Optimal. Leaf size=129 \[ -\frac {x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} \left (a+b \tan ^{-1}(c x)\right )}{2 d (p+1)}-\frac {b c x^{-2 p-1} \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} F_1\left (\frac {1}{2} (-2 p-1);1,-p-1;\frac {1}{2} (1-2 p);-c^2 x^2,-\frac {e x^2}{d}\right )}{2 \left (2 p^2+3 p+1\right )} \]
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Rubi [A] time = 0.17, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {264, 4976, 12, 511, 510} \[ -\frac {x^{-2 (p+1)} \left (d+e x^2\right )^{p+1} \left (a+b \tan ^{-1}(c x)\right )}{2 d (p+1)}-\frac {b c x^{-2 p-1} \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} F_1\left (\frac {1}{2} (-2 p-1);1,-p-1;\frac {1}{2} (1-2 p);-c^2 x^2,-\frac {e x^2}{d}\right )}{2 \left (2 p^2+3 p+1\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 510
Rule 511
Rule 4976
Rubi steps
\begin {align*} \int x^{-3-2 p} \left (d+e x^2\right )^p \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac {x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (1+p)}-(b c) \int -\frac {x^{-2 (1+p)} \left (d+e x^2\right )^{1+p}}{2 d (1+p) \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (1+p)}+\frac {(b c) \int \frac {x^{-2 (1+p)} \left (d+e x^2\right )^{1+p}}{1+c^2 x^2} \, dx}{2 d (1+p)}\\ &=-\frac {x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (1+p)}+\frac {\left (b c \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p}\right ) \int \frac {x^{-2 (1+p)} \left (1+\frac {e x^2}{d}\right )^{1+p}}{1+c^2 x^2} \, dx}{2 (1+p)}\\ &=-\frac {b c x^{-1-2 p} \left (d+e x^2\right )^p \left (1+\frac {e x^2}{d}\right )^{-p} F_1\left (\frac {1}{2} (-1-2 p);1,-1-p;\frac {1}{2} (1-2 p);-c^2 x^2,-\frac {e x^2}{d}\right )}{2 \left (1+3 p+2 p^2\right )}-\frac {x^{-2 (1+p)} \left (d+e x^2\right )^{1+p} \left (a+b \tan ^{-1}(c x)\right )}{2 d (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 166, normalized size = 1.29 \[ -\frac {x^{-2 (p+1)} \left (d+e x^2\right )^p \left (\frac {e x^2}{d}+1\right )^{-p} \left (c (2 p+1) \left (d+e x^2\right ) \left (\frac {e x^2}{d}+1\right )^p \left (a+b \tan ^{-1}(c x)\right )+b x \left (c^2 d-e\right ) F_1\left (-p-\frac {1}{2};-p,1;\frac {1}{2}-p;-\frac {e x^2}{d},-c^2 x^2\right )+b e x \, _2F_1\left (-p-\frac {1}{2},-p;\frac {1}{2}-p;-\frac {e x^2}{d}\right )\right )}{2 c d (p+1) (2 p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x\right ) + a\right )} {\left (e x^{2} + d\right )}^{p} x^{-2 \, p - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.52, size = 0, normalized size = 0.00 \[ \int x^{-3-2 p} \left (e \,x^{2}+d \right )^{p} \left (a +b \arctan \left (c x \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\arctan \left (c x\right ) e^{\left (p \log \left (e x^{2} + d\right ) - 2 \, p \log \relax (x)\right )}}{x^{3}}\,{d x} - \frac {{\left (e x^{2} + d\right )} a e^{\left (p \log \left (e x^{2} + d\right ) - 2 \, p \log \relax (x)\right )}}{2 \, d {\left (p + 1\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^p}{x^{2\,p+3}} \,d x \]
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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