Optimal. Leaf size=151 \[ \frac{e \sqrt{a+c x^2} \left (c d^2-2 a e^2\right )}{a (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0844861, antiderivative size = 151, 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 = {741, 807, 725, 206} \[ \frac{e \sqrt{a+c x^2} \left (c d^2-2 a e^2\right )}{a (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )}-\frac{3 c d e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 741
Rule 807
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+c x^2\right )^{3/2}} \, dx &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt{a+c x^2}}-\frac{\int \frac{-2 a e^2-c d e x}{(d+e x)^2 \sqrt{a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt{a+c x^2}}+\frac{e \left (c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}+\frac{\left (3 c d e^2\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt{a+c x^2}}+\frac{e \left (c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}-\frac{\left (3 c d e^2\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 )}{\left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x) \sqrt{a+c x^2}}+\frac{e \left (c d^2-2 a e^2\right ) \sqrt{a+c x^2}}{a \left (c d^2+a e^2\right )^2 (d+e x)}-\frac{3 c d e^2 \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 )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.125062, size = 139, normalized size = 0.92 \[ \frac{-a^2 e^3+a c e \left (2 d^2+d e x-2 e^2 x^2\right )+c^2 d^2 x (d+e x)}{a \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}-\frac{3 c d e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.194, size = 400, normalized size = 2.7 \begin{align*} -{\frac{1}{a{e}^{2}+c{d}^{2}} \left ({\frac{d}{e}}+x \right ) ^{-1}{\frac{1}{\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}}}}}}}+3\,{\frac{ced}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}{\frac{1}{\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}}}}}}}+3\,{\frac{{c}^{2}{d}^{2}x}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}{\frac{1}{\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}}}}}}}-3\,{\frac{ced}{ \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}}}}}}}-2\,{\frac{cx}{ \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\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}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.41863, size = 1777, normalized size = 11.77 \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}{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]