Optimal. Leaf size=145 \[ -\frac{3 c d e \sqrt{a+c x^2}}{2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{c \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}} \]
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Rubi [A] time = 0.0707614, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {745, 807, 725, 206} \[ -\frac{3 c d e \sqrt{a+c x^2}}{2 (d+e x) \left (a e^2+c d^2\right )^2}-\frac{e \sqrt{a+c x^2}}{2 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{c \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 745
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \sqrt{a+c x^2}} \, dx &=-\frac{e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac{c \int \frac{-2 d+e x}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )}\\ &=-\frac{e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac{3 c d e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}+\frac{\left (c \left (2 c d^2-a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac{3 c d e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac{\left (c \left (2 c d^2-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 )}{2 \left (c d^2+a e^2\right )^2}\\ &=-\frac{e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right ) (d+e x)^2}-\frac{3 c d e \sqrt{a+c x^2}}{2 \left (c d^2+a e^2\right )^2 (d+e x)}-\frac{c \left (2 c d^2-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 )}{2 \left (c d^2+a e^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.139823, size = 161, normalized size = 1.11 \[ \frac{-e \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (a e^2+c d (4 d+3 e x)\right )-c (d+e x)^2 \left (2 c d^2-a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+c (d+e x)^2 \left (2 c d^2-a e^2\right ) \log (d+e x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.193, size = 426, normalized size = 2.9 \begin{align*} -{\frac{1}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-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 ) ^{-2}}-{\frac{3\,cd}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}-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{3\,{c}^{2}{d}^{2}}{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\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{c \left ({\frac{d}{e}}+x \right ) ^{2}-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{c}{2\,e \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{c \left ({\frac{d}{e}}+x \right ) ^{2}-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 [B] time = 4.10872, size = 1432, normalized size = 9.88 \begin{align*} \left [-\frac{{\left (2 \, c^{2} d^{4} - a c d^{2} e^{2} +{\left (2 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{c d^{2} + a e^{2}} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (4 \, c^{2} d^{4} e + 5 \, a c d^{2} e^{3} + a^{2} e^{5} + 3 \,{\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt{c x^{2} + a}}{4 \,{\left (c^{3} d^{8} + 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} + a^{3} d^{2} e^{6} +{\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x\right )}}, -\frac{{\left (2 \, c^{2} d^{4} - a c d^{2} e^{2} +{\left (2 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} x^{2} + 2 \,{\left (2 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) +{\left (4 \, c^{2} d^{4} e + 5 \, a c d^{2} e^{3} + a^{2} e^{5} + 3 \,{\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (c^{3} d^{8} + 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} + a^{3} d^{2} e^{6} +{\left (c^{3} d^{6} e^{2} + 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e + 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x\right )}}\right ] \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}{\sqrt{a + c x^{2}} \left (d + e x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.42993, size = 466, normalized size = 3.21 \begin{align*} -c{\left (\frac{{\left (2 \, c d^{2} - a e^{2}\right )} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-c d^{2} - a e^{2}}} + \frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c d^{2} e + 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} c^{\frac{3}{2}} d^{3} - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a c d^{2} e - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a \sqrt{c} d e^{2} -{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} a e^{3} + 3 \, a^{2} \sqrt{c} d e^{2} -{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a^{2} e^{3}}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} e + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} \sqrt{c} d - a e\right )}^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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