3*c*(c*d - b*e)*(8*A*c^2*d + b^2*B*e - 4
*b*c*(B*d + A*e))*x))/(4*b^4*c*(c*d - b*e)*(b*x + c*x^2)) - (3*(16*A*c^2*d^2 + b^2*e*(4*B*d + A*e) - 4*b*c*d*(
2*B*d + 3*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*Sqrt[d]) + (3*(16*A*c^3*d^2 - b^3*B*e^2 - 4*b*c^2*d*(2*
B*d + 5*A*e) + b^2*c*e*(8*B*d + 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*Sqrt[c]*Sqrt[
c*d - b*e])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac {\int \frac {-\frac {1}{2} d \left (12 A c^2 d+2 b^2 B e-b c (6 B d+7 A e)\right )-\frac {1}{2} e \left (10 A c^2 d+b^2 B e-5 b c (B d+A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2 c}\\ &=-\frac {\sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (6 b B c d-12 A c^2 d-2 b^2 B e+7 A b c e\right )-3 c (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) x\right )}{4 b^4 c (c d-b e) \left (b x+c x^2\right )}-\frac {\int \frac {-\frac {3}{4} c d (c d-b e) \left (16 A c^2 d^2+b^2 e (4 B d+A e)-4 b c d (2 B d+3 A e)\right )-\frac {3}{4} c d e (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (6 b B c d-12 A c^2 d-2 b^2 B e+7 A b c e\right )-3 c (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) x\right )}{4 b^4 c (c d-b e) \left (b x+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{4} c d^2 e (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right )-\frac {3}{4} c d e (c d-b e) \left (16 A c^2 d^2+b^2 e (4 B d+A e)-4 b c d (2 B d+3 A e)\right )-\frac {3}{4} c d e (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 c d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (6 b B c d-12 A c^2 d-2 b^2 B e+7 A b c e\right )-3 c (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) x\right )}{4 b^4 c (c d-b e) \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 A c^2 d^2+b^2 e (4 B d+A e)-4 b c d (2 B d+3 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}-\frac {\left (3 \left (16 A c^3 d^2-b^3 B e^2-4 b c^2 d (2 B d+5 A e)+b^2 c e (8 B d+5 A e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5}\\ &=-\frac {\sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (6 b B c d-12 A c^2 d-2 b^2 B e+7 A b c e\right )-3 c (c d-b e) \left (8 A c^2 d+b^2 B e-4 b c (B d+A e)\right ) x\right )}{4 b^4 c (c d-b e) \left (b x+c x^2\right )}-\frac {3 \left (16 A c^2 d^2+b^2 e (4 B d+A e)-4 b c d (2 B d+3 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 \sqrt {d}}+\frac {3 \left (16 A c^3 d^2-b^3 B e^2-4 b c^2 d (2 B d+5 A e)+b^2 c e (8 B d+5 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 \sqrt {c} \sqrt {c d-b e}}\\ \end {align*}

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Mathematica [A]  time = 2.85, size = 472, normalized size = 1.36 \begin {gather*} \frac {\frac {6 b^2 c^{7/2} (d+e x)^{5/2} (c d-b e) \left (b^2 e (A e+4 B d)-b c d (11 A e+6 B d)+12 A c^2 d^2\right )+(b+c x) \left ((b+c x) \left (9 c^{5/2} (c d-b e)^2 \left (\frac {2}{3} \sqrt {d+e x} (4 d+e x)-2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right ) \left (b^2 e (A e+4 B d)-4 b c d (3 A e+2 B d)+16 A c^2 d^2\right )-6 c^2 d^2 \left (b^2 c e (5 A e+8 B d)-4 b c^2 d (5 A e+2 B d)+16 A c^3 d^2+b^3 (-B) e^2\right ) \left (\sqrt {c} \sqrt {d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )\right )-6 b c^{7/2} (d+e x)^{5/2} \left (b^3 e^2 (A e+4 B d)-b^2 c d e (14 A e+15 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{5/2} d (c d-b e)^2}-\frac {6 (d+e x)^{5/2} (A b e-8 A c d+4 b B d)}{b d x}-\frac {12 A (d+e x)^{5/2}}{x^2}}{24 b d (b+c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(3/2))/(b*x + c*x^2)^3,x]

[Out]

((-12*A*(d + e*x)^(5/2))/x^2 - (6*(4*b*B*d - 8*A*c*d + A*b*e)*(d + e*x)^(5/2))/(b*d*x) + (6*b^2*c^(7/2)*(c*d -
b*e)*(12*A*c^2*d^2 + b^2*e*(4*B*d + A*e) - b*c*d*(6*B*d + 11*A*e))*(d + e*x)^(5/2) + (b + c*x)*(-6*b*c^(7/2)*
(-24*A*c^3*d^3 + b^3*e^2*(4*B*d + A*e) + 12*b*c^2*d^2*(B*d + 3*A*e) - b^2*c*d*e*(15*B*d + 14*A*e))*(d + e*x)^(
5/2) + (b + c*x)*(9*c^(5/2)*(c*d - b*e)^2*(16*A*c^2*d^2 + b^2*e*(4*B*d + A*e) - 4*b*c*d*(2*B*d + 3*A*e))*((2*S
qrt[d + e*x]*(4*d + e*x))/3 - 2*d^(3/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 6*c^2*d^2*(16*A*c^3*d^2 - b^3*B*e^2
- 4*b*c^2*d*(2*B*d + 5*A*e) + b^2*c*e*(8*B*d + 5*A*e))*(Sqrt[c]*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d
- b*e)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]]))))/(b^4*c^(5/2)*d*(c*d - b*e)^2))/(24*b*d*(b +
c*x)^2)

