Optimal. Leaf size=110 \[ \frac{5 e \sqrt{d+e x} (b d-a e)}{b^3}-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{5 e (d+e x)^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0575072, antiderivative size = 110, 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, 50, 63, 208} \[ \frac{5 e \sqrt{d+e x} (b d-a e)}{b^3}-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{5 e (d+e x)^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^{5/2}}{(a+b x)^2} \, dx\\ &=-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{(5 e) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{2 b}\\ &=\frac{5 e (d+e x)^{3/2}}{3 b^2}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{(5 e (b d-a e)) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{2 b^2}\\ &=\frac{5 e (b d-a e) \sqrt{d+e x}}{b^3}+\frac{5 e (d+e x)^{3/2}}{3 b^2}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{\left (5 e (b d-a e)^2\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 b^3}\\ &=\frac{5 e (b d-a e) \sqrt{d+e x}}{b^3}+\frac{5 e (d+e x)^{3/2}}{3 b^2}-\frac{(d+e x)^{5/2}}{b (a+b x)}+\frac{\left (5 (b d-a e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b d}{e}+\frac{b x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{b^3}\\ &=\frac{5 e (b d-a e) \sqrt{d+e x}}{b^3}+\frac{5 e (d+e x)^{3/2}}{3 b^2}-\frac{(d+e x)^{5/2}}{b (a+b x)}-\frac{5 e (b d-a e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0166566, size = 50, normalized size = 0.45 \[ \frac{2 e (d+e x)^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{b (d+e x)}{a e-b d}\right )}{7 (a e-b d)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.202, size = 258, normalized size = 2.4 \begin{align*}{\frac{2\,e}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-4\,{\frac{a{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+4\,{\frac{e\sqrt{ex+d}d}{{b}^{2}}}-{\frac{{a}^{2}{e}^{3}}{{b}^{3} \left ( bxe+ae \right ) }\sqrt{ex+d}}+2\,{\frac{\sqrt{ex+d}ad{e}^{2}}{{b}^{2} \left ( bxe+ae \right ) }}-{\frac{e{d}^{2}}{b \left ( bxe+ae \right ) }\sqrt{ex+d}}+5\,{\frac{{a}^{2}{e}^{3}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-10\,{\frac{ad{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+5\,{\frac{e{d}^{2}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \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 = 1.84785, size = 707, normalized size = 6.43 \begin{align*} \left [-\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\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 (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{15 \,{\left (a b d e - a^{2} e^{2} +{\left (b^{2} d e - a b e^{2}\right )} x\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 (2 \, b^{2} e^{2} x^{2} - 3 \, b^{2} d^{2} + 20 \, a b d e - 15 \, a^{2} e^{2} + 2 \,{\left (7 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (b^{4} x + a 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 [B] time = 1.18858, size = 258, normalized size = 2.35 \begin{align*} \frac{5 \,{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} - \frac{\sqrt{x e + d} b^{2} d^{2} e - 2 \, \sqrt{x e + d} a b d e^{2} + \sqrt{x e + d} a^{2} e^{3}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e + 6 \, \sqrt{x e + d} b^{4} d e - 6 \, \sqrt{x e + d} a b^{3} e^{2}\right )}}{3 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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