Optimal. Leaf size=197 \[ -\frac {568 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 x^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {48 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 x \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {8 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac {568 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 x^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {48 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 x \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {8 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rule 94
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx &=\frac {(c-a c x)^{7/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{7/2} x^{7/2} \, dx}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}}{x^{11/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (4 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{3 a \left (1-\frac {1}{a x}\right )^{7/2}}\\ &=-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (8 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {9}{a}+\frac {7 x}{2 a^2}\right ) \sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}\\ &=-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {48 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (284 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{105 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\\ &=-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {568 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 \left (1-\frac {1}{a x}\right )^{7/2} x^2}+\frac {48 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 75, normalized size = 0.38 \[ -\frac {2 c^3 \sqrt {\frac {1}{a x}+1} (a x+1) \left (35 a^3 x^3-165 a^2 x^2+321 a x-319\right ) \sqrt {c-a c x}}{315 a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 94, normalized size = 0.48 \[ -\frac {2 \, {\left (35 \, a^{5} c^{3} x^{5} - 95 \, a^{4} c^{3} x^{4} + 26 \, a^{3} c^{3} x^{3} + 158 \, a^{2} c^{3} x^{2} - 317 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 124, normalized size = 0.63 \[ \frac {2 \, {\left (\frac {256 \, \sqrt {2} \sqrt {-c} c^{3}}{\mathrm {sgn}\relax (c)} - \frac {35 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} - 270 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c + 756 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{2} + 840 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{3}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{315 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 64, normalized size = 0.32 \[ \frac {2 \left (a x +1\right ) \left (35 x^{3} a^{3}-165 a^{2} x^{2}+321 a x -319\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{3} \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 83, normalized size = 0.42 \[ -\frac {2 \, {\left (35 \, a^{4} \sqrt {-c} c^{3} x^{4} - 130 \, a^{3} \sqrt {-c} c^{3} x^{3} + 156 \, a^{2} \sqrt {-c} c^{3} x^{2} + 2 \, a \sqrt {-c} c^{3} x - 319 \, \sqrt {-c} c^{3}\right )} \sqrt {a x + 1}}{315 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 102, normalized size = 0.52 \[ \frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (-35\,a^4\,x^4+60\,a^3\,x^3+34\,a^2\,x^2-124\,a\,x+193\right )}{315\,a}+\frac {1024\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{315\,a\,\left (a\,x-1\right )} \]
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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