Optimal. Leaf size=244 \[ \frac{\sqrt{a+c x^2} \left (13 c d^2-2 a e^2\right )}{3 c^2 e^4}-\frac{d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e^5}+\frac{d^5 \sqrt{a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac{d^4 \left (5 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^5 \left (a e^2+c d^2\right )^{3/2}}-\frac{5 d \sqrt{a+c x^2} (d+e x)}{3 c e^4}+\frac{\sqrt{a+c x^2} (d+e x)^2}{3 c e^4} \]
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Rubi [A] time = 0.889832, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1651, 1654, 844, 217, 206, 725} \[ \frac{\sqrt{a+c x^2} \left (13 c d^2-2 a e^2\right )}{3 c^2 e^4}-\frac{d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e^5}+\frac{d^5 \sqrt{a+c x^2}}{e^4 (d+e x) \left (a e^2+c d^2\right )}-\frac{d^4 \left (5 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^5 \left (a e^2+c d^2\right )^{3/2}}-\frac{5 d \sqrt{a+c x^2} (d+e x)}{3 c e^4}+\frac{\sqrt{a+c x^2} (d+e x)^2}{3 c e^4} \]
Antiderivative was successfully verified.
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Rule 1651
Rule 1654
Rule 844
Rule 217
Rule 206
Rule 725
Rubi steps
\begin{align*} \int \frac{x^5}{(d+e x)^2 \sqrt{a+c x^2}} \, dx &=\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac{\int \frac{-\frac{a d^4}{e^3}+\frac{d^3 \left (c d^2+a e^2\right ) x}{e^4}-\frac{d^2 \left (c d^2+a e^2\right ) x^2}{e^3}+d \left (a+\frac{c d^2}{e^2}\right ) x^3-\frac{\left (c d^2+a e^2\right ) x^4}{e}}{(d+e x) \sqrt{a+c x^2}} \, dx}{c d^2+a e^2}\\ &=\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}+\frac{(d+e x)^2 \sqrt{a+c x^2}}{3 c e^4}-\frac{\int \frac{-a d^2 e \left (c d^2-2 a e^2\right )+4 d \left (c d^2+a e^2\right )^2 x+2 e \left (c d^2+a e^2\right )^2 x^2+10 c d e^2 \left (c d^2+a e^2\right ) x^3}{(d+e x) \sqrt{a+c x^2}} \, dx}{3 c e^4 \left (c d^2+a e^2\right )}\\ &=\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac{5 d (d+e x) \sqrt{a+c x^2}}{3 c e^4}+\frac{(d+e x)^2 \sqrt{a+c x^2}}{3 c e^4}-\frac{\int \frac{-6 a c d^2 e^4 \left (2 c d^2+a e^2\right )-2 c d e^3 \left (c d^2+a e^2\right )^2 x-2 c e^4 \left (13 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x^2}{(d+e x) \sqrt{a+c x^2}} \, dx}{6 c^2 e^7 \left (c d^2+a e^2\right )}\\ &=\frac{\left (13 c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{3 c^2 e^4}+\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac{5 d (d+e x) \sqrt{a+c x^2}}{3 c e^4}+\frac{(d+e x)^2 \sqrt{a+c x^2}}{3 c e^4}-\frac{\int \frac{-6 a c^2 d^2 e^6 \left (2 c d^2+a e^2\right )+6 c^2 d e^5 \left (4 c d^2-a e^2\right ) \left (c d^2+a e^2\right ) x}{(d+e x) \sqrt{a+c x^2}} \, dx}{6 c^3 e^9 \left (c d^2+a e^2\right )}\\ &=\frac{\left (13 c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{3 c^2 e^4}+\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac{5 d (d+e x) \sqrt{a+c x^2}}{3 c e^4}+\frac{(d+e x)^2 \sqrt{a+c x^2}}{3 c e^4}-\frac{\left (d \left (4 c d^2-a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{c e^5}+\frac{\left (d^4 \left (4 c d^2+5 a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{e^5 \left (c d^2+a e^2\right )}\\ &=\frac{\left (13 c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{3 c^2 e^4}+\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac{5 d (d+e x) \sqrt{a+c x^2}}{3 c e^4}+\frac{(d+e x)^2 \sqrt{a+c x^2}}{3 c e^4}-\frac{\left (d \left (4 c d^2-a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{c e^5}-\frac{\left (d^4 \left (4 c d^2+5 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )}\\ &=\frac{\left (13 c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{3 c^2 e^4}+\frac{d^5 \sqrt{a+c x^2}}{e^4 \left (c d^2+a e^2\right ) (d+e x)}-\frac{5 d (d+e x) \sqrt{a+c x^2}}{3 c e^4}+\frac{(d+e x)^2 \sqrt{a+c x^2}}{3 c e^4}-\frac{d \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e^5}-\frac{d^4 \left (4 c d^2+5 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{e^5 \left (c d^2+a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.571505, size = 230, normalized size = 0.94 \[ \frac{e \sqrt{a+c x^2} \left (-\frac{2 a e^2}{c^2}+\frac{3 d^5}{(d+e x) \left (a e^2+c d^2\right )}+\frac{9 d^2-3 d e x+e^2 x^2}{c}\right )-\frac{3 d \left (4 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}-\frac{3 d^4 \left (5 a e^2+4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{3 d^4 \left (5 a e^2+4 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{3 e^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.253, size = 474, normalized size = 1.9 \begin{align*}{\frac{{x}^{2}}{3\,c{e}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{2\,a}{3\,{c}^{2}{e}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{dx}{{e}^{3}c}\sqrt{c{x}^{2}+a}}+{\frac{ad}{{e}^{3}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+3\,{\frac{{d}^{2}\sqrt{c{x}^{2}+a}}{{e}^{4}c}}-4\,{\frac{{d}^{3}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) }{{e}^{5}\sqrt{c}}}-5\,{\frac{{d}^{4}}{{e}^{6}}\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{d}^{5}}{{e}^{5} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}+{\frac{{d}^{6}c}{{e}^{6} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \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 [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{a + c x^{2}} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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