Optimal. Leaf size=114 \[ \frac {e^{2 \text {ArcTan}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {20 e^{2 \text {ArcTan}(a x)} (2+3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {24 e^{2 \text {ArcTan}(a x)} (2+a x)}{377 a c^3 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {24 (a x+2) e^{2 \text {ArcTan}(a x)}}{377 a c^3 \sqrt {a^2 c x^2+c}}+\frac {20 (3 a x+2) e^{2 \text {ArcTan}(a x)}}{377 a c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {(5 a x+2) e^{2 \text {ArcTan}(a x)}}{29 a c \left (a^2 c x^2+c\right )^{5/2}} \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 )^{7/2}} \, dx &=\frac {e^{2 \tan ^{-1}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {20 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{29 c}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {20 e^{2 \tan ^{-1}(a x)} (2+3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {120 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{377 c^2}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (2+5 a x)}{29 a c \left (c+a^2 c x^2\right )^{5/2}}+\frac {20 e^{2 \tan ^{-1}(a x)} (2+3 a x)}{377 a c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {24 e^{2 \tan ^{-1}(a x)} (2+a x)}{377 a c^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 81, normalized size = 0.71 \begin {gather*} \frac {e^{2 \text {ArcTan}(a x)} \left (114+149 a x+136 a^2 x^2+108 a^3 x^3+48 a^4 x^4+24 a^5 x^5\right )}{377 a c^3 \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 72, normalized size = 0.63
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (24 a^{5} x^{5}+48 a^{4} x^{4}+108 a^{3} x^{3}+136 a^{2} x^{2}+149 a x +114\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{377 a \left (a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\) | \(72\) |
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 = 2.10, size = 99, normalized size = 0.87 \begin {gather*} \frac {{\left (24 \, a^{5} x^{5} + 48 \, a^{4} x^{4} + 108 \, a^{3} x^{3} + 136 \, a^{2} x^{2} + 149 \, a x + 114\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}}{377 \, {\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.69, size = 122, normalized size = 1.07 \begin {gather*} \frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {114}{377\,a^5\,c^3}+\frac {24\,x^5}{377\,c^3}+\frac {149\,x}{377\,a^4\,c^3}+\frac {48\,x^4}{377\,a\,c^3}+\frac {108\,x^3}{377\,a^2\,c^3}+\frac {136\,x^2}{377\,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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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