3.11.95 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=225 \[ -\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (5 A b e-4 A c d+2 b B d)}{b^3}-\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^2 c^2}-\frac {(c d-b e)^{3/2} \left (-b c (2 B d-A e)+4 A c^2 d-3 b^2 B e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}} \]

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Rubi [A]  time = 0.58, antiderivative size = 225, 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, 824, 826, 1166, 208} \begin {gather*} -\frac {(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac {e \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+3 b^2 B e\right )}{b^2 c^2}-\frac {(c d-b e)^{3/2} \left (-b c (2 B d-A e)+4 A c^2 d-3 b^2 B e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{5/2}}-\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (5 A b e-4 A c d+2 b B d)}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^2,x]

[Out]

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

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 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 0]

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

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Mathematica [A]  time = 1.23, size = 302, normalized size = 1.34 \begin {gather*} -\frac {\frac {\frac {2 d \left (b c (2 B d-A e)-4 A c^2 d+3 b^2 B e\right ) \left (\sqrt {c} \sqrt {d+e x} \left (15 b^2 e^2-5 b c e (7 d+e x)+c^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )-15 (c d-b e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )\right )}{c^{5/2} (c d-b e)}+2 \left (15 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-\sqrt {d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )\right ) (5 A b e-4 A c d+2 b B d)}{30 b^2}+\frac {c (d+e x)^{7/2} (A b e-2 A c d+b B d)}{b (b+c x) (b e-c d)}+\frac {A (d+e x)^{7/2}}{x (b+c x)}}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^2,x]

[Out]

-(((c*(b*B*d - 2*A*c*d + A*b*e)*(d + e*x)^(7/2))/(b*(-(c*d) + b*e)*(b + c*x)) + (A*(d + e*x)^(7/2))/(x*(b + c*
x)) + (2*(2*b*B*d - 4*A*c*d + 5*A*b*e)*(-(Sqrt[d + e*x]*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) + 15*d^(5/2)*ArcTanh[
Sqrt[d + e*x]/Sqrt[d]]) + (2*d*(-4*A*c^2*d + 3*b^2*B*e + b*c*(2*B*d - A*e))*(Sqrt[c]*Sqrt[d + e*x]*(15*b^2*e^2
- 5*b*c*e*(7*d + e*x) + c^2*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) - 15*(c*d - b*e)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d - b*e]]))/(c^(5/2)*(c*d - b*e)))/(30*b^2))/(b*d))

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IntegrateAlgebraic [A]  time = 1.05, size = 387, normalized size = 1.72 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (-5 A b d^{3/2} e+4 A c d^{5/2}-2 b B d^{5/2}\right )}{b^3}+\frac {\left (-4 A c^2 d (b e-c d)^{3/2}-A b c e (b e-c d)^{3/2}+3 b^2 B e (b e-c d)^{3/2}+2 b B c d (b e-c d)^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 c^{5/2}}+\frac {\sqrt {d+e x} \left (-A b^2 c e^3 (d+e x)+A b^2 c d e^3-3 A b c^2 d^2 e^2+2 A b c^2 d e^2 (d+e x)+2 A c^3 d^3 e-2 A c^3 d^2 e (d+e x)+3 b^3 B e^3 (d+e x)-3 b^3 B d e^3+4 b^2 B c d^2 e^2+2 b^2 B c e^2 (d+e x)^2-6 b^2 B c d e^2 (d+e x)-b B c^2 d^3 e+b B c^2 d^2 e (d+e x)\right )}{b^2 c^2 e x (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^2,x]

[Out]

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

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fricas [A]  time = 10.83, size = 1589, normalized size = 7.06

