Optimal. Leaf size=108 \[ \frac {3 (a x+1) e^{\tan ^{-1}(a x)}}{13 a c^3 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+1) e^{\tan ^{-1}(a x)}}{13 a c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {(5 a x+1) e^{\tan ^{-1}(a x)}}{26 a c \left (a^2 c x^2+c\right )^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5069} \[ \frac {3 (a x+1) e^{\tan ^{-1}(a x)}}{13 a c^3 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+1) e^{\tan ^{-1}(a x)}}{13 a c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {(5 a x+1) e^{\tan ^{-1}(a x)}}{26 a c \left (a^2 c x^2+c\right )^{5/2}} \]
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 )^{7/2}} \, dx &=\frac {e^{\tan ^{-1}(a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {10 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{13 c}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {e^{\tan ^{-1}(a x)} (1+3 a x)}{13 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 \int \frac {e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{13 c^2}\\ &=\frac {e^{\tan ^{-1}(a x)} (1+5 a x)}{26 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {e^{\tan ^{-1}(a x)} (1+3 a x)}{13 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {3 e^{\tan ^{-1}(a x)} (1+a x)}{13 a c^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 79, normalized size = 0.73 \[ \frac {\left (6 a^5 x^5+6 a^4 x^4+18 a^3 x^3+14 a^2 x^2+17 a x+9\right ) e^{\tan ^{-1}(a x)}}{26 a c^3 \left (a^2 x^2+1\right )^2 \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 97, normalized size = 0.90 \[ \frac {{\left (6 \, a^{5} x^{5} + 6 \, a^{4} x^{4} + 18 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 17 \, a x + 9\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (\arctan \left (a x\right )\right )}}{26 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]
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 = 70, normalized size = 0.65 \[ \frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{5} x^{5}+6 a^{4} x^{4}+18 a^{3} x^{3}+14 a^{2} x^{2}+17 a x +9\right ) {\mathrm e}^{\arctan \left (a x \right )}}{26 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}} \]
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 \[ \int \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 120, normalized size = 1.11 \[ \frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (\frac {9}{26\,a^5\,c^3}+\frac {3\,x^5}{13\,c^3}+\frac {17\,x}{26\,a^4\,c^3}+\frac {3\,x^4}{13\,a\,c^3}+\frac {9\,x^3}{13\,a^2\,c^3}+\frac {7\,x^2}{13\,a^3\,c^3}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^4}+x^4\,\sqrt {c\,a^2\,x^2+c}+\frac {2\,x^2\,\sqrt {c\,a^2\,x^2+c}}{a^2}} \]
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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