3.1250 \(\int \frac{(A+B x) (d+e x)^{3/2}}{(b x+c x^2)^3} \, dx\)

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

[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])

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Rubi [A]  time = 0.868746, antiderivative size = 346, normalized size of antiderivative = 1., 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} \[ \frac{3 \left (b^2 c e (5 A e+8 B d)-4 b c^2 d (5 A e+2 B d)+16 A c^3 d^2+b^3 (-B) e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 \sqrt{c} \sqrt{c d-b e}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 e (A e+4 B d)-4 b c d (3 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5 \sqrt{d}}-\frac{\sqrt{d+e x} \left (b (c d-b e) \left (7 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-3 c x (c d-b e) \left (-4 b c (A e+B d)+8 A c^2 d+b^2 B e\right )\right )}{4 b^4 c \left (b x+c x^2\right ) (c d-b e)}-\frac{\sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

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 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]

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]

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*}

Mathematica [A]  time = 3.23934, size = 472, normalized size = 1.36 \[ \frac{\frac{(b+c x) \left ((b+c x) \left (9 c^{5/2} (c d-b e)^2 \left (\frac{2}{3} \sqrt{d+e x} (4 d+e x)-2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right ) \left (b^2 e (A e+4 B d)-4 b c d (3 A e+2 B d)+16 A c^2 d^2\right )-6 c^2 d^2 \left (b^2 c e (5 A e+8 B d)-4 b c^2 d (5 A e+2 B d)+16 A c^3 d^2+b^3 (-B) e^2\right ) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )\right )-6 b c^{7/2} (d+e x)^{5/2} \left (-b^2 c d e (14 A e+15 B d)+b^3 e^2 (A e+4 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )+6 b^2 c^{7/2} (d+e x)^{5/2} (c d-b e) \left (b^2 e (A e+4 B d)-b c d (11 A e+6 B d)+12 A c^2 d^2\right )}{b^4 c^{5/2} d (c d-b e)^2}-\frac{6 (d+e x)^{5/2} (A b e-8 A c d+4 b B d)}{b d x}-\frac{12 A (d+e x)^{5/2}}{x^2}}{24 b d (b+c x)^2} \]

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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Maple [B]  time = 0.023, size = 785, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

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]

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

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

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

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Fricas [B]  time = 13.8398, size = 7127, normalized size = 20.6 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.29837, size = 892, normalized size = 2.58 \begin{align*} \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{align*}

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)