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

[1/2*(((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*b^2*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A*b^2*c^2)*e^2)*x^2 + (2*(B*b^2
*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*b^2*c^2)*d*e - (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e
*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((5*A*b*c^3*d*e + 2*(B*b*c^3 - 2*A*c^4)
*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) +
2*d)/x) + 2*(2*B*b^3*c*e^2*x^2 - A*b^2*c^2*d^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e +
(3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), 1/2*(2*((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*
b^2*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A*b^2*c^2)*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*
b^2*c^2)*d*e - (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(
c*d - b*e)) + ((5*A*b*c^3*d*e + 2*(B*b*c^3 - 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^2 - 2*A*b*c^3)*
d^2)*x)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*B*b^3*c*e^2*x^2 - A*b^2*c^2*d^2 + ((B*b^2*
c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e + (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 +
b^4*c^2*x), 1/2*(2*((5*A*b*c^3*d*e + 2*(B*b*c^3 - 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^2 - 2*A*b
*c^3)*d^2)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*b^2*c^2 + 3*A*b*c^3
)*d*e - (3*B*b^3*c - A*b^2*c^2)*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*b^2*c^2)*d*e - (3*B
*b^4 - A*b^3*c)*e^2)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/
(c*x + b)) + 2*(2*B*b^3*c*e^2*x^2 - A*b^2*c^2*d^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e
+ (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x), (((2*(B*b*c^3 - 2*A*c^4)*d^2 + (B*b^2
*c^2 + 3*A*b*c^3)*d*e - (3*B*b^3*c - A*b^2*c^2)*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^2 + (B*b^3*c + 3*A*b^2
*c^2)*d*e - (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d
- b*e)) + ((5*A*b*c^3*d*e + 2*(B*b*c^3 - 2*A*c^4)*d^2)*x^2 + (5*A*b^2*c^2*d*e + 2*(B*b^2*c^2 - 2*A*b*c^3)*d^2
)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*B*b^3*c*e^2*x^2 - A*b^2*c^2*d^2 + ((B*b^2*c^2 - 2*A*b*c^3)
*d^2 - 2*(B*b^3*c - A*b^2*c^2)*d*e + (3*B*b^4 - A*b^3*c)*e^2)*x)*sqrt(e*x + d))/(b^3*c^3*x^2 + b^4*c^2*x)]

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giac [B]  time = 0.24, size = 468, normalized size = 2.08 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{2}}{c^{2}} + \frac {{\left (2 \, B b d^{3} - 4 \, A c d^{3} + 5 \, A b d^{2} e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{3} d^{3} - 4 \, A c^{4} d^{3} - B b^{2} c^{2} d^{2} e + 7 \, A b c^{3} d^{2} e - 4 \, B b^{3} c d e^{2} - 2 \, A b^{2} c^{2} d e^{2} + 3 \, B b^{4} e^{3} - A b^{3} c e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c^{2}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c^{2} d^{2} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} d^{2} e - \sqrt {x e + d} B b c^{2} d^{3} e + 2 \, \sqrt {x e + d} A c^{3} d^{3} e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} c d e^{2} + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A b c^{2} d e^{2} + 2 \, \sqrt {x e + d} B b^{2} c d^{2} e^{2} - 3 \, \sqrt {x e + d} A b c^{2} d^{2} e^{2} + {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} e^{3} - {\left (x e + d\right )}^{\frac {3}{2}} A b^{2} c e^{3} - \sqrt {x e + d} B b^{3} d e^{3} + \sqrt {x e + d} A b^{2} c d e^{3}}{{\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 )} b^{2} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^2/c^2 + (2*B*b*d^3 - 4*A*c*d^3 + 5*A*b*d^2*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)
) - (2*B*b*c^3*d^3 - 4*A*c^4*d^3 - B*b^2*c^2*d^2*e + 7*A*b*c^3*d^2*e - 4*B*b^3*c*d*e^2 - 2*A*b^2*c^2*d*e^2 + 3
*B*b^4*e^3 - A*b^3*c*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^3*c^2) + ((x*e
+ d)^(3/2)*B*b*c^2*d^2*e - 2*(x*e + d)^(3/2)*A*c^3*d^2*e - sqrt(x*e + d)*B*b*c^2*d^3*e + 2*sqrt(x*e + d)*A*c^3
*d^3*e - 2*(x*e + d)^(3/2)*B*b^2*c*d*e^2 + 2*(x*e + d)^(3/2)*A*b*c^2*d*e^2 + 2*sqrt(x*e + d)*B*b^2*c*d^2*e^2 -
3*sqrt(x*e + d)*A*b*c^2*d^2*e^2 + (x*e + d)^(3/2)*B*b^3*e^3 - (x*e + d)^(3/2)*A*b^2*c*e^3 - sqrt(x*e + d)*B*b
^3*d*e^3 + sqrt(x*e + d)*A*b^2*c*d*e^3)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2
*c^2)

