Optimal. Leaf size=77 \[ \frac {8 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{15 (c-a c x)^{3/2}}+\frac {2 a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}{5 \sqrt {c-a c x}} \]
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Rubi [A]
time = 0.10, antiderivative size = 89, normalized size of antiderivative = 1.16, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6311, 6316, 79,
37} \begin {gather*} \frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac {(c-a c x)^{3/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\left (7 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {14 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.74 \begin {gather*} -\frac {2 c \sqrt {1+\frac {1}{a x}} (1+a x) (-7+3 a x) \sqrt {c-a c x}}{15 a \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 43, normalized size = 0.56
method | result | size |
default | \(-\frac {2 \sqrt {-c \left (a x -1\right )}\, c \left (a x +1\right ) \left (3 a x -7\right )}{15 \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(43\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (3 a x -7\right ) \left (-a c x +c \right )^{\frac {3}{2}}}{15 a \left (a x -1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(48\) |
risch | \(\frac {2 c^{2} \left (a x -1\right ) \left (3 a^{2} x^{2}-4 a x -7\right )}{15 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 45, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, a^{2} \sqrt {-c} c x^{2} - 4 \, a \sqrt {-c} c x - 7 \, \sqrt {-c} c\right )} \sqrt {a x + 1}}{15 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 64, normalized size = 0.83 \begin {gather*} -\frac {2 \, {\left (3 \, a^{3} c x^{3} - a^{2} c x^{2} - 11 \, a c x - 7 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{2} x - a\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 {\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 50, normalized size = 0.65 \begin {gather*} -\frac {2\,c\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\left (3\,a\,x-7\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{15\,a\,\left (a\,x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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