Optimal. Leaf size=157 \[ \frac {2 c^2 (1-a x)^{3/2} (a x+1)^{5/2}}{5 a^2 (c-a c x)^{3/2}}-\frac {10 c^2 (1-a x)^{3/2} (a x+1)^{3/2}}{3 a^2 (c-a c x)^{3/2}}+\frac {16 c^2 (1-a x)^{3/2} \sqrt {a x+1}}{a^2 (c-a c x)^{3/2}}+\frac {8 c^2 (1-a x)^{3/2}}{a^2 \sqrt {a x+1} (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6130, 23, 77} \[ \frac {2 c^2 (1-a x)^{3/2} (a x+1)^{5/2}}{5 a^2 (c-a c x)^{3/2}}-\frac {10 c^2 (1-a x)^{3/2} (a x+1)^{3/2}}{3 a^2 (c-a c x)^{3/2}}+\frac {16 c^2 (1-a x)^{3/2} \sqrt {a x+1}}{a^2 (c-a c x)^{3/2}}+\frac {8 c^2 (1-a x)^{3/2}}{a^2 \sqrt {a x+1} (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 77
Rule 6130
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} x \sqrt {c-a c x} \, dx &=\int \frac {x (1-a x)^{3/2} \sqrt {c-a c x}}{(1+a x)^{3/2}} \, dx\\ &=\frac {(1-a x)^{3/2} \int \frac {x (c-a c x)^2}{(1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=\frac {(1-a x)^{3/2} \int \left (-\frac {4 c^2}{a (1+a x)^{3/2}}+\frac {8 c^2}{a \sqrt {1+a x}}-\frac {5 c^2 \sqrt {1+a x}}{a}+\frac {c^2 (1+a x)^{3/2}}{a}\right ) \, dx}{(c-a c x)^{3/2}}\\ &=\frac {8 c^2 (1-a x)^{3/2}}{a^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {16 c^2 (1-a x)^{3/2} \sqrt {1+a x}}{a^2 (c-a c x)^{3/2}}-\frac {10 c^2 (1-a x)^{3/2} (1+a x)^{3/2}}{3 a^2 (c-a c x)^{3/2}}+\frac {2 c^2 (1-a x)^{3/2} (1+a x)^{5/2}}{5 a^2 (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.38 \[ \frac {2 c \sqrt {1-a x} \left (3 a^3 x^3-16 a^2 x^2+79 a x+158\right )}{15 a^2 \sqrt {a x+1} \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 60, normalized size = 0.38 \[ -\frac {2 \, {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 79 \, a x + 158\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{15 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 0.40 \[ \frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {-a c x +c}\, \left (3 x^{3} a^{3}-16 a^{2} x^{2}+79 a x +158\right )}{15 \left (a x +1\right )^{2} \left (a x -1\right )^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 64, normalized size = 0.41 \[ \frac {2 \, {\left (3 \, a^{3} \sqrt {c} x^{3} - 16 \, a^{2} \sqrt {c} x^{2} + 79 \, a \sqrt {c} x + 158 \, \sqrt {c}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{15 \, {\left (a^{4} x^{2} - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 97, normalized size = 0.62 \[ \frac {\sqrt {c-a\,c\,x}\,\left (\frac {316\,\sqrt {1-a^2\,x^2}}{15\,a^4}+\frac {158\,x\,\sqrt {1-a^2\,x^2}}{15\,a^3}+\frac {2\,x^3\,\sqrt {1-a^2\,x^2}}{5\,a}-\frac {32\,x^2\,\sqrt {1-a^2\,x^2}}{15\,a^2}\right )}{\frac {1}{a^2}-x^2} \]
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 \[ \int \frac {x \sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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