Optimal. Leaf size=208 \[ -\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (a+b x) (b d-a e)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (a+b x)^2 (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (a+b x)^3 (b d-a e)}+\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{7/2}}-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5} \]
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Rubi [A] time = 0.123246, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (a+b x) (b d-a e)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (a+b x)^2 (b d-a e)^2}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (a+b x)^3 (b d-a e)}+\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{7/2}}-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5} \]
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 )^3} \, dx &=\int \frac{(d+e x)^{3/2}}{(a+b x)^6} \, dx\\ &=-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}+\frac{(3 e) \int \frac{\sqrt{d+e x}}{(a+b x)^5} \, dx}{10 b}\\ &=-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}+\frac{\left (3 e^2\right ) \int \frac{1}{(a+b x)^4 \sqrt{d+e x}} \, dx}{80 b^2}\\ &=-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (b d-a e) (a+b x)^3}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}-\frac{e^3 \int \frac{1}{(a+b x)^3 \sqrt{d+e x}} \, dx}{32 b^2 (b d-a e)}\\ &=-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (b d-a e) (a+b x)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (b d-a e)^2 (a+b x)^2}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}+\frac{\left (3 e^4\right ) \int \frac{1}{(a+b x)^2 \sqrt{d+e x}} \, dx}{128 b^2 (b d-a e)^2}\\ &=-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (b d-a e) (a+b x)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (b d-a e)^2 (a+b x)^2}-\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (b d-a e)^3 (a+b x)}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}-\frac{\left (3 e^5\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{256 b^2 (b d-a e)^3}\\ &=-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (b d-a e) (a+b x)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (b d-a e)^2 (a+b x)^2}-\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (b d-a e)^3 (a+b x)}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}-\frac{\left (3 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{128 b^2 (b d-a e)^3}\\ &=-\frac{3 e \sqrt{d+e x}}{40 b^2 (a+b x)^4}-\frac{e^2 \sqrt{d+e x}}{80 b^2 (b d-a e) (a+b x)^3}+\frac{e^3 \sqrt{d+e x}}{64 b^2 (b d-a e)^2 (a+b x)^2}-\frac{3 e^4 \sqrt{d+e x}}{128 b^2 (b d-a e)^3 (a+b x)}-\frac{(d+e x)^{3/2}}{5 b (a+b x)^5}+\frac{3 e^5 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0181231, size = 52, normalized size = 0.25 \[ \frac{2 e^5 (d+e x)^{5/2} \, _2F_1\left (\frac{5}{2},6;\frac{7}{2};-\frac{b (d+e x)}{a e-b d}\right )}{5 (a e-b d)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.232, size = 300, normalized size = 1.4 \begin{align*}{\frac{3\,{e}^{5}{b}^{2}}{128\, \left ( bxe+ae \right ) ^{5} \left ({a}^{3}{e}^{3}-3\,d{e}^{2}{a}^{2}b+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) } \left ( ex+d \right ) ^{{\frac{9}{2}}}}+{\frac{7\,{e}^{5}b}{64\, \left ( bxe+ae \right ) ^{5} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{5}}{5\, \left ( bxe+ae \right ) ^{5} \left ( ae-bd \right ) } \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{5}}{64\, \left ( bxe+ae \right ) ^{5}b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{6}a}{128\, \left ( bxe+ae \right ) ^{5}{b}^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{5}d}{128\, \left ( bxe+ae \right ) ^{5}b}\sqrt{ex+d}}+{\frac{3\,{e}^{5}}{ \left ( 128\,{a}^{3}{e}^{3}-384\,d{e}^{2}{a}^{2}b+384\,a{b}^{2}{d}^{2}e-128\,{b}^{3}{d}^{3} \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 = 1.86769, size = 3065, normalized size = 14.74 \begin{align*} \text{result too large to display} \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 [B] time = 1.24847, size = 556, normalized size = 2.67 \begin{align*} -\frac{3 \, \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{5}}{128 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \sqrt{-b^{2} d + a b e}} - \frac{15 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{4} e^{5} - 70 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} d e^{5} + 128 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d^{2} e^{5} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{3} e^{5} - 15 \, \sqrt{x e + d} b^{4} d^{4} e^{5} + 70 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{3} e^{6} - 256 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} d e^{6} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d^{2} e^{6} + 60 \, \sqrt{x e + d} a b^{3} d^{3} e^{6} + 128 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{2} e^{7} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} d e^{7} - 90 \, \sqrt{x e + d} a^{2} b^{2} d^{2} e^{7} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b e^{8} + 60 \, \sqrt{x e + d} a^{3} b d e^{8} - 15 \, \sqrt{x e + d} a^{4} e^{9}}{640 \,{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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