Optimal. Leaf size=101 \[ \frac {64 c^2 \sqrt {1-a^2 x^2}}{15 a \sqrt {c-a c x}}+\frac {16 c \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{15 a}+\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6262, 671, 663}
\begin {gather*} \frac {64 c^2 \sqrt {1-a^2 x^2}}{15 a \sqrt {c-a c x}}+\frac {16 c \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{15 a}+\frac {2 \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a} \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 &=\frac {\int \frac {(c-a c x)^{5/2}}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a}+\frac {8}{5} \int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {16 c \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{15 a}+\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a}+\frac {1}{15} (32 c) \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {64 c^2 \sqrt {1-a^2 x^2}}{15 a \sqrt {c-a c x}}+\frac {16 c \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{15 a}+\frac {2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{5 a}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 49, normalized size = 0.49 \begin {gather*} \frac {2 c^2 \sqrt {1-a^2 x^2} \left (43-14 a x+3 a^2 x^2\right )}{15 a \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.40, size = 50, normalized size = 0.50
method | result | size |
gosper | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \left (-c x a +c \right )^{\frac {3}{2}} \left (3 a^{2} x^{2}-14 a x +43\right )}{15 a \left (a x -1\right )^{2}}\) | \(48\) |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, c \left (3 a^{2} x^{2}-14 a x +43\right )}{15 \left (a x -1\right ) a}\) | \(50\) |
risch | \(-\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{2} \left (3 a^{2} x^{2}-14 a x +43\right ) \left (a x +1\right )}{15 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {\left (a x +1\right ) c}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 49, normalized size = 0.49 \begin {gather*} \frac {2 \, {\left (3 \, a^{2} c^{\frac {3}{2}} x^{2} - 14 \, a c^{\frac {3}{2}} x + 43 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{15 \, {\left (a^{2} x - a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 52, normalized size = 0.51 \begin {gather*} -\frac {2 \, {\left (3 \, a^{2} c x^{2} - 14 \, a c x + 43 \, 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}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{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 = 0.94, size = 76, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {c-a\,c\,x}\,\left (\frac {86\,c\,\sqrt {1-a^2\,x^2}}{15\,a^2}+\frac {2\,c\,x^2\,\sqrt {1-a^2\,x^2}}{5}-\frac {28\,c\,x\,\sqrt {1-a^2\,x^2}}{15\,a}\right )}{x-\frac {1}{a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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