Optimal. Leaf size=100 \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{b d-a e}}-\frac{3 e \sqrt{d+e x}}{4 b^2 (a+b x)}-\frac{(d+e x)^{3/2}}{2 b (a+b x)^2} \]
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Rubi [A] time = 0.0489792, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {27, 47, 63, 208} \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{b d-a e}}-\frac{3 e \sqrt{d+e x}}{4 b^2 (a+b x)}-\frac{(d+e x)^{3/2}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^{3/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{3/2}}{(a+b x)^3} \, dx\\ &=-\frac{(d+e x)^{3/2}}{2 b (a+b x)^2}+\frac{(3 e) \int \frac{\sqrt{d+e x}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac{3 e \sqrt{d+e x}}{4 b^2 (a+b x)}-\frac{(d+e x)^{3/2}}{2 b (a+b x)^2}+\frac{\left (3 e^2\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 b^2}\\ &=-\frac{3 e \sqrt{d+e x}}{4 b^2 (a+b x)}-\frac{(d+e x)^{3/2}}{2 b (a+b x)^2}+\frac{(3 e) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 b^2}\\ &=-\frac{3 e \sqrt{d+e x}}{4 b^2 (a+b x)}-\frac{(d+e x)^{3/2}}{2 b (a+b x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{5/2} \sqrt{b d-a e}}\\ \end{align*}
Mathematica [A] time = 0.101774, size = 90, normalized size = 0.9 \[ \frac{3 e^2 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{a e-b d}}\right )}{4 b^{5/2} \sqrt{a e-b d}}-\frac{\sqrt{d+e x} (3 a e+2 b d+5 b e x)}{4 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 121, normalized size = 1.2 \begin{align*} -{\frac{5\,{e}^{2}}{4\, \left ( bex+ae \right ) ^{2}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{3}a}{4\, \left ( bex+ae \right ) ^{2}{b}^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{2}d}{4\, \left ( bex+ae \right ) ^{2}b}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\,{b}^{2}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \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 [B] time = 1.03861, size = 795, normalized size = 7.95 \begin{align*} \left [\frac{3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{b^{2} d - a b e} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d}}{b x + a}\right ) - 2 \,{\left (2 \, b^{3} d^{2} + a b^{2} d e - 3 \, a^{2} b e^{2} + 5 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (a^{2} b^{4} d - a^{3} b^{3} e +{\left (b^{6} d - a b^{5} e\right )} x^{2} + 2 \,{\left (a b^{5} d - a^{2} b^{4} e\right )} x\right )}}, \frac{3 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{-b^{2} d + a b e} \arctan \left (\frac{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}{b e x + b d}\right ) -{\left (2 \, b^{3} d^{2} + a b^{2} d e - 3 \, a^{2} b e^{2} + 5 \,{\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (a^{2} b^{4} d - a^{3} b^{3} e +{\left (b^{6} d - a b^{5} e\right )} x^{2} + 2 \,{\left (a b^{5} d - a^{2} b^{4} e\right )} x\right )}}\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 = 1.14707, size = 151, normalized size = 1.51 \begin{align*} \frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \, \sqrt{-b^{2} d + a b e} b^{2}} - \frac{5 \,{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} - 3 \, \sqrt{x e + d} b d e^{2} + 3 \, \sqrt{x e + d} a e^{3}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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