3.1757 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=438 \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]

[Out]

-((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)*(d + e*x)^12) + ((b*d - a*e)^4*(6
*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) - (b*(b*d - a*e)^3*(3
*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^10) + (10*b^2*(b*d - a*e)^
2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(d + e*x)^9) - (5*b^3*(b*d - a*e)*
(3*b*B*d - A*b*e - 2*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)*(d + e*x)^8) + (b^4*(6*b*B*d - A*b
*e - 5*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^7) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(6*e^7*(a + b*x)*(d + e*x)^6)

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Rubi [A]  time = 0.295949, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {770, 77} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{8 e^7 (a+b x) (d+e x)^8}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{9 e^7 (a+b x) (d+e x)^9}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{12 e^7 (a+b x) (d+e x)^{12}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^13,x]

[Out]

-((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*e^7*(a + b*x)*(d + e*x)^12) + ((b*d - a*e)^4*(6
*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) - (b*(b*d - a*e)^3*(3
*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^10) + (10*b^2*(b*d - a*e)^
2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(d + e*x)^9) - (5*b^3*(b*d - a*e)*
(3*b*B*d - A*b*e - 2*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)*(d + e*x)^8) + (b^4*(6*b*B*d - A*b
*e - 5*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^7) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(6*e^7*(a + b*x)*(d + e*x)^6)

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{13}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{13}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{13}}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{12}}-\frac{5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{11}}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^{10}}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^9}+\frac{b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^8}+\frac{b^{10} B}{e^6 (d+e x)^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{(b d-a e)^5 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{12 e^7 (a+b x) (d+e x)^{12}}+\frac{(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^{10}}+\frac{10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}-\frac{5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x) (d+e x)^8}+\frac{b^4 (6 b B d-A b e-5 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.22869, size = 465, normalized size = 1.06 \[ -\frac{\sqrt{(a+b x)^2} \left (14 a^2 b^3 e^2 \left (2 A e \left (12 d^2 e x+d^3+66 d e^2 x^2+220 e^3 x^3\right )+B \left (66 d^2 e^2 x^2+12 d^3 e x+d^4+220 d e^3 x^3+495 e^4 x^4\right )\right )+28 a^3 b^2 e^3 \left (3 A e \left (d^2+12 d e x+66 e^2 x^2\right )+B \left (12 d^2 e x+d^3+66 d e^2 x^2+220 e^3 x^3\right )\right )+42 a^4 b e^4 \left (5 A e (d+12 e x)+B \left (d^2+12 d e x+66 e^2 x^2\right )\right )+42 a^5 e^5 (11 A e+B (d+12 e x))+a b^4 e \left (7 A e \left (66 d^2 e^2 x^2+12 d^3 e x+d^4+220 d e^3 x^3+495 e^4 x^4\right )+5 B \left (66 d^3 e^2 x^2+220 d^2 e^3 x^3+12 d^4 e x+d^5+495 d e^4 x^4+792 e^5 x^5\right )\right )+b^5 \left (A e \left (66 d^3 e^2 x^2+220 d^2 e^3 x^3+12 d^4 e x+d^5+495 d e^4 x^4+792 e^5 x^5\right )+B \left (66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+12 d^5 e x+d^6+792 d e^5 x^5+924 e^6 x^6\right )\right )\right )}{5544 e^7 (a+b x) (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^13,x]

[Out]

-(Sqrt[(a + b*x)^2]*(42*a^5*e^5*(11*A*e + B*(d + 12*e*x)) + 42*a^4*b*e^4*(5*A*e*(d + 12*e*x) + B*(d^2 + 12*d*e
*x + 66*e^2*x^2)) + 28*a^3*b^2*e^3*(3*A*e*(d^2 + 12*d*e*x + 66*e^2*x^2) + B*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 +
220*e^3*x^3)) + 14*a^2*b^3*e^2*(2*A*e*(d^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + B*(d^4 + 12*d^3*e*x +
66*d^2*e^2*x^2 + 220*d*e^3*x^3 + 495*e^4*x^4)) + a*b^4*e*(7*A*e*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^
3*x^3 + 495*e^4*x^4) + 5*B*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5)
) + b^5*(A*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*x^5) + B*(d^6 + 12
*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^6))))/(5544*e^7*(a +
b*x)*(d + e*x)^12)

