Optimal. Leaf size=75 \[ b \text {Int}\left (\frac {x^2 \tan ^{-1}(c x)}{\sqrt {d+e x^2}},x\right )-\frac {a d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}+\frac {a x \sqrt {d+e x^2}}{2 e} \]
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Rubi [A] time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx &=a \int \frac {x^2}{\sqrt {d+e x^2}} \, dx+b \int \frac {x^2 \tan ^{-1}(c x)}{\sqrt {d+e x^2}} \, dx\\ &=\frac {a x \sqrt {d+e x^2}}{2 e}+b \int \frac {x^2 \tan ^{-1}(c x)}{\sqrt {d+e x^2}} \, dx-\frac {(a d) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{2 e}\\ &=\frac {a x \sqrt {d+e x^2}}{2 e}+b \int \frac {x^2 \tan ^{-1}(c x)}{\sqrt {d+e x^2}} \, dx-\frac {(a d) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{2 e}\\ &=\frac {a x \sqrt {d+e x^2}}{2 e}-\frac {a d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 e^{3/2}}+b \int \frac {x^2 \tan ^{-1}(c x)}{\sqrt {d+e x^2}} \, dx\\ \end {align*}
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Mathematica [A] time = 9.38, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \arctan \left (c x\right ) + a x^{2}}{\sqrt {e x^{2} + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a +b \arctan \left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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