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IntegrateAlgebraic [A]  time = 2.19, size = 568, normalized size = 1.64 \begin {gather*} -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (A b^2 e^2-12 A b c d e+16 A c^2 d^2+4 b^2 B d e-8 b B c d^2\right )}{4 b^5 \sqrt {d}}+\frac {3 \left (5 A b^2 c e^2-20 A b c^2 d e+16 A c^3 d^2+b^3 (-B) e^2+8 b^2 B c d e-8 b B c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{4 b^5 \sqrt {c} \sqrt {b e-c d}}-\frac {\sqrt {d+e x} \left (5 A b^3 e^3 (d+e x)-3 A b^3 d e^3+27 A b^2 c d^2 e^2-46 A b^2 c d e^2 (d+e x)+19 A b^2 c e^2 (d+e x)^2-48 A b c^2 d^3 e+108 A b c^2 d^2 e (d+e x)-72 A b c^2 d e (d+e x)^2+12 A b c^2 e (d+e x)^3+24 A c^3 d^4-72 A c^3 d^3 (d+e x)+72 A c^3 d^2 (d+e x)^2-24 A c^3 d (d+e x)^3-9 b^3 B d^2 e^2+14 b^3 B d e^2 (d+e x)-5 b^3 B e^2 (d+e x)^2+21 b^2 B c d^3 e-45 b^2 B c d^2 e (d+e x)+27 b^2 B c d e (d+e x)^2-3 b^2 B c e (d+e x)^3-12 b B c^2 d^4+36 b B c^2 d^3 (d+e x)-36 b B c^2 d^2 (d+e x)^2+12 b B c^2 d (d+e x)^3\right )}{4 b^4 e x^2 (b e+c (d+e x)-c d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(3/2))/(b*x + c*x^2)^3,x]

[Out]

-1/4*(Sqrt[d + e*x]*(-12*b*B*c^2*d^4 + 24*A*c^3*d^4 + 21*b^2*B*c*d^3*e - 48*A*b*c^2*d^3*e - 9*b^3*B*d^2*e^2 +
27*A*b^2*c*d^2*e^2 - 3*A*b^3*d*e^3 + 36*b*B*c^2*d^3*(d + e*x) - 72*A*c^3*d^3*(d + e*x) - 45*b^2*B*c*d^2*e*(d +
e*x) + 108*A*b*c^2*d^2*e*(d + e*x) + 14*b^3*B*d*e^2*(d + e*x) - 46*A*b^2*c*d*e^2*(d + e*x) + 5*A*b^3*e^3*(d +
e*x) - 36*b*B*c^2*d^2*(d + e*x)^2 + 72*A*c^3*d^2*(d + e*x)^2 + 27*b^2*B*c*d*e*(d + e*x)^2 - 72*A*b*c^2*d*e*(d
+ e*x)^2 - 5*b^3*B*e^2*(d + e*x)^2 + 19*A*b^2*c*e^2*(d + e*x)^2 + 12*b*B*c^2*d*(d + e*x)^3 - 24*A*c^3*d*(d +
e*x)^3 - 3*b^2*B*c*e*(d + e*x)^3 + 12*A*b*c^2*e*(d + e*x)^3))/(b^4*e*x^2*(-(c*d) + b*e + c*(d + e*x))^2) + (3*
(-8*b*B*c^2*d^2 + 16*A*c^3*d^2 + 8*b^2*B*c*d*e - 20*A*b*c^2*d*e - b^3*B*e^2 + 5*A*b^2*c*e^2)*ArcTan[(Sqrt[c]*S
qrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*Sqrt[c]*Sqrt[-(c*d) + b*e]) - (3*(-8*b*B*c*d^2 + 16*A*c^
2*d^2 + 4*b^2*B*d*e - 12*A*b*c*d*e + A*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*Sqrt[d])

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fricas [B]  time = 2.07, size = 3471, normalized size = 10.03

