Optimal. Leaf size=74 \[ b \text {Int}\left (\frac {x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}},x\right )+\frac {a x^{m+1} \, _2F_1\left (1,\frac {m}{2};\frac {m+3}{2};-\frac {e x^2}{d}\right )}{d (m+1) \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.17, 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 = {} \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=a \int \frac {x^m}{\left (d+e x^2\right )^{3/2}} \, dx+b \int \frac {x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx\\ &=b \int \frac {x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx+\frac {\left (a \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {x^m}{\left (1+\frac {e x^2}{d}\right )^{3/2}} \, dx}{d \sqrt {d+e x^2}}\\ &=\frac {a x^{1+m} \sqrt {1+\frac {e x^2}{d}} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right )}{d (1+m) \sqrt {d+e x^2}}+b \int \frac {x^m \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx\\ \end {align*}
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Mathematica [A] time = 4.63, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arctan \left (c x\right ) + a\right )} x^{m}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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