Optimal. Leaf size=71 \[ \frac {8 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}} \]
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Rubi [A]
time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6262, 671, 663}
\begin {gather*} \frac {8 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 663
Rule 671
Rule 6262
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=c \int \sqrt {c-a c x} \sqrt {1-a^2 x^2} \, dx\\ &=\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}+\frac {1}{5} \left (4 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}} \, dx\\ &=\frac {8 c^3 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.62 \begin {gather*} -\frac {2 c (1+a x)^{3/2} (-7+3 a x) \sqrt {c-a c x}}{15 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 47, normalized size = 0.66
method | result | size |
gosper | \(\frac {2 \left (a x +1\right )^{2} \left (3 a x -7\right ) \left (-c x a +c \right )^{\frac {3}{2}}}{15 a \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}}\) | \(47\) |
default | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (a x +1\right ) c \left (3 a x -7\right )}{15 \left (a x -1\right ) a}\) | \(47\) |
risch | \(\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{2} \left (3 a^{2} x^{2}-4 a x -7\right ) \left (a x +1\right )}{15 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {\left (a x +1\right ) c}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 82, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (a^{3} c^{\frac {3}{2}} x^{3} - 2 \, a^{2} c^{\frac {3}{2}} x^{2} + 3 \, a c^{\frac {3}{2}} x + 6 \, c^{\frac {3}{2}}\right )}}{5 \, \sqrt {a x + 1} a} - \frac {2 \, {\left (a^{2} c^{\frac {3}{2}} x^{2} - 4 \, a c^{\frac {3}{2}} x - 5 \, c^{\frac {3}{2}}\right )}}{3 \, \sqrt {a x + 1} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 52, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} c x^{2} - 4 \, a c x - 7 \, c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{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}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 = 0.93, size = 49, normalized size = 0.69 \begin {gather*} \frac {\sqrt {c-a\,c\,x}\,\left (\frac {22\,c\,x}{15}+\frac {14\,c}{15\,a}-\frac {2\,a^2\,c\,x^3}{5}+\frac {2\,a\,c\,x^2}{15}\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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