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*((8*(B*b*c^4 - 2*A*c^5)*d^3 - 4*(2*B*b^2*c^3 - 5*A*b*c^4)*d^2*e + (B*b^3*c^2 - 5*A*b^2*c^3)*d*e^2)*x^4
+ 2*(8*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 4*(2*B*b^3*c^2 - 5*A*b^2*c^3)*d^2*e + (B*b^4*c - 5*A*b^3*c^2)*d*e^2)*x^3
+ (8*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 - 4*(2*B*b^4*c - 5*A*b^3*c^2)*d^2*e + (B*b^5 - 5*A*b^4*c)*d*e^2)*x^2)*sqrt
(c^2*d - b*c*e)*log((c*e*x + 2*c*d - b*e - 2*sqrt(c^2*d - b*c*e)*sqrt(e*x + d))/(c*x + b)) - 3*((A*b^3*c^3*e^3
+ 8*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(3*B*b^2*c^4 - 7*A*b*c^5)*d^2*e + (4*B*b^3*c^3 - 13*A*b^2*c^4)*d*e^2)*x^4 + 2
*(A*b^4*c^2*e^3 + 8*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(3*B*b^3*c^3 - 7*A*b^2*c^4)*d^2*e + (4*B*b^4*c^2 - 13*A*b^
3*c^3)*d*e^2)*x^3 + (A*b^5*c*e^3 + 8*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(3*B*b^4*c^2 - 7*A*b^3*c^3)*d^2*e + (4*
B*b^5*c - 13*A*b^4*c^2)*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*A*b^4*c^2*d^3
- 2*A*b^5*c*d^2*e + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (5*B*b^3*c^3 - 12*A*b^2*c^4)*d^2*e + (B*b^4*c^2 - 4*A*b
^3*c^3)*d*e^2)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (23*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + (5*B*b^5*c - 19
*A*b^4*c^2)*d*e^2)*x^2 - (5*A*b^5*c*d*e^2 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3 + (4*B*b^5*c - 13*A*b^4*c^2)*d^2*e
)*x)*sqrt(e*x + d))/((b^5*c^4*d^2 - b^6*c^3*d*e)*x^4 + 2*(b^6*c^3*d^2 - b^7*c^2*d*e)*x^3 + (b^7*c^2*d^2 - b^8*
c*d*e)*x^2), 1/8*(6*((8*(B*b*c^4 - 2*A*c^5)*d^3 - 4*(2*B*b^2*c^3 - 5*A*b*c^4)*d^2*e + (B*b^3*c^2 - 5*A*b^2*c^3
)*d*e^2)*x^4 + 2*(8*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 4*(2*B*b^3*c^2 - 5*A*b^2*c^3)*d^2*e + (B*b^4*c - 5*A*b^3*c^2
)*d*e^2)*x^3 + (8*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 - 4*(2*B*b^4*c - 5*A*b^3*c^2)*d^2*e + (B*b^5 - 5*A*b^4*c)*d*e^
2)*x^2)*sqrt(-c^2*d + b*c*e)*arctan(sqrt(-c^2*d + b*c*e)*sqrt(e*x + d)/(c*e*x + c*d)) - 3*((A*b^3*c^3*e^3 + 8*
(B*b*c^5 - 2*A*c^6)*d^3 - 4*(3*B*b^2*c^4 - 7*A*b*c^5)*d^2*e + (4*B*b^3*c^3 - 13*A*b^2*c^4)*d*e^2)*x^4 + 2*(A*b
^4*c^2*e^3 + 8*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(3*B*b^3*c^3 - 7*A*b^2*c^4)*d^2*e + (4*B*b^4*c^2 - 13*A*b^3*c^3
)*d*e^2)*x^3 + (A*b^5*c*e^3 + 8*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(3*B*b^4*c^2 - 7*A*b^3*c^3)*d^2*e + (4*B*b^5
*c - 13*A*b^4*c^2)*d*e^2)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*A*b^4*c^2*d^3 - 2*A
*b^5*c*d^2*e + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (5*B*b^3*c^3 - 12*A*b^2*c^4)*d^2*e + (B*b^4*c^2 - 4*A*b^3*c^
3)*d*e^2)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (23*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + (5*B*b^5*c - 19*A*b^
4*c^2)*d*e^2)*x^2 - (5*A*b^5*c*d*e^2 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3 + (4*B*b^5*c - 13*A*b^4*c^2)*d^2*e)*x)*
sqrt(e*x + d))/((b^5*c^4*d^2 - b^6*c^3*d*e)*x^4 + 2*(b^6*c^3*d^2 - b^7*c^2*d*e)*x^3 + (b^7*c^2*d^2 - b^8*c*d*e
)*x^2), -1/8*(6*((A*b^3*c^3*e^3 + 8*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(3*B*b^2*c^4 - 7*A*b*c^5)*d^2*e + (4*B*b^3*c^3
- 13*A*b^2*c^4)*d*e^2)*x^4 + 2*(A*b^4*c^2*e^3 + 8*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(3*B*b^3*c^3 - 7*A*b^2*c^4)
*d^2*e + (4*B*b^4*c^2 - 13*A*b^3*c^3)*d*e^2)*x^3 + (A*b^5*c*e^3 + 8*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(3*B*b^4
*c^2 - 7*A*b^3*c^3)*d^2*e + (4*B*b^5*c - 13*A*b^4*c^2)*d*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) -
3*((8*(B*b*c^4 - 2*A*c^5)*d^3 - 4*(2*B*b^2*c^3 - 5*A*b*c^4)*d^2*e + (B*b^3*c^2 - 5*A*b^2*c^3)*d*e^2)*x^4 + 2*
(8*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 4*(2*B*b^3*c^2 - 5*A*b^2*c^3)*d^2*e + (B*b^4*c - 5*A*b^3*c^2)*d*e^2)*x^3 + (8
*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 - 4*(2*B*b^4*c - 5*A*b^3*c^2)*d^2*e + (B*b^5 - 5*A*b^4*c)*d*e^2)*x^2)*sqrt(c^2*
d - b*c*e)*log((c*e*x + 2*c*d - b*e - 2*sqrt(c^2*d - b*c*e)*sqrt(e*x + d))/(c*x + b)) + 2*(2*A*b^4*c^2*d^3 - 2
*A*b^5*c*d^2*e + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (5*B*b^3*c^3 - 12*A*b^2*c^4)*d^2*e + (B*b^4*c^2 - 4*A*b^3*
c^3)*d*e^2)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (23*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + (5*B*b^5*c - 19*A*
b^4*c^2)*d*e^2)*x^2 - (5*A*b^5*c*d*e^2 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3 + (4*B*b^5*c - 13*A*b^4*c^2)*d^2*e)*x
)*sqrt(e*x + d))/((b^5*c^4*d^2 - b^6*c^3*d*e)*x^4 + 2*(b^6*c^3*d^2 - b^7*c^2*d*e)*x^3 + (b^7*c^2*d^2 - b^8*c*d
*e)*x^2), 1/4*(3*((8*(B*b*c^4 - 2*A*c^5)*d^3 - 4*(2*B*b^2*c^3 - 5*A*b*c^4)*d^2*e + (B*b^3*c^2 - 5*A*b^2*c^3)*d
*e^2)*x^4 + 2*(8*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 4*(2*B*b^3*c^2 - 5*A*b^2*c^3)*d^2*e + (B*b^4*c - 5*A*b^3*c^2)*d
*e^2)*x^3 + (8*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3 - 4*(2*B*b^4*c - 5*A*b^3*c^2)*d^2*e + (B*b^5 - 5*A*b^4*c)*d*e^2)*
x^2)*sqrt(-c^2*d + b*c*e)*arctan(sqrt(-c^2*d + b*c*e)*sqrt(e*x + d)/(c*e*x + c*d)) - 3*((A*b^3*c^3*e^3 + 8*(B*
b*c^5 - 2*A*c^6)*d^3 - 4*(3*B*b^2*c^4 - 7*A*b*c^5)*d^2*e + (4*B*b^3*c^3 - 13*A*b^2*c^4)*d*e^2)*x^4 + 2*(A*b^4*
c^2*e^3 + 8*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(3*B*b^3*c^3 - 7*A*b^2*c^4)*d^2*e + (4*B*b^4*c^2 - 13*A*b^3*c^3)*d
*e^2)*x^3 + (A*b^5*c*e^3 + 8*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(3*B*b^4*c^2 - 7*A*b^3*c^3)*d^2*e + (4*B*b^5*c
- 13*A*b^4*c^2)*d*e^2)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (2*A*b^4*c^2*d^3 - 2*A*b^5*c*d^2*e + 3
*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (5*B*b^3*c^3 - 12*A*b^2*c^4)*d^2*e + (B*b^4*c^2 - 4*A*b^3*c^3)*d*e^2)*x^3 +
(18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (23*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + (5*B*b^5*c - 19*A*b^4*c^2)*d*e^2)*x^
2 - (5*A*b^5*c*d*e^2 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3 + (4*B*b^5*c - 13*A*b^4*c^2)*d^2*e)*x)*sqrt(e*x + d))/(
(b^5*c^4*d^2 - b^6*c^3*d*e)*x^4 + 2*(b^6*c^3*d^2 - b^7*c^2*d*e)*x^3 + (b^7*c^2*d^2 - b^8*c*d*e)*x^2)]

