Optimal. Leaf size=72 \[ \frac {3 (a x+1) e^{\tan ^{-1}(a x)}}{10 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+1) e^{\tan ^{-1}(a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5069} \[ \frac {3 (a x+1) e^{\tan ^{-1}(a x)}}{10 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+1) e^{\tan ^{-1}(a x)}}{10 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5069
Rule 5070
Rubi steps
\begin {align*} \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {e^{\tan ^{-1}(a x)} (1+3 a x)}{10 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {3 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{5 c}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+3 a x)}{10 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {3 e^{\tan ^{-1}(a x)} (1+a x)}{10 a c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.83 \[ \frac {\left (3 a^3 x^3+3 a^2 x^2+6 a x+4\right ) e^{\tan ^{-1}(a x)}}{10 c^2 \left (a^3 x^2+a\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 70, normalized size = 0.97 \[ \frac {{\left (3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 4\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (\arctan \left (a x\right )\right )}}{10 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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 = 54, normalized size = 0.75 \[ \frac {\left (a^{2} x^{2}+1\right ) \left (3 a^{3} x^{3}+3 a^{2} x^{2}+6 a x +4\right ) {\mathrm e}^{\arctan \left (a x \right )}}{10 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}} \]
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 \[ \int \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 78, normalized size = 1.08 \[ \frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {2}{5\,a^3\,c^2}+\frac {3\,x^3}{10\,c^2}+\frac {3\,x}{5\,a^2\,c^2}+\frac {3\,x^2}{10\,a\,c^2}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^2}+x^2\,\sqrt {c\,a^2\,x^2+c}} \]
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 \[ \int \frac {e^{\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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