Optimal. Leaf size=81 \[ -\frac {a x^{-3-2 p} \left (d+e x^2\right )^{1+p} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2} (-1-2 p);-\frac {e x^2}{d}\right )}{d (3+2 p)}+b \text {Int}\left (x^{-4-2 p} \left (d+e x^2\right )^p \text {ArcTan}(c x),x\right ) \]
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Rubi [A]
time = 0.10, 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 = {}
\begin {gather*} \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \text {ArcTan}(c x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int x^{-4-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^{-4-2 p} \left (d+e x^2\right )^p \, dx+b \int x^{-4-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ &=b \int x^{-4-2 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^{-4-2 p} \left (1+\frac {e x^2}{d}\right )^p \, dx\\ &=-\frac {a x^{-3-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} \, _2F_1\left (\frac {1}{2} (-3-2 p),-p;\frac {1}{2} (-1-2 p);-\frac {e x^2}{d}\right )}{3+2 p}+b \int x^{-4-2 p} \left (d+e x^2\right )^p \tan ^{-1}(c x) \, dx\\ \end {align*}
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Mathematica [A]
time = 2.53, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-4-2 p} \left (d+e x^2\right )^p (a+b \text {ArcTan}(c x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 1.23, size = 0, normalized size = 0.00 \[\int x^{-4-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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