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giac [B]  time = 0.28, size = 661, normalized size = 1.91 \begin {gather*} \frac {3 \, {\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 8 \, B b^{2} c d e + 20 \, A b c^{2} d e + B b^{3} e^{2} - 5 \, A b^{2} c e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, \sqrt {-c^{2} d + b c e} b^{5}} - \frac {3 \, {\left (8 \, B b c d^{2} - 16 \, A c^{2} d^{2} - 4 \, B b^{2} d e + 12 \, A b c d e - A b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d}} - \frac {12 \, {\left (x e + d\right )}^{\frac {7}{2}} B b c^{2} d e - 24 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{3} d e - 36 \, {\left (x e + d\right )}^{\frac {5}{2}} B b c^{2} d^{2} e + 72 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{3} d^{2} e + 36 \, {\left (x e + d\right )}^{\frac {3}{2}} B b c^{2} d^{3} e - 72 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} d^{3} e - 12 \, \sqrt {x e + d} B b c^{2} d^{4} e + 24 \, \sqrt {x e + d} A c^{3} d^{4} e - 3 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{2} c e^{2} + 12 \, {\left (x e + d\right )}^{\frac {7}{2}} A b c^{2} e^{2} + 27 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{2} c d e^{2} - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} A b c^{2} d e^{2} - 45 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c d^{2} e^{2} + 108 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{2} d^{2} e^{2} + 21 \, \sqrt {x e + d} B b^{2} c d^{3} e^{2} - 48 \, \sqrt {x e + d} A b c^{2} d^{3} e^{2} - 5 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} e^{3} + 19 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{2} c e^{3} + 14 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d e^{3} - 46 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c d e^{3} - 9 \, \sqrt {x e + d} B b^{3} d^{2} e^{3} + 27 \, \sqrt {x e + d} A b^{2} c d^{2} e^{3} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} e^{4} - 3 \, \sqrt {x e + d} A b^{3} d e^{4}}{4 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

