Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{4 e^5 (a+b x) (d+e x)^4}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x) (d+e x)^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x) (d+e x)^7}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3} \]
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Rubi [A] time = 0.2184, antiderivative size = 298, 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} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{4 e^5 (a+b x) (d+e x)^4}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x) (d+e x)^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x) (d+e x)^7}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3} \]
Antiderivative was successfully verified.
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Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3 (A+B x)}{(d+e x)^8} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 (-B d+A e)}{e^4 (d+e x)^8}+\frac{b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^7}-\frac{3 b^4 (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^6}+\frac{b^5 (-4 b B d+A b e+3 a B e)}{e^4 (d+e x)^5}+\frac{b^6 B}{e^4 (d+e x)^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{(b d-a e)^3 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7}+\frac{(b d-a e)^2 (4 b B d-3 A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6}-\frac{3 b (b d-a e) (2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x) (d+e x)^5}+\frac{b^2 (4 b B d-A b e-3 a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}\\ \end{align*}
Mathematica [A] time = 0.121257, size = 233, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+10 a^3 e^3 (6 A e+B (d+7 e x))+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{420 e^5 (a+b x) (d+e x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 317, normalized size = 1.1 \begin{align*} -{\frac{140\,B{x}^{4}{b}^{3}{e}^{4}+105\,A{x}^{3}{b}^{3}{e}^{4}+315\,B{x}^{3}a{b}^{2}{e}^{4}+140\,B{x}^{3}{b}^{3}d{e}^{3}+252\,A{x}^{2}a{b}^{2}{e}^{4}+63\,A{x}^{2}{b}^{3}d{e}^{3}+252\,B{x}^{2}{a}^{2}b{e}^{4}+189\,B{x}^{2}a{b}^{2}d{e}^{3}+84\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+210\,Ax{a}^{2}b{e}^{4}+84\,Axa{b}^{2}d{e}^{3}+21\,Ax{b}^{3}{d}^{2}{e}^{2}+70\,Bx{a}^{3}{e}^{4}+84\,Bx{a}^{2}bd{e}^{3}+63\,Bxa{b}^{2}{d}^{2}{e}^{2}+28\,Bx{b}^{3}{d}^{3}e+60\,A{a}^{3}{e}^{4}+30\,Ad{e}^{3}{a}^{2}b+12\,Aa{b}^{2}{d}^{2}{e}^{2}+3\,A{b}^{3}{d}^{3}e+10\,Bd{e}^{3}{a}^{3}+12\,B{a}^{2}b{d}^{2}{e}^{2}+9\,Ba{b}^{2}{d}^{3}e+4\,B{b}^{3}{d}^{4}}{420\,{e}^{5} \left ( ex+d \right ) ^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.59879, size = 706, normalized size = 2.37 \begin{align*} -\frac{140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \,{\left (4 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \,{\left (4 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \,{\left (4 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.16133, size = 576, normalized size = 1.93 \begin{align*} -\frac{{\left (140 \, B b^{3} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 140 \, B b^{3} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 84 \, B b^{3} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 28 \, B b^{3} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + 4 \, B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) + 315 \, B a b^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 105 \, A b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 189 \, B a b^{2} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 63 \, A b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 63 \, B a b^{2} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 21 \, A b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 9 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 3 \, A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 252 \, B a^{2} b x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 252 \, A a b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, B a^{2} b d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 84 \, A a b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 12 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 12 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 70 \, B a^{3} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 210 \, A a^{2} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 10 \, B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) + 30 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 60 \, A a^{3} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{420 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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