Optimal. Leaf size=254 \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^5 (a+b x) (d+e x)^7}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^5 (a+b x) (d+e x)^8}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^5 (a+b x) (d+e x)^9}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{10 e^5 (a+b x) (d+e x)^{10}} \]
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Rubi [A] time = 0.136427, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6}+\frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{7 e^5 (a+b x) (d+e x)^7}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^5 (a+b x) (d+e x)^8}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{9 e^5 (a+b x) (d+e x)^9}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{10 e^5 (a+b x) (d+e x)^{10}} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
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)^{11}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^3}{(d+e x)^{11}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^4}{(d+e x)^{11}} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{11}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^{10}}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^9}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^8}+\frac{b^4}{e^4 (d+e x)^7}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^5 (a+b x) (d+e x)^{10}}+\frac{4 b (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x) (d+e x)^9}-\frac{3 b^2 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^8}+\frac{4 b^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^4 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^5 (a+b x) (d+e x)^6}\\ \end{align*}
Mathematica [A] time = 0.0591835, size = 162, normalized size = 0.64 \[ -\frac{\sqrt{(a+b x)^2} \left (21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+56 a^3 b e^3 (d+10 e x)+126 a^4 e^4+6 a b^3 e \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )\right )}{1260 e^5 (a+b x) (d+e x)^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 201, normalized size = 0.8 \begin{align*} -{\frac{210\,{x}^{4}{b}^{4}{e}^{4}+720\,{x}^{3}a{b}^{3}{e}^{4}+120\,{x}^{3}{b}^{4}d{e}^{3}+945\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+270\,{x}^{2}a{b}^{3}d{e}^{3}+45\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+560\,x{a}^{3}b{e}^{4}+210\,x{a}^{2}{b}^{2}d{e}^{3}+60\,xa{b}^{3}{d}^{2}{e}^{2}+10\,x{b}^{4}{d}^{3}e+126\,{a}^{4}{e}^{4}+56\,d{e}^{3}{a}^{3}b+21\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+6\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4}}{1260\,{e}^{5} \left ( ex+d \right ) ^{10} \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 = 1.51465, size = 608, normalized size = 2.39 \begin{align*} -\frac{210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \,{\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \,{\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \,{\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} 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 = 1.12161, size = 356, normalized size = 1.4 \begin{align*} -\frac{{\left (210 \, b^{4} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 120 \, b^{4} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 45 \, b^{4} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 10 \, b^{4} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) + 720 \, a b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 270 \, a b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 60 \, a b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 945 \, a^{2} b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 210 \, a^{2} b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 21 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 560 \, a^{3} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 56 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 126 \, a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{1260 \,{\left (x e + d\right )}^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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