Optimal. Leaf size=281 \[ \frac {1312 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {656 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {164 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.29, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6176, 6181, 89, 78, 45, 37} \[ \frac {164 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {656 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {1312 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 89
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{7/2} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{11/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {14}{a}+\frac {9 x}{2 a^2}}{x^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{9 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (41 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{9 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (82 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (328 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{15 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {656 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (656 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{45 a^4 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {1312 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{45 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {656 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{45 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {82 x^2 \sqrt {c-a c x}}{9 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {164 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {8 x^3 \sqrt {c-a c x}}{9 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 0.26 \[ \frac {2 \left (5 a^5 x^5-20 a^4 x^4+41 a^3 x^3-82 a^2 x^2+328 a x+656\right ) \sqrt {c-a c x}}{45 a^5 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 77, normalized size = 0.27 \[ \frac {2 \, {\left (5 \, a^{5} x^{5} - 20 \, a^{4} x^{4} + 41 \, a^{3} x^{3} - 82 \, a^{2} x^{2} + 328 \, a x + 656\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{45 \, {\left (a^{5} x - a^{4}\right )}} \]
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 = 0.04, size = 80, normalized size = 0.28 \[ \frac {2 \left (a x +1\right ) \left (5 x^{5} a^{5}-20 x^{4} a^{4}+41 x^{3} a^{3}-82 a^{2} x^{2}+328 a x +656\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{45 a^{4} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 117, normalized size = 0.42 \[ \frac {2 \, {\left (5 \, a^{6} \sqrt {-c} x^{6} - 15 \, a^{5} \sqrt {-c} x^{5} + 21 \, a^{4} \sqrt {-c} x^{4} - 41 \, a^{3} \sqrt {-c} x^{3} + 246 \, a^{2} \sqrt {-c} x^{2} + 984 \, a \sqrt {-c} x + 656 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{45 \, {\left (a^{6} x^{2} - 2 \, a^{5} x + a^{4}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 74, normalized size = 0.26 \[ \frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^5\,x^5-20\,a^4\,x^4+41\,a^3\,x^3-82\,a^2\,x^2+328\,a\,x+656\right )}{45\,a^4\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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