Optimal. Leaf size=183 \[ -\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)} \]
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Rubi [A] time = 0.0922766, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {646, 43} \[ -\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{(d+e x)^2} \, 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^5 (2 b d-3 a e)}{e^3}+\frac{b^6 x}{e^2}-\frac{b^3 (b d-a e)^3}{e^3 (d+e x)^2}+\frac{3 b^4 (b d-a e)^2}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{b^2 (2 b d-3 a e) x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)}+\frac{(b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{e^4 (a+b x) (d+e x)}+\frac{3 b (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0900179, size = 132, normalized size = 0.72 \[ \frac{\sqrt{(a+b x)^2} \left (6 a^2 b d e^2-2 a^3 e^3+6 a b^2 e \left (-d^2+d e x+e^2 x^2\right )+6 b (d+e x) (b d-a e)^2 \log (d+e x)+b^3 \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )}{2 e^4 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.202, size = 216, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}{b}^{3}{e}^{3}+6\,\ln \left ( ex+d \right ) x{a}^{2}b{e}^{3}-12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}+6\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e+6\,{x}^{2}a{b}^{2}{e}^{3}-3\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ){a}^{2}bd{e}^{2}-12\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e+6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}+6\,xa{b}^{2}d{e}^{2}-4\,x{b}^{3}{d}^{2}e-2\,{a}^{3}{e}^{3}+6\,d{e}^{2}{a}^{2}b-6\,a{b}^{2}{d}^{2}e+2\,{b}^{3}{d}^{3}}{2\, \left ( bx+a \right ) ^{3}{e}^{4} \left ( ex+d \right ) } \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 = 1.49036, size = 354, normalized size = 1.93 \begin{align*} \frac{b^{3} e^{3} x^{3} + 2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} - 3 \,{\left (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \,{\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\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 = 1.17053, size = 236, normalized size = 1.29 \begin{align*} 3 \,{\left (b^{3} d^{2} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{2} d e \mathrm{sgn}\left (b x + a\right ) + a^{2} b e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{3} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 4 \, b^{3} d x e \mathrm{sgn}\left (b x + a\right ) + 6 \, a b^{2} x e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )} + \frac{{\left (b^{3} d^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm{sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-4\right )}}{x e + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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