Optimal. Leaf size=254 \[ \frac {9088 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 x^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^4 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {768 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {128 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 x \left (1-\frac {1}{a x}\right )^{9/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {768 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9088 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 x^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^4 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {128 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 x \left (1-\frac {1}{a x}\right )^{9/2}} \]
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)^{9/2} \, dx &=\frac {(c-a c x)^{9/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{9/2} x^{9/2} \, dx}{\left (1-\frac {1}{a x}\right )^{9/2} x^{9/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4 \sqrt {1+\frac {x}{a}}}{x^{13/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (16 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\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 )}{11 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (64 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\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 )}{33 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (128 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\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 )}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {768 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (4544 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{1155 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9088 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 \left (1-\frac {1}{a x}\right )^{9/2} x^3}-\frac {768 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.33 \[ \frac {2 c^4 \sqrt {\frac {1}{a x}+1} (a x+1) \left (315 a^4 x^4-1820 a^3 x^3+4530 a^2 x^2-6396 a x+5419\right ) \sqrt {c-a c x}}{3465 a \sqrt {1-\frac {1}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 105, normalized size = 0.41 \[ \frac {2 \, {\left (315 \, a^{6} c^{4} x^{6} - 1190 \, a^{5} c^{4} x^{5} + 1205 \, a^{4} c^{4} x^{4} + 844 \, a^{3} c^{4} x^{3} - 2843 \, a^{2} c^{4} x^{2} + 4442 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3465 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 147, normalized size = 0.58 \[ \frac {2 \, {\left (\frac {4096 \, \sqrt {2} \sqrt {-c} c^{4}}{\mathrm {sgn}\relax (c)} + \frac {315 \, {\left (a c x + c\right )}^{5} \sqrt {-a c x - c} - 3080 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} c + 11880 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c^{2} - 22176 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{3} - 18480 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{4}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{3465 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 72, normalized size = 0.28 \[ \frac {2 \left (a x +1\right ) \left (315 x^{4} a^{4}-1820 x^{3} a^{3}+4530 a^{2} x^{2}-6396 a x +5419\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{3465 a \left (a x -1\right )^{4} \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 99, normalized size = 0.39 \[ \frac {2 \, {\left (315 \, a^{5} \sqrt {-c} c^{4} x^{5} - 1505 \, a^{4} \sqrt {-c} c^{4} x^{4} + 2710 \, a^{3} \sqrt {-c} c^{4} x^{3} - 1866 \, a^{2} \sqrt {-c} c^{4} x^{2} - 977 \, a \sqrt {-c} c^{4} x + 5419 \, \sqrt {-c} c^{4}\right )} \sqrt {a x + 1}}{3465 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 76, normalized size = 0.30 \[ \frac {2\,c^4\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (315\,a^4\,x^4-1820\,a^3\,x^3+4530\,a^2\,x^2-6396\,a\,x+5419\right )}{3465\,a\,\left (a\,x-1\right )} \]
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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