Optimal. Leaf size=166 \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x)}-\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^4 (a+b x)}+\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e} \]
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Rubi [A] time = 0.0678044, antiderivative size = 166, 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} \[ \frac{b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^3 (a+b x)}-\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 e^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x)}{e^4 (a+b x)}+\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e} \]
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} \, 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} \, 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^4 (b d-a e)^2}{e^3}-\frac{b^3 (b d-a e) \left (a b+b^2 x\right )}{e^2}+\frac{b^2 \left (a b+b^2 x\right )^2}{e}-\frac{b^3 (b d-a e)^3}{e^3 (d+e x)}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{b (b d-a e)^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{e^3 (a+b x)}-\frac{(b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2}+\frac{(a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e}-\frac{(b d-a e)^3 \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 = 0.0464589, size = 92, normalized size = 0.55 \[ \frac{\sqrt{(a+b x)^2} \left (b e x \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (b d-a e)^3 \log (d+e x)\right )}{6 e^4 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.199, size = 149, normalized size = 0.9 \begin{align*}{\frac{2\,{x}^{3}{b}^{3}{e}^{3}+9\,{x}^{2}a{b}^{2}{e}^{3}-3\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ){a}^{3}{e}^{3}-18\,\ln \left ( ex+d \right ){a}^{2}bd{e}^{2}+18\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}+18\,x{a}^{2}b{e}^{3}-18\,xa{b}^{2}d{e}^{2}+6\,x{b}^{3}{d}^{2}e}{6\, \left ( bx+a \right ) ^{3}{e}^{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 = 1.61041, size = 238, normalized size = 1.43 \begin{align*} \frac{2 \, b^{3} e^{3} x^{3} - 3 \,{\left (b^{3} d e^{2} - 3 \, a b^{2} e^{3}\right )} x^{2} + 6 \,{\left (b^{3} d^{2} e - 3 \, a b^{2} d e^{2} + 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21871, size = 234, normalized size = 1.41 \begin{align*} -{\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 )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, b^{3} x^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 3 \, b^{3} d x^{2} e \mathrm{sgn}\left (b x + a\right ) + 6 \, b^{3} d^{2} x \mathrm{sgn}\left (b x + a\right ) + 9 \, a b^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) - 18 \, a b^{2} d x e \mathrm{sgn}\left (b x + a\right ) + 18 \, a^{2} b x e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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