3/4*(8*B*b*c^2*d^2 - 16*A*c^3*d^2 - 8*B*b^2*c*d*e + 20*A*b*c^2*d*e + B*b^3*e^2 - 5*A*b^2*c*e^2)*arctan(sqrt(x*
e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5) - 3/4*(8*B*b*c*d^2 - 16*A*c^2*d^2 - 4*B*b^2*d*e + 12
*A*b*c*d*e - A*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^2*d*e -
24*(x*e + d)^(7/2)*A*c^3*d*e - 36*(x*e + d)^(5/2)*B*b*c^2*d^2*e + 72*(x*e + d)^(5/2)*A*c^3*d^2*e + 36*(x*e + d
)^(3/2)*B*b*c^2*d^3*e - 72*(x*e + d)^(3/2)*A*c^3*d^3*e - 12*sqrt(x*e + d)*B*b*c^2*d^4*e + 24*sqrt(x*e + d)*A*c
^3*d^4*e - 3*(x*e + d)^(7/2)*B*b^2*c*e^2 + 12*(x*e + d)^(7/2)*A*b*c^2*e^2 + 27*(x*e + d)^(5/2)*B*b^2*c*d*e^2 -
72*(x*e + d)^(5/2)*A*b*c^2*d*e^2 - 45*(x*e + d)^(3/2)*B*b^2*c*d^2*e^2 + 108*(x*e + d)^(3/2)*A*b*c^2*d^2*e^2 +
21*sqrt(x*e + d)*B*b^2*c*d^3*e^2 - 48*sqrt(x*e + d)*A*b*c^2*d^3*e^2 - 5*(x*e + d)^(5/2)*B*b^3*e^3 + 19*(x*e +
d)^(5/2)*A*b^2*c*e^3 + 14*(x*e + d)^(3/2)*B*b^3*d*e^3 - 46*(x*e + d)^(3/2)*A*b^2*c*d*e^3 - 9*sqrt(x*e + d)*B*
b^3*d^2*e^3 + 27*sqrt(x*e + d)*A*b^2*c*d^2*e^3 + 5*(x*e + d)^(3/2)*A*b^3*e^4 - 3*sqrt(x*e + d)*A*b^3*d*e^4)/((
(x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4)

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maple [B]  time = 0.08, size = 785, normalized size = 2.27 \begin {gather*} -\frac {9 \sqrt {e x +d}\, A c \,e^{3}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {21 \sqrt {e x +d}\, A \,c^{2} d \,e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, A \,c^{3} d^{2} e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {5 \sqrt {e x +d}\, B \,e^{3}}{4 \left (c e x +b e \right )^{2} b}-\frac {13 \sqrt {e x +d}\, B c d \,e^{2}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {2 \sqrt {e x +d}\, B \,c^{2} d^{2} e}{\left (c e x +b e \right )^{2} b^{3}}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} A \,c^{2} e^{2}}{4 \left (c e x +b e \right )^{2} b^{3}}-\frac {15 A c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} A \,c^{3} d e}{\left (c e x +b e \right )^{2} b^{4}}+\frac {15 A \,c^{2} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {12 A \,c^{3} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{5}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} B c \,e^{2}}{4 \left (c e x +b e \right )^{2} b^{2}}+\frac {3 B \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,c^{2} d e}{\left (c e x +b e \right )^{2} b^{3}}-\frac {6 B c d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {6 B \,c^{2} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {3 A \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} \sqrt {d}}+\frac {9 A c \sqrt {d}\, e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}-\frac {12 A \,c^{2} d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5}}-\frac {3 B \sqrt {d}\, e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}+\frac {6 B c \,d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4}}+\frac {3 \sqrt {e x +d}\, A d}{4 b^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, A c \,d^{2}}{b^{4} e \,x^{2}}+\frac {\sqrt {e x +d}\, B \,d^{2}}{b^{3} e \,x^{2}}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} A}{4 b^{3} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} A c d}{b^{4} e \,x^{2}}-\frac {\left (e x +d \right )^{\frac {3}{2}} B d}{b^{3} e \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x)^3,x)

[Out]

