Optimal. Leaf size=119 \[ -\frac{15 e^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 b^3} \]
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Rubi [A] time = 0.0555334, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {27, 47, 50, 63, 208} \[ -\frac{15 e^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2}}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{15 e^2 \sqrt{d+e x}}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{5/2}}{(a+b x)^3} \, dx\\ &=-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{(5 e) \int \frac{(d+e x)^{3/2}}{(a+b x)^2} \, dx}{4 b}\\ &=-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{\left (15 e^2\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{8 b^2}\\ &=\frac{15 e^2 \sqrt{d+e x}}{4 b^3}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{\left (15 e^2 (b d-a e)\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{8 b^3}\\ &=\frac{15 e^2 \sqrt{d+e x}}{4 b^3}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}+\frac{(15 e (b d-a 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^3}\\ &=\frac{15 e^2 \sqrt{d+e x}}{4 b^3}-\frac{5 e (d+e x)^{3/2}}{4 b^2 (a+b x)}-\frac{(d+e x)^{5/2}}{2 b (a+b x)^2}-\frac{15 e^2 \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0156911, size = 52, normalized size = 0.44 \[ \frac{2 e^2 (d+e x)^{7/2} \, _2F_1\left (3,\frac{7}{2};\frac{9}{2};-\frac{b (d+e x)}{a e-b d}\right )}{7 (a e-b d)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 238, normalized size = 2. \begin{align*} 2\,{\frac{{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+{\frac{9\,{e}^{3}a}{4\,{b}^{2} \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{e}^{2}d}{4\,b \left ( bex+ae \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{e}^{4}{a}^{2}}{4\,{b}^{3} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{7\,{e}^{3}ad}{2\,{b}^{2} \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}+{\frac{7\,{e}^{2}{d}^{2}}{4\,b \left ( bex+ae \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,{e}^{3}a}{4\,{b}^{3}}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}+{\frac{15\,{e}^{2}d}{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 [A] time = 1.06151, size = 728, normalized size = 6.12 \begin{align*} \left [\frac{15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (8 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} - 5 \, a b d e + 15 \, a^{2} e^{2} -{\left (9 \, b^{2} d e - 25 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}}, -\frac{15 \,{\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (-\frac{\sqrt{e x + d} b \sqrt{-\frac{b d - a e}{b}}}{b d - a e}\right ) -{\left (8 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} - 5 \, a b d e + 15 \, a^{2} e^{2} -{\left (9 \, b^{2} d e - 25 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\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 = 1.18544, size = 235, normalized size = 1.97 \begin{align*} \frac{15 \,{\left (b d e^{2} - a e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{4 \, \sqrt{-b^{2} d + a b e} b^{3}} + \frac{2 \, \sqrt{x e + d} e^{2}}{b^{3}} - \frac{9 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} d e^{2} - 7 \, \sqrt{x e + d} b^{2} d^{2} e^{2} - 9 \,{\left (x e + d\right )}^{\frac{3}{2}} a b e^{3} + 14 \, \sqrt{x e + d} a b d e^{3} - 7 \, \sqrt{x e + d} a^{2} e^{4}}{4 \,{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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