Optimal. Leaf size=136 \[ -\frac{e^2 \sqrt{d+e x}}{8 b^2 (a+b x) (b d-a e)}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3} \]
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Rubi [A] time = 0.0719368, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {27, 47, 51, 63, 208} \[ -\frac{e^2 \sqrt{d+e x}}{8 b^2 (a+b x) (b d-a e)}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(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)^4} \, dx\\ &=-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac{e \int \frac{\sqrt{d+e x}}{(a+b x)^3} \, dx}{2 b}\\ &=-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac{e^2 \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{8 b^2}\\ &=-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{e^2 \sqrt{d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3}-\frac{e^3 \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{e^2 \sqrt{d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{8 b^2 (b d-a e)}\\ &=-\frac{e \sqrt{d+e x}}{4 b^2 (a+b x)^2}-\frac{e^2 \sqrt{d+e x}}{8 b^2 (b d-a e) (a+b x)}-\frac{(d+e x)^{3/2}}{3 b (a+b x)^3}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 b^{5/2} (b d-a e)^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0162283, size = 52, normalized size = 0.38 \[ \frac{2 e^3 (d+e x)^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};-\frac{b (d+e x)}{a e-b d}\right )}{5 (a e-b d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.207, size = 163, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}}{8\, \left ( bxe+ae \right ) ^{3} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}}{3\, \left ( bxe+ae \right ) ^{3}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}a}{8\, \left ( bxe+ae \right ) ^{3}{b}^{2}}\sqrt{ex+d}}+{\frac{{e}^{3}d}{8\, \left ( bxe+ae \right ) ^{3}b}\sqrt{ex+d}}+{\frac{{e}^{3}}{ \left ( 8\,ae-8\,bd \right ){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 = 1.858, size = 1368, normalized size = 10.06 \begin{align*} \left [-\frac{3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\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 (8 \, b^{4} d^{3} - 10 \, a b^{3} d^{2} e - a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3} + 3 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (7 \, b^{4} d^{2} e - 11 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{48 \,{\left (a^{3} b^{5} d^{2} - 2 \, a^{4} b^{4} d e + a^{5} b^{3} e^{2} +{\left (b^{8} d^{2} - 2 \, a b^{7} d e + a^{2} b^{6} e^{2}\right )} x^{3} + 3 \,{\left (a b^{7} d^{2} - 2 \, a^{2} b^{6} d e + a^{3} b^{5} e^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d^{2} - 2 \, a^{3} b^{5} d e + a^{4} b^{4} e^{2}\right )} x\right )}}, -\frac{3 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\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 (8 \, b^{4} d^{3} - 10 \, a b^{3} d^{2} e - a^{2} b^{2} d e^{2} + 3 \, a^{3} b e^{3} + 3 \,{\left (b^{4} d e^{2} - a b^{3} e^{3}\right )} x^{2} + 2 \,{\left (7 \, b^{4} d^{2} e - 11 \, a b^{3} d e^{2} + 4 \, a^{2} b^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{24 \,{\left (a^{3} b^{5} d^{2} - 2 \, a^{4} b^{4} d e + a^{5} b^{3} e^{2} +{\left (b^{8} d^{2} - 2 \, a b^{7} d e + a^{2} b^{6} e^{2}\right )} x^{3} + 3 \,{\left (a b^{7} d^{2} - 2 \, a^{2} b^{6} d e + a^{3} b^{5} e^{2}\right )} x^{2} + 3 \,{\left (a^{2} b^{6} d^{2} - 2 \, a^{3} b^{5} d e + a^{4} b^{4} e^{2}\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 = 1.19194, size = 258, normalized size = 1.9 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{3}}{8 \,{\left (b^{3} d - a b^{2} e\right )} \sqrt{-b^{2} d + a b e}} - \frac{3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{2} e^{3} + 8 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{3} - 3 \, \sqrt{x e + d} b^{2} d^{2} e^{3} - 8 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{4} + 6 \, \sqrt{x e + d} a b d e^{4} - 3 \, \sqrt{x e + d} a^{2} e^{5}}{24 \,{\left (b^{3} d - a b^{2} e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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