-12/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*c^3*d^2+6/b^4/((b*e-c*d)*c)^(1/2)*ar
ctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*c^2*d^2-15/4*e^2/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*
e-c*d)*c)^(1/2)*c)*A*c-7/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^2+3/4*e^2/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B
*c-1/e/b^3/x^2*(e*x+d)^(3/2)*B*d-9/4*e^3/b^2/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*c+9*e/b^4*d^(1/2)*arctanh((e*x+d)^(
1/2)/d^(1/2))*A*c+1/e/b^3/x^2*(e*x+d)^(1/2)*B*d^2-5/4/b^3/x^2*(e*x+d)^(3/2)*A-12/b^5*d^(3/2)*arctanh((e*x+d)^(
1/2)/d^(1/2))*A*c^2+5/4*e^3/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-3/4*e^2/b^3/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))
*A-3*e/b^3*d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B+6/b^4*d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c+3/4/b^3/x
^2*(e*x+d)^(1/2)*A*d+3/4*e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+3/e/b^4/x^2
*(e*x+d)^(3/2)*A*c*d-3/e/b^4/x^2*(e*x+d)^(1/2)*A*c*d^2+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^3*d-3*e/b^4/(c*
e*x+b*e)^2*A*(e*x+d)^(1/2)*c^3*d^2-13/4*e^2/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*c*d-6*e/b^3/((b*e-c*d)*c)^(1/2)*
arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*c*d-2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*c^2*d+21/4*e^2/b^3/(c*
e*x+b*e)^2*A*(e*x+d)^(1/2)*c^2*d+2*e/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*c^2*d^2+15*e/b^4/((b*e-c*d)*c)^(1/2)*ar
ctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*c^2*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 4.46, size = 5796, normalized size = 16.75

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(3/2))/(b*x + c*x^2)^3,x)

[Out]

