Optimal. Leaf size=179 \[ \frac{e^2 \sqrt{a+c x^2}}{d (d+e x) \left (a e^2+c d^2\right )}+\frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \sqrt{a e^2+c d^2}}+\frac{c e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.137258, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {961, 266, 63, 208, 731, 725, 206} \[ \frac{e^2 \sqrt{a+c x^2}}{d (d+e x) \left (a e^2+c d^2\right )}+\frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^2 \sqrt{a e^2+c d^2}}+\frac{c e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 961
Rule 266
Rule 63
Rule 208
Rule 731
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x (d+e x)^2 \sqrt{a+c x^2}} \, dx &=\int \left (\frac{1}{d^2 x \sqrt{a+c x^2}}-\frac{e}{d (d+e x)^2 \sqrt{a+c x^2}}-\frac{e}{d^2 (d+e x) \sqrt{a+c x^2}}\right ) \, dx\\ &=\frac{\int \frac{1}{x \sqrt{a+c x^2}} \, dx}{d^2}-\frac{e \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^2}-\frac{e \int \frac{1}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{d}\\ &=\frac{e^2 \sqrt{a+c x^2}}{d \left (c d^2+a e^2\right ) (d+e x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac{e \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}-\frac{(c e) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{c d^2+a e^2}\\ &=\frac{e^2 \sqrt{a+c x^2}}{d \left (c d^2+a e^2\right ) (d+e x)}+\frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^2 \sqrt{c d^2+a e^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c d^2}+\frac{(c e) \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 )}{c d^2+a e^2}\\ &=\frac{e^2 \sqrt{a+c x^2}}{d \left (c d^2+a e^2\right ) (d+e x)}+\frac{c e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}}+\frac{e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{d^2 \sqrt{c d^2+a e^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{\sqrt{a} d^2}\\ \end{align*}
Mathematica [A] time = 0.251733, size = 178, normalized size = 0.99 \[ \frac{\frac{d e^2 \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}+\frac{e \left (a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{e \left (a e^2+2 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}-\frac{\log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )}{\sqrt{a}}+\frac{\log (x)}{\sqrt{a}}}{d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.256, size = 364, normalized size = 2. \begin{align*} -{\frac{1}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{1}{{d}^{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{ \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}{d \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}{a{e}^{2}+c{d}^{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{ \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.
[In]
[Out]
________________________________________________________________________________________
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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 4.89092, size = 2619, normalized size = 14.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{a + c x^{2}} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]