Optimal. Leaf size=137 \[ \frac{7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}+\frac{7 e \sqrt{d+e x} (b d-a e)^2}{b^4}-\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{7 e (d+e x)^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0757445, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}+\frac{7 e \sqrt{d+e x} (b d-a e)^2}{b^4}-\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{7 e (d+e x)^{5/2}}{5 b^2} \]
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)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^{7/2}}{(a+b x)^2} \, dx\\ &=-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{(7 e) \int \frac{(d+e x)^{5/2}}{a+b x} \, dx}{2 b}\\ &=\frac{7 e (d+e x)^{5/2}}{5 b^2}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{(7 e (b d-a e)) \int \frac{(d+e x)^{3/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac{7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac{7 e (d+e x)^{5/2}}{5 b^2}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{\left (7 e (b d-a e)^2\right ) \int \frac{\sqrt{d+e x}}{a+b x} \, dx}{2 b^3}\\ &=\frac{7 e (b d-a e)^2 \sqrt{d+e x}}{b^4}+\frac{7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac{7 e (d+e x)^{5/2}}{5 b^2}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{\left (7 e (b d-a e)^3\right ) \int \frac{1}{(a+b x) \sqrt{d+e x}} \, dx}{2 b^4}\\ &=\frac{7 e (b d-a e)^2 \sqrt{d+e x}}{b^4}+\frac{7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac{7 e (d+e x)^{5/2}}{5 b^2}-\frac{(d+e x)^{7/2}}{b (a+b x)}+\frac{\left (7 (b d-a e)^3\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^4}\\ &=\frac{7 e (b d-a e)^2 \sqrt{d+e x}}{b^4}+\frac{7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac{7 e (d+e x)^{5/2}}{5 b^2}-\frac{(d+e x)^{7/2}}{b (a+b x)}-\frac{7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0173275, size = 50, normalized size = 0.36 \[ \frac{2 e (d+e x)^{9/2} \, _2F_1\left (2,\frac{9}{2};\frac{11}{2};-\frac{b (d+e x)}{a e-b d}\right )}{9 (a e-b d)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.202, size = 387, normalized size = 2.8 \begin{align*}{\frac{2\,e}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{4\,a{e}^{2}}{3\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{4\,de}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+6\,{\frac{{a}^{2}{e}^{3}\sqrt{ex+d}}{{b}^{4}}}-12\,{\frac{ad{e}^{2}\sqrt{ex+d}}{{b}^{3}}}+6\,{\frac{e{d}^{2}\sqrt{ex+d}}{{b}^{2}}}+{\frac{{a}^{3}{e}^{4}}{{b}^{4} \left ( bxe+ae \right ) }\sqrt{ex+d}}-3\,{\frac{\sqrt{ex+d}d{e}^{3}{a}^{2}}{{b}^{3} \left ( bxe+ae \right ) }}+3\,{\frac{\sqrt{ex+d}a{d}^{2}{e}^{2}}{{b}^{2} \left ( bxe+ae \right ) }}-{\frac{e{d}^{3}}{b \left ( bxe+ae \right ) }\sqrt{ex+d}}-7\,{\frac{{a}^{3}{e}^{4}}{{b}^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+21\,{\frac{d{e}^{3}{a}^{2}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-21\,{\frac{a{d}^{2}{e}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{b\sqrt{ex+d}}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+7\,{\frac{e{d}^{3}}{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 [B] time = 1.92009, size = 1049, normalized size = 7.66 \begin{align*} \left [\frac{105 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\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 (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \,{\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{105 \,{\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\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 (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \,{\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (b^{5} x + a b^{4}\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 = 1.2795, size = 379, normalized size = 2.77 \begin{align*} \frac{7 \,{\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\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^{4}} - \frac{\sqrt{x e + d} b^{3} d^{3} e - 3 \, \sqrt{x e + d} a b^{2} d^{2} e^{2} + 3 \, \sqrt{x e + d} a^{2} b d e^{3} - \sqrt{x e + d} a^{3} e^{4}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{4}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{8} e + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{8} d e + 45 \, \sqrt{x e + d} b^{8} d^{2} e - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{7} e^{2} - 90 \, \sqrt{x e + d} a b^{7} d e^{2} + 45 \, \sqrt{x e + d} a^{2} b^{6} e^{3}\right )}}{15 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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