((3*(d + e*x)^(1/2)*(A*b^3*d*e^4 - 8*A*c^3*d^4*e + 3*B*b^3*d^2*e^3 + 16*A*b*c^2*d^3*e^2 - 9*A*b^2*c*d^2*e^3 -
7*B*b^2*c*d^3*e^2 + 4*B*b*c^2*d^4*e))/(4*b^4) - ((d + e*x)^(3/2)*(5*A*b^3*e^4 - 72*A*c^3*d^3*e + 14*B*b^3*d*e^
3 + 108*A*b*c^2*d^2*e^2 - 45*B*b^2*c*d^2*e^2 - 46*A*b^2*c*d*e^3 + 36*B*b*c^2*d^3*e))/(4*b^4) + ((d + e*x)^(5/2
)*(5*B*b^3*e^3 - 19*A*b^2*c*e^3 - 72*A*c^3*d^2*e + 72*A*b*c^2*d*e^2 + 36*B*b*c^2*d^2*e - 27*B*b^2*c*d*e^2))/(4
*b^4) + (3*c*(d + e*x)^(7/2)*(B*b^2*e^2 - 4*A*b*c*e^2 + 8*A*c^2*d*e - 4*B*b*c*d*e))/(4*b^4))/(c^2*(d + e*x)^4
- (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 +
6*c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) + (atan((((-c*(b*e - c*d))^(1/2)*(((d + e*x)^(1
/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6 + 6624*A^2*b^2*c^5*d^2*e^4 + 1152*B^2*b^2*c^
5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 90*A*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d^3*e^3 -
2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d^4*e^2 + 1152*A*B*b^4*c^3*d*e^5 + 8064*A*B*b
^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4))/(8*b^8) - (3*(-c*(b*e - c*d))^(1/2)*((3*A*b^12*c^2*e^5 - 24*A*b^11
*c^3*d*e^4 + 9*B*b^12*c^2*d*e^4 + 24*A*b^10*c^4*d^2*e^3 - 12*B*b^11*c^3*d^2*e^3)/b^12 - (3*(64*b^11*c^2*e^3 -
128*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c
^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(64*b^8*(b^5*c^2*d - b^6*c*e)))*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*
c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*(b^5*c^2*d - b^6*c*e)))*(16*A*c^3*d^2 - B*b^3*e^2
+ 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e)*3i)/(8*(b^5*c^2*d - b^6*c*e)) + ((-c*(b*e -
c*d))^(1/2)*(((d + e*x)^(1/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6 + 6624*A^2*b^2*c^5
*d^2*e^4 + 1152*B^2*b^2*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 90*A*B*b^5*c^2*e^6
- 9216*A^2*b*c^6*d^3*e^3 - 2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d^4*e^2 + 1152*A*B*
b^4*c^3*d*e^5 + 8064*A*B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4))/(8*b^8) + (3*(-c*(b*e - c*d))^(1/2)*((3*
A*b^12*c^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*B*b^12*c^2*d*e^4 + 24*A*b^10*c^4*d^2*e^3 - 12*B*b^11*c^3*d^2*e^3)/b^1
2 + (3*(64*b^11*c^2*e^3 - 128*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(16*A*c^3*d^2 - B*b^3*e^2
+ 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(64*b^8*(b^5*c^2*d - b^6*c*e)))*(16*A*c^3*
d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*(b^5*c^2*d - b^6*c*e)))*
(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e)*3i)/(8*(b^5*c^2*d
- b^6*c*e)))/(((135*A^3*b^5*c^3*e^8)/8 - 1728*A^3*c^8*d^5*e^3 - 3996*A^3*b^2*c^6*d^3*e^5 + 1674*A^3*b^3*c^5*d^
2*e^6 + 216*B^3*b^3*c^5*d^5*e^3 - 378*B^3*b^4*c^4*d^4*e^4 + 216*B^3*b^5*c^3*d^3*e^5 - (189*B^3*b^6*c^2*d^2*e^6
)/4 + (27*A*B^2*b^7*c*e^8)/32 + (27*B^3*b^7*c*d*e^7)/8 - (243*A^2*B*b^6*c^2*e^8)/32 + 4320*A^3*b*c^7*d^4*e^4 -
(1215*A^3*b^4*c^4*d*e^7)/4 - 1296*A*B^2*b^2*c^6*d^5*e^3 + 2592*A*B^2*b^3*c^5*d^4*e^4 - 1782*A*B^2*b^4*c^4*d^3
*e^5 + (999*A*B^2*b^5*c^3*d^2*e^6)/2 - 5832*A^2*B*b^2*c^6*d^4*e^4 + 4698*A^2*B*b^3*c^5*d^3*e^5 - (3267*A^2*B*b
^4*c^4*d^2*e^6)/2 - (405*A*B^2*b^6*c^2*d*e^7)/8 + 2592*A^2*B*b*c^7*d^5*e^3 + (1809*A^2*B*b^5*c^3*d*e^7)/8)/b^1
2 - (3*(-c*(b*e - c*d))^(1/2)*(((d + e*x)^(1/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6
+ 6624*A^2*b^2*c^5*d^2*e^4 + 1152*B^2*b^2*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 9
0*A*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d^3*e^3 - 2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d
^4*e^2 + 1152*A*B*b^4*c^3*d*e^5 + 8064*A*B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4))/(8*b^8) - (3*(-c*(b*e
- c*d))^(1/2)*((3*A*b^12*c^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*B*b^12*c^2*d*e^4 + 24*A*b^10*c^4*d^2*e^3 - 12*B*b^1
1*c^3*d^2*e^3)/b^12 - (3*(64*b^11*c^2*e^3 - 128*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(16*A*c
^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(64*b^8*(b^5*c^2*d - b^6
*c*e)))*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))/(8*(b^5*c
^2*d - b^6*c*e)))*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))
/(8*(b^5*c^2*d - b^6*c*e)) + (3*(-c*(b*e - c*d))^(1/2)*(((d + e*x)^(1/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d
^4*e^2 + 9*B^2*b^6*c*e^6 + 6624*A^2*b^2*c^5*d^2*e^4 + 1152*B^2*b^2*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 86
4*B^2*b^4*c^3*d^2*e^4 - 90*A*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d^3*e^3 - 2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2
*d*e^5 - 4608*A*B*b*c^6*d^4*e^2 + 1152*A*B*b^4*c^3*d*e^5 + 8064*A*B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4
))/(8*b^8) + (3*(-c*(b*e - c*d))^(1/2)*((3*A*b^12*c^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*B*b^12*c^2*d*e^4 + 24*A*b^
10*c^4*d^2*e^3 - 12*B*b^11*c^3*d^2*e^3)/b^12 + (3*(64*b^11*c^2*e^3 - 128*b^10*c^3*d*e^2)*(-c*(b*e - c*d))^(1/2