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Maple [A]  time = 0.013, size = 687, normalized size = 1.6 \begin{align*} -{\frac{924\,B{x}^{6}{b}^{5}{e}^{6}+792\,A{x}^{5}{b}^{5}{e}^{6}+3960\,B{x}^{5}a{b}^{4}{e}^{6}+792\,B{x}^{5}{b}^{5}d{e}^{5}+3465\,A{x}^{4}a{b}^{4}{e}^{6}+495\,A{x}^{4}{b}^{5}d{e}^{5}+6930\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+2475\,B{x}^{4}a{b}^{4}d{e}^{5}+495\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+6160\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+1540\,A{x}^{3}a{b}^{4}d{e}^{5}+220\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+6160\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+3080\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+1100\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+220\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+5544\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+1848\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+462\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+66\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+2772\,B{x}^{2}{a}^{4}b{e}^{6}+1848\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+924\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+330\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+66\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+2520\,Ax{a}^{4}b{e}^{6}+1008\,Ax{a}^{3}{b}^{2}d{e}^{5}+336\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+84\,Axa{b}^{4}{d}^{3}{e}^{3}+12\,Ax{b}^{5}{d}^{4}{e}^{2}+504\,Bx{a}^{5}{e}^{6}+504\,Bx{a}^{4}bd{e}^{5}+336\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+168\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+60\,Bxa{b}^{4}{d}^{4}{e}^{2}+12\,Bx{b}^{5}{d}^{5}e+462\,A{a}^{5}{e}^{6}+210\,Ad{e}^{5}{a}^{4}b+84\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+28\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+7\,Aa{b}^{4}{d}^{4}{e}^{2}+A{b}^{5}{d}^{5}e+42\,Bd{e}^{5}{a}^{5}+42\,B{a}^{4}b{d}^{2}{e}^{4}+28\,B{a}^{3}{b}^{2}{d}^{3}{e}^{3}+14\,B{a}^{2}{b}^{3}{d}^{4}{e}^{2}+5\,Ba{b}^{4}{d}^{5}e+B{b}^{5}{d}^{6}}{5544\,{e}^{7} \left ( ex+d \right ) ^{12} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^13,x)

[Out]

-1/5544/e^7*(924*B*b^5*e^6*x^6+792*A*b^5*e^6*x^5+3960*B*a*b^4*e^6*x^5+792*B*b^5*d*e^5*x^5+3465*A*a*b^4*e^6*x^4
+495*A*b^5*d*e^5*x^4+6930*B*a^2*b^3*e^6*x^4+2475*B*a*b^4*d*e^5*x^4+495*B*b^5*d^2*e^4*x^4+6160*A*a^2*b^3*e^6*x^
3+1540*A*a*b^4*d*e^5*x^3+220*A*b^5*d^2*e^4*x^3+6160*B*a^3*b^2*e^6*x^3+3080*B*a^2*b^3*d*e^5*x^3+1100*B*a*b^4*d^
2*e^4*x^3+220*B*b^5*d^3*e^3*x^3+5544*A*a^3*b^2*e^6*x^2+1848*A*a^2*b^3*d*e^5*x^2+462*A*a*b^4*d^2*e^4*x^2+66*A*b
^5*d^3*e^3*x^2+2772*B*a^4*b*e^6*x^2+1848*B*a^3*b^2*d*e^5*x^2+924*B*a^2*b^3*d^2*e^4*x^2+330*B*a*b^4*d^3*e^3*x^2
+66*B*b^5*d^4*e^2*x^2+2520*A*a^4*b*e^6*x+1008*A*a^3*b^2*d*e^5*x+336*A*a^2*b^3*d^2*e^4*x+84*A*a*b^4*d^3*e^3*x+1
2*A*b^5*d^4*e^2*x+504*B*a^5*e^6*x+504*B*a^4*b*d*e^5*x+336*B*a^3*b^2*d^2*e^4*x+168*B*a^2*b^3*d^3*e^3*x+60*B*a*b
^4*d^4*e^2*x+12*B*b^5*d^5*e*x+462*A*a^5*e^6+210*A*a^4*b*d*e^5+84*A*a^3*b^2*d^2*e^4+28*A*a^2*b^3*d^3*e^3+7*A*a*
b^4*d^4*e^2+A*b^5*d^5*e+42*B*a^5*d*e^5+42*B*a^4*b*d^2*e^4+28*B*a^3*b^2*d^3*e^3+14*B*a^2*b^3*d^4*e^2+5*B*a*b^4*
d^5*e+B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^12/(b*x+a)^5

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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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68525, size = 1428, normalized size = 3.26 \begin{align*} -\frac{924 \, B b^{5} e^{6} x^{6} + B b^{5} d^{6} + 462 \, A a^{5} e^{6} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 42 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 792 \,{\left (B b^{5} d e^{5} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 495 \,{\left (B b^{5} d^{2} e^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 220 \,{\left (B b^{5} d^{3} e^{3} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 66 \,{\left (B b^{5} d^{4} e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 12 \,{\left (B b^{5} d^{5} e +{\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 7 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 28 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 42 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^13,x, algorithm="fricas")

