Optimal. Leaf size=76 \[ \frac {e^{2 \text {ArcTan}(a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 e^{2 \text {ArcTan}(a x)} (2+a x)}{65 a c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.06, 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 = {5178, 5177}
\begin {gather*} \frac {6 (a x+2) e^{2 \text {ArcTan}(a x)}}{65 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+2) e^{2 \text {ArcTan}(a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5177
Rule 5178
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 = 0.02, size = 62, normalized size = 0.82 \begin {gather*} \frac {e^{2 \text {ArcTan}(a x)} \left (22+21 a x+12 a^2 x^2+6 a^3 x^3\right )}{65 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 56, normalized size = 0.74
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{3} x^{3}+12 a^{2} x^{2}+21 a x +22\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{65 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(56\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.27, size = 72, normalized size = 0.95 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \frac {e^{2 \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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