)*(d + e*x)^(1/2)*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e))
/(64*b^8*(b^5*c^2*d - b^6*c*e)))*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e +
8*B*b^2*c*d*e))/(8*(b^5*c^2*d - b^6*c*e)))*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*
c^2*d*e + 8*B*b^2*c*d*e))/(8*(b^5*c^2*d - b^6*c*e))))*(-c*(b*e - c*d))^(1/2)*(16*A*c^3*d^2 - B*b^3*e^2 + 5*A*b
^2*c*e^2 - 8*B*b*c^2*d^2 - 20*A*b*c^2*d*e + 8*B*b^2*c*d*e)*3i)/(4*(b^5*c^2*d - b^6*c*e)) + (atan((((((d + e*x)
^(1/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6 + 6624*A^2*b^2*c^5*d^2*e^4 + 1152*B^2*b^2
*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 90*A*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d^3*e^
3 - 2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d^4*e^2 + 1152*A*B*b^4*c^3*d*e^5 + 8064*A*
B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4))/(8*b^8) - (3*((3*A*b^12*c^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*B*b^1
2*c^2*d*e^4 + 24*A*b^10*c^4*d^2*e^3 - 12*B*b^11*c^3*d^2*e^3)/b^12 - (3*(64*b^11*c^2*e^3 - 128*b^10*c^3*d*e^2)*
(d + e*x)^(1/2)*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(64*b^13*d^(1/2)))*(A*b
^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(8*b^5*d^(1/2)))*(A*b^2*e^2 + 16*A*c^2*d^2
- 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e)*3i)/(8*b^5*d^(1/2)) + ((((d + e*x)^(1/2)*(234*A^2*b^4*c^3*e^6 + 46
08*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6 + 6624*A^2*b^2*c^5*d^2*e^4 + 1152*B^2*b^2*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*
d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 90*A*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d^3*e^3 - 2016*A^2*b^3*c^4*d*e^5 - 144
*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d^4*e^2 + 1152*A*B*b^4*c^3*d*e^5 + 8064*A*B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3
*c^4*d^2*e^4))/(8*b^8) + (3*((3*A*b^12*c^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*B*b^12*c^2*d*e^4 + 24*A*b^10*c^4*d^2*
e^3 - 12*B*b^11*c^3*d^2*e^3)/b^12 + (3*(64*b^11*c^2*e^3 - 128*b^10*c^3*d*e^2)*(d + e*x)^(1/2)*(A*b^2*e^2 + 16*
A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(64*b^13*d^(1/2)))*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*
d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(8*b^5*d^(1/2)))*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12
*A*b*c*d*e)*3i)/(8*b^5*d^(1/2)))/(((135*A^3*b^5*c^3*e^8)/8 - 1728*A^3*c^8*d^5*e^3 - 3996*A^3*b^2*c^6*d^3*e^5 +
1674*A^3*b^3*c^5*d^2*e^6 + 216*B^3*b^3*c^5*d^5*e^3 - 378*B^3*b^4*c^4*d^4*e^4 + 216*B^3*b^5*c^3*d^3*e^5 - (189
*B^3*b^6*c^2*d^2*e^6)/4 + (27*A*B^2*b^7*c*e^8)/32 + (27*B^3*b^7*c*d*e^7)/8 - (243*A^2*B*b^6*c^2*e^8)/32 + 4320
*A^3*b*c^7*d^4*e^4 - (1215*A^3*b^4*c^4*d*e^7)/4 - 1296*A*B^2*b^2*c^6*d^5*e^3 + 2592*A*B^2*b^3*c^5*d^4*e^4 - 17
82*A*B^2*b^4*c^4*d^3*e^5 + (999*A*B^2*b^5*c^3*d^2*e^6)/2 - 5832*A^2*B*b^2*c^6*d^4*e^4 + 4698*A^2*B*b^3*c^5*d^3
*e^5 - (3267*A^2*B*b^4*c^4*d^2*e^6)/2 - (405*A*B^2*b^6*c^2*d*e^7)/8 + 2592*A^2*B*b*c^7*d^5*e^3 + (1809*A^2*B*b
^5*c^3*d*e^7)/8)/b^12 - (3*(((d + e*x)^(1/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6 + 6
624*A^2*b^2*c^5*d^2*e^4 + 1152*B^2*b^2*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 90*A
*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d^3*e^3 - 2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d^4*
e^2 + 1152*A*B*b^4*c^3*d*e^5 + 8064*A*B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4))/(8*b^8) - (3*((3*A*b^12*c
^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*B*b^12*c^2*d*e^4 + 24*A*b^10*c^4*d^2*e^3 - 12*B*b^11*c^3*d^2*e^3)/b^12 - (3*(
64*b^11*c^2*e^3 - 128*b^10*c^3*d*e^2)*(d + e*x)^(1/2)*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e -
12*A*b*c*d*e))/(64*b^13*d^(1/2)))*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(8*b^
5*d^(1/2)))*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(8*b^5*d^(1/2)) + (3*(((d +
e*x)^(1/2)*(234*A^2*b^4*c^3*e^6 + 4608*A^2*c^7*d^4*e^2 + 9*B^2*b^6*c*e^6 + 6624*A^2*b^2*c^5*d^2*e^4 + 1152*B^
2*b^2*c^5*d^4*e^2 - 1728*B^2*b^3*c^4*d^3*e^3 + 864*B^2*b^4*c^3*d^2*e^4 - 90*A*B*b^5*c^2*e^6 - 9216*A^2*b*c^6*d
^3*e^3 - 2016*A^2*b^3*c^4*d*e^5 - 144*B^2*b^5*c^2*d*e^5 - 4608*A*B*b*c^6*d^4*e^2 + 1152*A*B*b^4*c^3*d*e^5 + 80
64*A*B*b^2*c^5*d^3*e^3 - 4896*A*B*b^3*c^4*d^2*e^4))/(8*b^8) + (3*((3*A*b^12*c^2*e^5 - 24*A*b^11*c^3*d*e^4 + 9*
B*b^12*c^2*d*e^4 + 24*A*b^10*c^4*d^2*e^3 - 12*B*b^11*c^3*d^2*e^3)/b^12 + (3*(64*b^11*c^2*e^3 - 128*b^10*c^3*d*
e^2)*(d + e*x)^(1/2)*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(64*b^13*d^(1/2)))
*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(8*b^5*d^(1/2)))*(A*b^2*e^2 + 16*A*c^2
*d^2 - 8*B*b*c*d^2 + 4*B*b^2*d*e - 12*A*b*c*d*e))/(8*b^5*d^(1/2))))*(A*b^2*e^2 + 16*A*c^2*d^2 - 8*B*b*c*d^2 +
4*B*b^2*d*e - 12*A*b*c*d*e)*3i)/(4*b^5*d^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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