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maple [B]  time = 0.07, size = 614, normalized size = 2.73 \begin {gather*} \frac {2 A d \,e^{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}-\frac {7 A c \,d^{2} 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^{2}}+\frac {4 A \,c^{2} d^{3} \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 {A \,e^{3} \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}\, c}-\frac {3 B b \,e^{3} \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}\, c^{2}}+\frac {B \,d^{2} 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}-\frac {2 B c \,d^{3} \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^{2}}+\frac {4 B d \,e^{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}\, c}+\frac {2 \sqrt {e x +d}\, A d \,e^{2}}{\left (c e x +b e \right ) b}-\frac {\sqrt {e x +d}\, A c \,d^{2} e}{\left (c e x +b e \right ) b^{2}}-\frac {\sqrt {e x +d}\, A \,e^{3}}{\left (c e x +b e \right ) c}+\frac {\sqrt {e x +d}\, B b \,e^{3}}{\left (c e x +b e \right ) c^{2}}+\frac {\sqrt {e x +d}\, B \,d^{2} e}{\left (c e x +b e \right ) b}-\frac {2 \sqrt {e x +d}\, B d \,e^{2}}{\left (c e x +b e \right ) c}-\frac {5 A \,d^{\frac {3}{2}} e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 A c \,d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}-\frac {2 B \,d^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {2 \sqrt {e x +d}\, B \,e^{2}}{c^{2}}-\frac {\sqrt {e x +d}\, A \,d^{2}}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^2,x)

[Out]

2*e^2*B/c^2*(e*x+d)^(1/2)-e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A+2*e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d-e/b^2*c*(e*x
+d)^(1/2)/(c*e*x+b*e)*A*d^2+e^3*b/c^2*(e*x+d)^(1/2)/(c*e*x+b*e)*B-2*e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d+e/b*(e
*x+d)^(1/2)/(c*e*x+b*e)*B*d^2+e^3/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+2*e^2/b/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d-7*e/b^2*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+
d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2+4/b^3*c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c
)*A*d^3-3*e^3*b/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B+4*e^2/c/((b*e-c*d)*c)^(1
/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d+e/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)
^(1/2)*c)*B*d^2-2/b^2*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*d^3-d^2/b^2*A*(e*x+d
)^(1/2)/x-5*e*d^(3/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4*d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-2*
d^(5/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*B

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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)^(5/2)/(c*x^2+b*x)^2,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 = 2.70, size = 5878, normalized size = 26.12

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(5/2))/(b*x + c*x^2)^2,x)

[Out]

