Optimal. Leaf size=268 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d^2}+\frac{e^4 \sqrt{a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^4 \sqrt{a e^2+c d^2}}+\frac{c e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^4}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2} \]
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Rubi [A] time = 0.222282, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {961, 266, 51, 63, 208, 264, 731, 725, 206} \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d^2}+\frac{e^4 \sqrt{a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^4 \sqrt{a e^2+c d^2}}+\frac{c e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^4}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2} \]
Antiderivative was successfully verified.
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Rule 961
Rule 266
Rule 51
Rule 63
Rule 208
Rule 264
Rule 731
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 (d+e x)^2 \sqrt{a+c x^2}} \, dx &=\int \left (\frac{1}{d^2 x^3 \sqrt{a+c x^2}}-\frac{2 e}{d^3 x^2 \sqrt{a+c x^2}}+\frac{3 e^2}{d^4 x \sqrt{a+c x^2}}-\frac{e^3}{d^3 (d+e x)^2 \sqrt{a+c x^2}}-\frac{3 e^3}{d^4 (d+e x) \sqrt{a+c x^2}}\right ) \, dx\\ &=\frac{\int \frac{1}{x^3 \sqrt{a+c x^2}} \, dx}{d^2}-\frac{(2 e) \int \frac{1}{x^2 \sqrt{a+c x^2}} \, dx}{d^3}+\frac{\left (3 e^2\right ) \int \frac{1}{x \sqrt{a+c x^2}} \, dx}{d^4}-\frac{\left (3 e^3\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^4}-\frac{e^3 \int \frac{1}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{d^3}\\ &=\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{e^4 \sqrt{a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{2 d^4}+\frac{\left (3 e^3\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 )}{d^4}-\frac{\left (c e^3\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^2 \left (c d^2+a e^2\right )}\\ &=-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{e^4 \sqrt{a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^4 \sqrt{c d^2+a e^2}}-\frac{c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{4 a d^2}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c d^4}+\frac{\left (c e^3\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 )}{d^2 \left (c d^2+a e^2\right )}\\ &=-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{e^4 \sqrt{a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac{c e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^4 \sqrt{c d^2+a e^2}}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^4}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{2 a d^2}\\ &=-\frac{\sqrt{a+c x^2}}{2 a d^2 x^2}+\frac{2 e \sqrt{a+c x^2}}{a d^3 x}+\frac{e^4 \sqrt{a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac{c e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac{3 e^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^4 \sqrt{c d^2+a e^2}}+\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 a^{3/2} d^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^4}\\ \end{align*}
Mathematica [A] time = 0.530588, size = 229, normalized size = 0.85 \[ \frac{\frac{\left (c d^2-6 a e^2\right ) \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{a^{3/2}}+\frac{\log (x) \left (6 a e^2-c d^2\right )}{a^{3/2}}+d \sqrt{a+c x^2} \left (\frac{2 e^4}{(d+e x) \left (a e^2+c d^2\right )}-\frac{d-4 e x}{a x^2}\right )+\frac{2 e^3 \left (3 a e^2+4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{2 e^3 \left (3 a e^2+4 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{2 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.249, size = 452, normalized size = 1.7 \begin{align*} -3\,{\frac{{e}^{2}}{{d}^{4}\sqrt{a}}\ln \left ({\frac{2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a}}{x}} \right ) }+3\,{\frac{{e}^{2}}{{d}^{4}}\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{{e}^{3}}{{d}^{3} \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{c{e}^{2}}{{d}^{2} \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}}}}}}}-{\frac{1}{2\,a{d}^{2}{x}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{c}{2\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}+2\,{\frac{e\sqrt{c{x}^{2}+a}}{a{d}^{3}x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 9.00832, size = 3771, normalized size = 14.07 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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