Optimal. Leaf size=81 \[ b \text {Int}\left (x^{-2 p-8} \tan ^{-1}(c x) \left (d+e x^2\right )^p,x\right )-\frac {a x^{-2 p-7} \left (d+e x^2\right )^{p+1} \, _2F_1\left (-\frac {5}{2},1;\frac {1}{2} (-2 p-5);-\frac {e x^2}{d}\right )}{d (2 p+7)} \]
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Rubi [A] time = 0.14, 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 = {} \[ \int x^{-8-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^{-8-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^{-8-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-8-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ &=b \int x^{-8-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx+\left (a \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p}\right ) \int x^{-8-2 p} \left (1+\frac {e x^2}{d}\right )^p \, dx\\ &=-\frac {a x^{-7-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \, _2F_1\left (\frac {1}{2} (-7-2 p),-p;\frac {1}{2} (-5-2 p);-\frac {e x^2}{d}\right )}{7+2 p}+b \int x^{-8-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ \end {align*}
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Mathematica [A] time = 3.22, size = 0, normalized size = 0.00 \[ \int x^{-8-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 8}, x\right ) \]
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 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.44, size = 0, normalized size = 0.00 \[ \int x^{-8-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )\, dx \]
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 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 8}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+8}} \,d x \]
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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