(atan(((((2*(d + e*x)^(1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A^2*b^2*c^6*d^4*e^4 - 2
0*A^2*b^3*c^5*d^3*e^5 - 10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^
4*d^4*e^4 + 20*B^2*b^5*c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^
2*b^5*c^3*d*e^7 - 6*A*B*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*b^2*c^6*d^5*e^3 + 10*A
*B*b^3*c^5*d^4*e^4 - 80*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) + ((d^3)^(1/2)*((4*A*b^8*c^4*
d*e^5 - 12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^3*e^3 + 16*B*b^8*c^4*d
^2*e^4)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(d^3)^(1/2)*(d + e*x)^(1/2)*(5*A*b*e - 4*A*c*d + 2*B*b*
d))/(b^7*c^3))*(5*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(d^3)^(1/2)*(5*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3) +
(((2*(d + e*x)^(1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A^2*b^2*c^6*d^4*e^4 - 20*A^2*
b^3*c^5*d^3*e^5 - 10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^4*d^4*
e^4 + 20*B^2*b^5*c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^2*b^5*
c^3*d*e^7 - 6*A*B*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*b^2*c^6*d^5*e^3 + 10*A*B*b^3
*c^5*d^4*e^4 - 80*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) - ((d^3)^(1/2)*((4*A*b^8*c^4*d*e^5
- 12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^3*e^3 + 16*B*b^8*c^4*d^2*e^4
)/(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(d^3)^(1/2)*(d + e*x)^(1/2)*(5*A*b*e - 4*A*c*d + 2*B*b*d))/(b
^7*c^3))*(5*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(d^3)^(1/2)*(5*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^3))/((2*(3
2*A^3*c^8*d^8*e^3 + 18*B^3*b^8*d^3*e^8 + 166*A^3*b^2*c^6*d^6*e^5 - 50*A^3*b^3*c^5*d^5*e^6 - 41*A^3*b^4*c^4*d^4
*e^7 + 16*A^3*b^5*c^3*d^3*e^8 + 5*A^3*b^6*c^2*d^2*e^9 - 4*B^3*b^3*c^5*d^8*e^3 - 14*B^3*b^4*c^4*d^7*e^4 + 28*B^
3*b^5*c^3*d^6*e^5 + 20*B^3*b^6*c^2*d^5*e^6 + 45*A*B^2*b^8*d^2*e^9 - 128*A^3*b*c^7*d^7*e^4 - 48*B^3*b^7*c*d^4*e
^7 + 24*A*B^2*b^2*c^6*d^8*e^3 + 24*A*B^2*b^3*c^5*d^7*e^4 - 183*A*B^2*b^4*c^4*d^6*e^5 + 120*A*B^2*b^5*c^3*d^5*e
^6 + 138*A*B^2*b^6*c^2*d^4*e^7 + 72*A^2*B*b^2*c^6*d^7*e^4 + 171*A^2*B*b^3*c^5*d^6*e^5 - 420*A^2*B*b^4*c^4*d^5*
e^6 + 249*A^2*B*b^5*c^3*d^4*e^7 + 6*A^2*B*b^6*c^2*d^3*e^8 - 168*A*B^2*b^7*c*d^3*e^8 - 48*A^2*B*b*c^7*d^8*e^3 -
30*A^2*B*b^7*c*d^2*e^9))/(b^6*c^3) - (((2*(d + e*x)^(1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e
^2 + 90*A^2*b^2*c^6*d^4*e^4 - 20*A^2*b^3*c^5*d^3*e^5 - 10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*
b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^4*d^4*e^4 + 20*B^2*b^5*c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^
7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^2*b^5*c^3*d*e^7 - 6*A*B*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7
+ 56*A*B*b^2*c^6*d^5*e^3 + 10*A*B*b^3*c^5*d^4*e^4 - 80*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3
) + ((d^3)^(1/2)*((4*A*b^8*c^4*d*e^5 - 12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b
^7*c^5*d^3*e^3 + 16*B*b^8*c^4*d^2*e^4)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(d^3)^(1/2)*(d + e*x)^(1
