Optimal. Leaf size=368 \[ -\frac {94208 (c-a c x)^{9/2}}{231 a^6 x^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}-\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 x^2 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^6 (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 x \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}-\frac {40960 (c-a c x)^{9/2}}{231 a^5 x^4 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {4096 (c-a c x)^{9/2}}{231 a^4 x^3 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 x^2 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {4096 (c-a c x)^{9/2}}{231 a^4 x^3 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}-\frac {40960 (c-a c x)^{9/2}}{231 a^5 x^4 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}-\frac {94208 (c-a c x)^{9/2}}{231 a^6 x^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^6 (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 x \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 78
Rule 89
Rule 94
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac {(c-a c x)^{9/2} \int e^{-3 \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 )^6}{x^{13/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (24 \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 )^5}{x^{11/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{11 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (160 \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}{x^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{33 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (2560 \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}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{231 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2048 \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}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{77 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}+\frac {4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (4096 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{231 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {40960 (c-a c x)^{9/2}}{231 a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^4}+\frac {4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (47104 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {16 \left (a-\frac {1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}-\frac {94208 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^5}-\frac {40960 (c-a c x)^{9/2}}{231 a^5 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^4}+\frac {4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {1024 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {320 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac {1}{a x}\right )^{9/2} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 84, normalized size = 0.23 \[ \frac {2 c^4 \left (21 a^6 x^6-182 a^5 x^5+755 a^4 x^4-2132 a^3 x^3+5419 a^2 x^2-23062 a x-46355\right ) \sqrt {c-a c x}}{231 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 105, normalized size = 0.29 \[ \frac {2 \, {\left (21 \, a^{6} c^{4} x^{6} - 182 \, a^{5} c^{4} x^{5} + 755 \, a^{4} c^{4} x^{4} - 2132 \, a^{3} c^{4} x^{3} + 5419 \, a^{2} c^{4} x^{2} - 23062 \, a c^{4} x - 46355 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{231 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 88, normalized size = 0.24 \[ \frac {2 \left (a x +1\right ) \left (21 x^{6} a^{6}-182 x^{5} a^{5}+755 x^{4} a^{4}-2132 x^{3} a^{3}+5419 a^{2} x^{2}-23062 a x -46355\right ) \left (-a c x +c \right )^{\frac {9}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{231 a \left (a x -1\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 152, normalized size = 0.41 \[ \frac {2 \, {\left (21 \, a^{7} \sqrt {-c} c^{4} x^{7} - 161 \, a^{6} \sqrt {-c} c^{4} x^{6} + 573 \, a^{5} \sqrt {-c} c^{4} x^{5} - 1377 \, a^{4} \sqrt {-c} c^{4} x^{4} + 3287 \, a^{3} \sqrt {-c} c^{4} x^{3} - 17643 \, a^{2} \sqrt {-c} c^{4} x^{2} - 69417 \, a \sqrt {-c} c^{4} x - 46355 \, \sqrt {-c} c^{4}\right )} {\left (a x - 1\right )}^{2}}{231 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.45, size = 110, normalized size = 0.30 \[ \frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (21\,a^5\,x^5-161\,a^4\,x^4+594\,a^3\,x^3-1538\,a^2\,x^2+3881\,a\,x-19181\right )}{231\,a}-\frac {131072\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{231\,a\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________