Optimal. Leaf size=136 \[ \frac {(1+a x)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}-\frac {4 (1+a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1+a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5} \]
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Rubi [A]
time = 0.13, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6328, 45}
\begin {gather*} \frac {(a x+1)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {4 (a x+1)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {(a x+1)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 x^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6327
Rule 6328
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac {\left (c-a^2 c x^2\right )^{5/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (c-a^2 c x^2\right )^{5/2} \int (-1+a x)^2 (1+a x)^3 \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {\left (c-a^2 c x^2\right )^{5/2} \int \left (4 (1+a x)^3-4 (1+a x)^4+(1+a x)^5\right ) \, dx}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ &=\frac {(1+a x)^4 \left (c-a^2 c x^2\right )^{5/2}}{a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}-\frac {4 (1+a x)^5 \left (c-a^2 c x^2\right )^{5/2}}{5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}+\frac {(1+a x)^6 \left (c-a^2 c x^2\right )^{5/2}}{6 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x^5}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 63, normalized size = 0.46 \begin {gather*} \frac {c^2 (1+a x)^4 \left (11-14 a x+5 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{30 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 81, normalized size = 0.60
method | result | size |
default | \(\frac {\left (5 a^{5} x^{5}+6 a^{4} x^{4}-15 a^{3} x^{3}-20 a^{2} x^{2}+15 a x +30\right ) x \,c^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}{30 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(81\) |
gosper | \(\frac {x \left (5 a^{5} x^{5}+6 a^{4} x^{4}-15 a^{3} x^{3}-20 a^{2} x^{2}+15 a x +30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{30 \left (a x +1\right )^{3} \left (a x -1\right )^{2} \sqrt {\frac {a x -1}{a x +1}}}\) | \(84\) |
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 = 0.36, size = 73, normalized size = 0.54 \begin {gather*} \frac {{\left (5 \, a^{5} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{5} - 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} + 15 \, a c^{2} x^{2} + 30 \, c^{2} x\right )} \sqrt {-a^{2} c}}{30 \, a} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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