/2)*(5*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c^3))*(5*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(d^3)^(1/2)*(5*A*b*e - 4
*A*c*d + 2*B*b*d))/(2*b^3) + (((2*(d + e*x)^(1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A
^2*b^2*c^6*d^4*e^4 - 20*A^2*b^3*c^5*d^3*e^5 - 10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d
^5*e^3 - 15*B^2*b^4*c^4*d^4*e^4 + 20*B^2*b^5*c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^
2*b*c^7*d^5*e^3 + 4*A^2*b^5*c^3*d*e^7 - 6*A*B*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*
b^2*c^6*d^5*e^3 + 10*A*B*b^3*c^5*d^4*e^4 - 80*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) - ((d^3
)^(1/2)*((4*A*b^8*c^4*d*e^5 - 12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^
3*e^3 + 16*B*b^8*c^4*d^2*e^4)/(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(d^3)^(1/2)*(d + e*x)^(1/2)*(5*A*
b*e - 4*A*c*d + 2*B*b*d))/(b^7*c^3))*(5*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(d^3)^(1/2)*(5*A*b*e - 4*A*c*d +
2*B*b*d))/(2*b^3)))*(d^3)^(1/2)*(5*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/b^3 - (((d + e*x)^(1/2)*(2*A*c^3*d^3*e - B*b
^3*d*e^3 - 3*A*b*c^2*d^2*e^2 + 2*B*b^2*c*d^2*e^2 + A*b^2*c*d*e^3 - B*b*c^2*d^3*e))/b^2 + ((d + e*x)^(3/2)*(B*b
^3*e^3 - A*b^2*c*e^3 - 2*A*c^3*d^2*e + 2*A*b*c^2*d*e^2 + B*b*c^2*d^2*e - 2*B*b^2*c*d*e^2))/b^2)/((2*c^3*d - b*
c^2*e)*(d + e*x) - c^3*(d + e*x)^2 - c^3*d^2 + b*c^2*d*e) + (2*B*e^2*(d + e*x)^(1/2))/c^2 + (atan(((((2*(d + e
*x)^(1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A^2*b^2*c^6*d^4*e^4 - 20*A^2*b^3*c^5*d^3*
e^5 - 10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^4*d^4*e^4 + 20*B^2
*b^5*c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^2*b^5*c^3*d*e^7 -
6*A*B*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*b^2*c^6*d^5*e^3 + 10*A*B*b^3*c^5*d^4*e^4
- 80*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) + ((-c^5*(b*e - c*d)^3)^(1/2)*((4*A*b^8*c^4*d*e
^5 - 12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^3*e^3 + 16*B*b^8*c^4*d^2*
e^4)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d - 3*
B*b^2*e + A*b*c*e - 2*B*b*c*d))/(b^7*c^8))*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(2*b^3*c^5))*(-c^5*(
b*e - c*d)^3)^(1/2)*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d)*1i)/(2*b^3*c^5) + (((2*(d + e*x)^(1/2)*(9*B^
2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A^2*b^2*c^6*d^4*e^4 - 20*A^2*b^3*c^5*d^3*e^5 - 10*A^2*b^
4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^4*d^4*e^4 + 20*B^2*b^5*c^3*d^3*e^
5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^2*b^5*c^3*d*e^7 - 6*A*B*b^7*c*e^8
- 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*b^2*c^6*d^5*e^3 + 10*A*B*b^3*c^5*d^4*e^4 - 80*A*B*b^4*c
^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) - ((-c^5*(b*e - c*d)^3)^(1/2)*((4*A*b^8*c^4*d*e^5 - 12*B*b^9*c
^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^3*e^3 + 16*B*b^8*c^4*d^2*e^4)/(b^6*c^3)
- ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d - 3*B*b^2*e + A*b*c
