Optimal. Leaf size=90 \[ b \text{Unintegrable}\left (\frac{x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}},x\right )-\frac{a x}{e^2 \sqrt{d+e x^2}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{e^{5/2}}-\frac{a x^3}{3 e \left (d+e x^2\right )^{3/2}} \]
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Rubi [A] time = 0.179546, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=a \int \frac{x^4}{\left (d+e x^2\right )^{5/2}} \, dx+b \int \frac{x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\\ &=-\frac{a x^3}{3 e \left (d+e x^2\right )^{3/2}}+b \int \frac{x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac{a \int \frac{x^2}{\left (d+e x^2\right )^{3/2}} \, dx}{e}\\ &=-\frac{a x^3}{3 e \left (d+e x^2\right )^{3/2}}-\frac{a x}{e^2 \sqrt{d+e x^2}}+b \int \frac{x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac{a \int \frac{1}{\sqrt{d+e x^2}} \, dx}{e^2}\\ &=-\frac{a x^3}{3 e \left (d+e x^2\right )^{3/2}}-\frac{a x}{e^2 \sqrt{d+e x^2}}+b \int \frac{x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{e^2}\\ &=-\frac{a x^3}{3 e \left (d+e x^2\right )^{3/2}}-\frac{a x}{e^2 \sqrt{d+e x^2}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{e^{5/2}}+b \int \frac{x^4 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\\ \end{align*}
Mathematica [A] time = 11.4376, size = 0, normalized size = 0. \[ \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.608, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ( a+b\arctan \left ( cx \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} \arctan \left (c x\right ) + a x^{4}\right )} \sqrt{e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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