Optimal. Leaf size=76 \[ b \text {Int}\left (x^m \tan ^{-1}(c x) \left (d+e x^2\right )^{3/2},x\right )+\frac {a x^{m+1} \left (d+e x^2\right )^{5/2} \, _2F_1\left (1,\frac {m+6}{2};\frac {m+3}{2};-\frac {e x^2}{d}\right )}{d (m+1)} \]
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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 = {} \[ \int x^m \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^m \left (d+e x^2\right )^{3/2} \, dx+b \int x^m \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx\\ &=b \int x^m \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx+\frac {\left (a d \sqrt {d+e x^2}\right ) \int x^m \left (1+\frac {e x^2}{d}\right )^{3/2} \, dx}{\sqrt {1+\frac {e x^2}{d}}}\\ &=\frac {a d x^{1+m} \sqrt {d+e x^2} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};-\frac {e x^2}{d}\right )}{(1+m) \sqrt {1+\frac {e x^2}{d}}}+b \int x^m \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx\\ \end {align*}
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Mathematica [A] time = 0.13, size = 0, normalized size = 0.00 \[ \int x^m \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d} x^{m}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.02, size = 0, normalized size = 0.00 \[ \int x^{m} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )\, dx \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{m}\,{d x} \]
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 \[ \int x^m\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2} \,d x \]
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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