*e - 2*B*b*c*d))/(b^7*c^8))*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(2*b^3*c^5))*(-c^5*(b*e - c*d)^3)^(
1/2)*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d)*1i)/(2*b^3*c^5))/((2*(32*A^3*c^8*d^8*e^3 + 18*B^3*b^8*d^3*e
^8 + 166*A^3*b^2*c^6*d^6*e^5 - 50*A^3*b^3*c^5*d^5*e^6 - 41*A^3*b^4*c^4*d^4*e^7 + 16*A^3*b^5*c^3*d^3*e^8 + 5*A^
3*b^6*c^2*d^2*e^9 - 4*B^3*b^3*c^5*d^8*e^3 - 14*B^3*b^4*c^4*d^7*e^4 + 28*B^3*b^5*c^3*d^6*e^5 + 20*B^3*b^6*c^2*d
^5*e^6 + 45*A*B^2*b^8*d^2*e^9 - 128*A^3*b*c^7*d^7*e^4 - 48*B^3*b^7*c*d^4*e^7 + 24*A*B^2*b^2*c^6*d^8*e^3 + 24*A
*B^2*b^3*c^5*d^7*e^4 - 183*A*B^2*b^4*c^4*d^6*e^5 + 120*A*B^2*b^5*c^3*d^5*e^6 + 138*A*B^2*b^6*c^2*d^4*e^7 + 72*
A^2*B*b^2*c^6*d^7*e^4 + 171*A^2*B*b^3*c^5*d^6*e^5 - 420*A^2*B*b^4*c^4*d^5*e^6 + 249*A^2*B*b^5*c^3*d^4*e^7 + 6*
A^2*B*b^6*c^2*d^3*e^8 - 168*A*B^2*b^7*c*d^3*e^8 - 48*A^2*B*b*c^7*d^8*e^3 - 30*A^2*B*b^7*c*d^2*e^9))/(b^6*c^3)
- (((2*(d + e*x)^(1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A^2*b^2*c^6*d^4*e^4 - 20*A^2
*b^3*c^5*d^3*e^5 - 10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^4*d^4
*e^4 + 20*B^2*b^5*c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^2*b^5
*c^3*d*e^7 - 6*A*B*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*b^2*c^6*d^5*e^3 + 10*A*B*b^
3*c^5*d^4*e^4 - 80*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) + ((-c^5*(b*e - c*d)^3)^(1/2)*((4*
A*b^8*c^4*d*e^5 - 12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^3*e^3 + 16*B
*b^8*c^4*d^2*e^4)/(b^6*c^3) + ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(4
*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(b^7*c^8))*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(2*b^3*
c^5))*(-c^5*(b*e - c*d)^3)^(1/2)*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(2*b^3*c^5) + (((2*(d + e*x)^(
1/2)*(9*B^2*b^8*e^8 + A^2*b^6*c^2*e^8 + 32*A^2*c^8*d^6*e^2 + 90*A^2*b^2*c^6*d^4*e^4 - 20*A^2*b^3*c^5*d^3*e^5 -
10*A^2*b^4*c^4*d^2*e^6 + 8*B^2*b^2*c^6*d^6*e^2 - 4*B^2*b^3*c^5*d^5*e^3 - 15*B^2*b^4*c^4*d^4*e^4 + 20*B^2*b^5*
c^3*d^3*e^5 + 10*B^2*b^6*c^2*d^2*e^6 - 24*B^2*b^7*c*d*e^7 - 96*A^2*b*c^7*d^5*e^3 + 4*A^2*b^5*c^3*d*e^7 - 6*A*B
*b^7*c*e^8 - 32*A*B*b*c^7*d^6*e^2 - 4*A*B*b^6*c^2*d*e^7 + 56*A*B*b^2*c^6*d^5*e^3 + 10*A*B*b^3*c^5*d^4*e^4 - 80
*A*B*b^4*c^4*d^3*e^5 + 60*A*B*b^5*c^3*d^2*e^6))/(b^4*c^3) - ((-c^5*(b*e - c*d)^3)^(1/2)*((4*A*b^8*c^4*d*e^5 -
12*B*b^9*c^3*d*e^5 + 8*A*b^6*c^6*d^3*e^3 - 12*A*b^7*c^5*d^2*e^4 - 4*B*b^7*c^5*d^3*e^3 + 16*B*b^8*c^4*d^2*e^4)/
(b^6*c^3) - ((4*b^7*c^5*e^3 - 8*b^6*c^6*d*e^2)*(-c^5*(b*e - c*d)^3)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d - 3*B*b^2
*e + A*b*c*e - 2*B*b*c*d))/(b^7*c^8))*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(2*b^3*c^5))*(-c^5*(b*e -
c*d)^3)^(1/2)*(4*A*c^2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d))/(2*b^3*c^5)))*(-c^5*(b*e - c*d)^3)^(1/2)*(4*A*c^
2*d - 3*B*b^2*e + A*b*c*e - 2*B*b*c*d)*1i)/(b^3*c^5)

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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)**(5/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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