Optimal. Leaf size=285 \[ \frac{b e x^{-2 p-3} \left (d+e x^2\right )^p \left (\frac{e x^2}{d}+1\right )^{-p} \text{Hypergeometric2F1}\left (\frac{1}{2} (-2 p-3),-p-1,\frac{1}{2} (-2 p-1),-\frac{e x^2}{d}\right )}{2 c d \left (2 p^3+9 p^2+13 p+6\right )}+\frac{e x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (p+1) (p+2)}-\frac{x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} \left (a+b \tan ^{-1}(c x)\right )}{2 d (p+2)}-\frac{b x^{-2 p-3} \left (c^2 d (p+1)+e\right ) \left (d+e x^2\right )^p \left (\frac{e x^2}{d}+1\right )^{-p} F_1\left (\frac{1}{2} (-2 p-3);1,-p-1;\frac{1}{2} (-2 p-1);-c^2 x^2,-\frac{e x^2}{d}\right )}{2 c d (p+1) (p+2) (2 p+3)} \]
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Rubi [A] time = 0.374795, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {271, 264, 4976, 12, 584, 365, 364, 511, 510} \[ \frac{e x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (p+1) (p+2)}-\frac{x^{-2 (p+2)} \left (d+e x^2\right )^{p+1} \left (a+b \tan ^{-1}(c x)\right )}{2 d (p+2)}-\frac{b x^{-2 p-3} \left (c^2 d (p+1)+e\right ) \left (d+e x^2\right )^p \left (\frac{e x^2}{d}+1\right )^{-p} F_1\left (\frac{1}{2} (-2 p-3);1,-p-1;\frac{1}{2} (-2 p-1);-c^2 x^2,-\frac{e x^2}{d}\right )}{2 c d (p+1) (p+2) (2 p+3)}+\frac{b e x^{-2 p-3} \left (d+e x^2\right )^p \left (\frac{e x^2}{d}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (-2 p-3),-p-1;\frac{1}{2} (-2 p-1);-\frac{e x^2}{d}\right )}{2 c d \left (2 p^3+9 p^2+13 p+6\right )} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rule 4976
Rule 12
Rule 584
Rule 365
Rule 364
Rule 511
Rule 510
Rubi steps
\begin{align*} \int x^{-5-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (1+p) (2+p)}-\frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (2+p)}-(b c) \int \frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (-d-d p+e x^2\right )}{2 d^2 (1+p) (2+p) \left (1+c^2 x^2\right )} \, dx\\ &=\frac{e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (1+p) (2+p)}-\frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (2+p)}-\frac{(b c) \int \frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (-d-d p+e x^2\right )}{1+c^2 x^2} \, dx}{2 d^2 (1+p) (2+p)}\\ &=\frac{e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (1+p) (2+p)}-\frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (2+p)}-\frac{(b c) \int \left (\frac{e x^{-2 (2+p)} \left (d+e x^2\right )^{1+p}}{c^2}+\frac{\left (-e+c^2 (-d-d p)\right ) x^{-2 (2+p)} \left (d+e x^2\right )^{1+p}}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^2 (1+p) (2+p)}\\ &=\frac{e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (1+p) (2+p)}-\frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (2+p)}-\frac{(b e) \int x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \, dx}{2 c d^2 (1+p) (2+p)}+\frac{\left (b \left (e+c^2 d (1+p)\right )\right ) \int \frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p}}{1+c^2 x^2} \, dx}{2 c d^2 (1+p) (2+p)}\\ &=\frac{e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (1+p) (2+p)}-\frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (2+p)}-\frac{\left (b e \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p}\right ) \int x^{-2 (2+p)} \left (1+\frac{e x^2}{d}\right )^{1+p} \, dx}{2 c d (1+p) (2+p)}+\frac{\left (b \left (e+c^2 d (1+p)\right ) \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p}\right ) \int \frac{x^{-2 (2+p)} \left (1+\frac{e x^2}{d}\right )^{1+p}}{1+c^2 x^2} \, dx}{2 c d (1+p) (2+p)}\\ &=-\frac{b \left (e+c^2 d (1+p)\right ) x^{-3-2 p} \left (d+e x^2\right )^p \left (1+\frac{e x^2}{d}\right )^{-p} F_1\left (\frac{1}{2} (-3-2 p);1,-1-p;\frac{1}{2} (-1-2 p);-c^2 x^2,-\frac{e x^2}{d}\right )}{2 c d (1+p) (2+p) (3+2 p)}+\frac{e x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d^2 (1+p) (2+p)}-\frac{x^{-2 (2+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (2+p)}+\frac{b e 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),-1-p;\frac{1}{2} (-1-2 p);-\frac{e x^2}{d}\right )}{2 c d \left (6+13 p+9 p^2+2 p^3\right )}\\ \end{align*}
Mathematica [F] time = 4.08405, size = 0, normalized size = 0. \[ \int x^{-5-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.898, size = 0, normalized size = 0. \begin{align*} \int{x}^{-5-2\,p} \left ( e{x}^{2}+d \right ) ^{p} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{\arctan \left (c x\right ) e^{\left (p \log \left (e x^{2} + d\right ) - 2 \, p \log \left (x\right )\right )}}{x^{5}}\,{d x} + \frac{{\left (e^{2} x^{4} - d e p x^{2} - d^{2}{\left (p + 1\right )}\right )} a e^{\left (p \log \left (e x^{2} + d\right ) - 2 \, p \log \left (x\right )\right )}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} d^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}{\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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