Optimal. Leaf size=246 \[ \frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^2}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^5 (a+b x) (d+e x)^4}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \]
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Rubi [A] time = 0.135284, antiderivative size = 246, 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} \[ \frac{4 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^5 (a+b x) (d+e x)}-\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^5 (a+b x) (d+e x)^2}+\frac{4 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^5 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^5 (a+b x) (d+e x)^4}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)} \]
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)^5} \, 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)^5} \, 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)^5} \, 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)^5}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^4}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^3}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^2}+\frac{b^4}{e^4 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^5 (a+b x) (d+e x)^4}+\frac{4 b (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3}-\frac{3 b^2 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)^2}+\frac{4 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^5 (a+b x) (d+e x)}+\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^5 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.104711, size = 144, normalized size = 0.59 \[ \frac{\sqrt{(a+b x)^2} \left ((b d-a e) \left (a^2 b e^2 (7 d+16 e x)+3 a^3 e^3+a b^2 e \left (13 d^2+40 d e x+36 e^2 x^2\right )+b^3 \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )+12 b^4 (d+e x)^4 \log (d+e x)\right )}{12 e^5 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 276, normalized size = 1.1 \begin{align*}{\frac{12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}+48\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+72\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}-48\,{x}^{3}a{b}^{3}{e}^{4}+48\,{x}^{3}{b}^{4}d{e}^{3}+48\,\ln \left ( ex+d \right ) x{b}^{4}{d}^{3}e-36\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-72\,{x}^{2}a{b}^{3}d{e}^{3}+108\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){b}^{4}{d}^{4}-16\,x{a}^{3}b{e}^{4}-24\,x{a}^{2}{b}^{2}d{e}^{3}-48\,xa{b}^{3}{d}^{2}{e}^{2}+88\,x{b}^{4}{d}^{3}e-3\,{a}^{4}{e}^{4}-4\,d{e}^{3}{a}^{3}b-6\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}-12\,a{b}^{3}{d}^{3}e+25\,{b}^{4}{d}^{4}}{12\, \left ( bx+a \right ) ^{3}{e}^{5} \left ( ex+d \right ) ^{4}} \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 = 1.63555, size = 544, normalized size = 2.21 \begin{align*} \frac{25 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 48 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} e - 6 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} - 2 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + 4 \, b^{4} d e^{3} x^{3} + 6 \, b^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} d^{3} e x + b^{4} d^{4}\right )} \log \left (e x + d\right )}{12 \,{\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} 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 = 1.1319, size = 362, normalized size = 1.47 \begin{align*} b^{4} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (48 \,{\left (b^{4} d e^{2} \mathrm{sgn}\left (b x + a\right ) - a b^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 36 \,{\left (3 \, b^{4} d^{2} e \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{3} d e^{2} \mathrm{sgn}\left (b x + a\right ) - a^{2} b^{2} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 8 \,{\left (11 \, b^{4} d^{3} \mathrm{sgn}\left (b x + a\right ) - 6 \, a b^{3} d^{2} e \mathrm{sgn}\left (b x + a\right ) - 3 \, a^{2} b^{2} d e^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a^{3} b e^{3} \mathrm{sgn}\left (b x + a\right )\right )} x +{\left (25 \, b^{4} d^{4} \mathrm{sgn}\left (b x + a\right ) - 12 \, a b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) - 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a^{4} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-4\right )}}{12 \,{\left (x e + d\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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