Optimal. Leaf size=76 \[ \frac {6 (a x+2) e^{2 \tan ^{-1}(a x)}}{65 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+2) e^{2 \tan ^{-1}(a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5070, 5069} \[ \frac {6 (a x+2) e^{2 \tan ^{-1}(a x)}}{65 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+2) e^{2 \tan ^{-1}(a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5069
Rule 5070
Rubi steps
\begin {align*} \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {e^{2 \tan ^{-1}(a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{13 c}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 e^{2 \tan ^{-1}(a x)} (2+a x)}{65 a c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.82 \[ \frac {\left (6 a^3 x^3+12 a^2 x^2+21 a x+22\right ) e^{2 \tan ^{-1}(a x)}}{65 c^2 \left (a^3 x^2+a\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 72, normalized size = 0.95 \[ \frac {{\left (6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 21 \, a x + 22\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}}{65 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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 [A] time = 0.04, size = 56, normalized size = 0.74 \[ \frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{3} x^{3}+12 a^{2} x^{2}+21 a x +22\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{65 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]
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 \[ \int \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 80, normalized size = 1.05 \[ \frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {22}{65\,a^3\,c^2}+\frac {6\,x^3}{65\,c^2}+\frac {21\,x}{65\,a^2\,c^2}+\frac {12\,x^2}{65\,a\,c^2}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^2}+x^2\,\sqrt {c\,a^2\,x^2+c}} \]
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 \[ \int \frac {e^{2 \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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