Optimal. Leaf size=80 \[ \frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{2} d^3 \sqrt{e}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0688886, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1150, 414, 527, 12, 377, 208} \[ \frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{2} d^3 \sqrt{e}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1150
Rule 414
Rule 527
Rule 12
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx &=\int \frac{1}{\left (d-e x^2\right ) \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac{\int \frac{-5 d e+2 e^2 x^2}{\left (d-e x^2\right ) \left (d+e x^2\right )^{3/2}} \, dx}{6 d^2 e}\\ &=\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\int \frac{3 d^2 e^2}{\left (d-e x^2\right ) \sqrt{d+e x^2}} \, dx}{12 d^4 e^2}\\ &=\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\int \frac{1}{\left (d-e x^2\right ) \sqrt{d+e x^2}} \, dx}{4 d^2}\\ &=\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{d-2 d e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{4 d^2}\\ &=\frac{x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac{7 x}{12 d^3 \sqrt{d+e x^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{4 \sqrt{2} d^3 \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 1.68192, size = 345, normalized size = 4.31 \[ \frac{\frac{384 e^4 x^8 \left (d+e x^2\right )^2 \text{HypergeometricPFQ}\left (\{2,2,2\},\left \{1,\frac{9}{2}\right \},-\frac{2 e x^2}{d-e x^2}\right )}{e x^2-d}+\frac{384 e^4 x^8 \left (4 d^2+7 d e x^2+3 e^2 x^4\right ) \, _2F_1\left (2,2;\frac{9}{2};-\frac{2 e x^2}{d-e x^2}\right )}{e x^2-d}+\frac{35 \sqrt{2} \sqrt{\frac{e x^2}{e x^2-d}} \left (-5 d^2 e x^2-15 d^3+12 d e^2 x^4+8 e^3 x^6\right ) \left (\sqrt{2} \sqrt{\frac{e x^2}{e x^2-d}} \sqrt{\frac{d+e x^2}{d-e x^2}} \left (-3 d^2-2 d e x^2+5 e^2 x^4\right )+3 \left (d+e x^2\right )^2 \sin ^{-1}\left (\sqrt{2} \sqrt{\frac{e x^2}{e x^2-d}}\right )\right )}{\sqrt{\frac{d+e x^2}{d-e x^2}}}}{2520 d^5 e^3 x^5 \sqrt{d+e x^2} \left (1-\frac{e^2 x^4}{d^2}\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.022, size = 911, normalized size = 11.4 \begin{align*} \text{result too large to display} \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}{{\left (e^{2} x^{4} - d^{2}\right )}{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.10236, size = 632, normalized size = 7.9 \begin{align*} \left [\frac{3 \, \sqrt{2}{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{e} \log \left (\frac{17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt{2}{\left (3 \, e x^{3} + d x\right )} \sqrt{e x^{2} + d} \sqrt{e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 8 \,{\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt{e x^{2} + d}}{96 \,{\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}, -\frac{3 \, \sqrt{2}{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{-e} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{4 \,{\left (e^{2} x^{3} + d e x\right )}}\right ) - 4 \,{\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt{e x^{2} + d}}{48 \,{\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}\right ] \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}{- d^{3} \sqrt{d + e x^{2}} - d^{2} e x^{2} \sqrt{d + e x^{2}} + d e^{2} x^{4} \sqrt{d + e x^{2}} + e^{3} x^{6} \sqrt{d + e x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, -1\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]