Optimal. Leaf size=90 \[ \frac {i 2^{p-i} (1-i a x)^{p+(1+i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (i-p,p+(1+i);p+(2+i);\frac {1}{2} (1-i a x)\right )}{a (p+(1+i))} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5076, 5073, 69} \[ \frac {i 2^{p-i} (1-i a x)^{p+(1+i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (i-p,p+(1+i);p+(2+i);\frac {1}{2} (1-i a x)\right )}{a (p+(1+i))} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int e^{2 \tan ^{-1}(a x)} \left (c+a^2 c x^2\right )^p \, dx &=\left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int e^{2 \tan ^{-1}(a x)} \left (1+a^2 x^2\right )^p \, dx\\ &=\left (\left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p\right ) \int (1-i a x)^{i+p} (1+i a x)^{-i+p} \, dx\\ &=\frac {i 2^{-i+p} (1-i a x)^{(1+i)+p} \left (1+a^2 x^2\right )^{-p} \left (c+a^2 c x^2\right )^p \, _2F_1\left (i-p,(1+i)+p;(2+i)+p;\frac {1}{2} (1-i a x)\right )}{a ((1+i)+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 90, normalized size = 1.00 \[ \frac {i 2^{p-i} (1-i a x)^{p+(1+i)} \left (a^2 x^2+1\right )^{-p} \left (a^2 c x^2+c\right )^p \, _2F_1\left (i-p,p+(1+i);p+(2+i);\frac {1}{2} (1-i a x)\right )}{a (p+(1+i))} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (2 \, \arctan \left (a x\right )\right )}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{2 \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{p}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{p} e^{\left (2 \, \arctan \left (a x\right )\right )}\,{d x} \]
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 \[ \int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{p} e^{2 \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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