[Out]

-1/5544*(924*B*b^5*e^6*x^6 + B*b^5*d^6 + 462*A*a^5*e^6 + (5*B*a*b^4 + A*b^5)*d^5*e + 7*(2*B*a^2*b^3 + A*a*b^4)
*d^4*e^2 + 28*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 42*(B*a^5 + 5*A*a^4*b)*d*
e^5 + 792*(B*b^5*d*e^5 + (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 495*(B*b^5*d^2*e^4 + (5*B*a*b^4 + A*b^5)*d*e^5 + 7*(2*
B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 220*(B*b^5*d^3*e^3 + (5*B*a*b^4 + A*b^5)*d^2*e^4 + 7*(2*B*a^2*b^3 + A*a*b^4)*d
*e^5 + 28*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 66*(B*b^5*d^4*e^2 + (5*B*a*b^4 + A*b^5)*d^3*e^3 + 7*(2*B*a^2*b^3
+ A*a*b^4)*d^2*e^4 + 28*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 42*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 12*(B*b^5*d^5*e
+ (5*B*a*b^4 + A*b^5)*d^4*e^2 + 7*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 28*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 42*(B
*a^4*b + 2*A*a^3*b^2)*d*e^5 + 42*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^19*x^12 + 12*d*e^18*x^11 + 66*d^2*e^17*x^10 +
220*d^3*e^16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 792*d^7*e^12*x^5 + 495*d^8*e^11*x^
4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11*e^8*x + d^12*e^7)

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**13,x)

[Out]

Timed out

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Giac [B]  time = 1.17718, size = 1238, normalized size = 2.83 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^13,x, algorithm="giac")

[Out]

-1/5544*(924*B*b^5*x^6*e^6*sgn(b*x + a) + 792*B*b^5*d*x^5*e^5*sgn(b*x + a) + 495*B*b^5*d^2*x^4*e^4*sgn(b*x + a
) + 220*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 66*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 12*B*b^5*d^5*x*e*sgn(b*x + a) + B
*b^5*d^6*sgn(b*x + a) + 3960*B*a*b^4*x^5*e^6*sgn(b*x + a) + 792*A*b^5*x^5*e^6*sgn(b*x + a) + 2475*B*a*b^4*d*x^
4*e^5*sgn(b*x + a) + 495*A*b^5*d*x^4*e^5*sgn(b*x + a) + 1100*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 220*A*b^5*d^2*
x^3*e^4*sgn(b*x + a) + 330*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 66*A*b^5*d^3*x^2*e^3*sgn(b*x + a) + 60*B*a*b^4*d
^4*x*e^2*sgn(b*x + a) + 12*A*b^5*d^4*x*e^2*sgn(b*x + a) + 5*B*a*b^4*d^5*e*sgn(b*x + a) + A*b^5*d^5*e*sgn(b*x +
a) + 6930*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 3465*A*a*b^4*x^4*e^6*sgn(b*x + a) + 3080*B*a^2*b^3*d*x^3*e^5*sgn(b
*x + a) + 1540*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 924*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 462*A*a*b^4*d^2*x^2*e
^4*sgn(b*x + a) + 168*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 84*A*a*b^4*d^3*x*e^3*sgn(b*x + a) + 14*B*a^2*b^3*d^4*
e^2*sgn(b*x + a) + 7*A*a*b^4*d^4*e^2*sgn(b*x + a) + 6160*B*a^3*b^2*x^3*e^6*sgn(b*x + a) + 6160*A*a^2*b^3*x^3*e
^6*sgn(b*x + a) + 1848*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 1848*A*a^2*b^3*d*x^2*e^5*sgn(b*x + a) + 336*B*a^3*b^
2*d^2*x*e^4*sgn(b*x + a) + 336*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 28*B*a^3*b^2*d^3*e^3*sgn(b*x + a) + 28*A*a^2
*b^3*d^3*e^3*sgn(b*x + a) + 2772*B*a^4*b*x^2*e^6*sgn(b*x + a) + 5544*A*a^3*b^2*x^2*e^6*sgn(b*x + a) + 504*B*a^
4*b*d*x*e^5*sgn(b*x + a) + 1008*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 42*B*a^4*b*d^2*e^4*sgn(b*x + a) + 84*A*a^3*b^
2*d^2*e^4*sgn(b*x + a) + 504*B*a^5*x*e^6*sgn(b*x + a) + 2520*A*a^4*b*x*e^6*sgn(b*x + a) + 42*B*a^5*d*e^5*sgn(b
*x + a) + 210*A*a^4*b*d*e^5*sgn(b*x + a) + 462*A*a^5*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^12