Optimal. Leaf size=115 \[ \frac {64 a^2 c^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{105 (c-a c x)^{3/2}}+\frac {16 a^2 c^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{35 \sqrt {c-a c x}}+\frac {2}{7} a^2 c^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-a c x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 137, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6176, 6181, 89, 78, 37} \[ \frac {142 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{5/2}}{105 a^2 x \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {36 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 37
Rule 78
Rule 89
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac {(c-a c x)^{5/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\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 )}{\left (1-\frac {1}{a x}\right )^{5/2}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {\left (2 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\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 )}{7 \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {36 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {\left (71 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {36 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{5/2}}{35 a \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {142 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{5/2}}{105 a^2 \left (1-\frac {1}{a x}\right )^{5/2} x}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{5/2}}{7 \left (1-\frac {1}{a x}\right )^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 67, normalized size = 0.58 \[ \frac {2 c^2 \sqrt {\frac {1}{a x}+1} (a x+1) \left (15 a^2 x^2-54 a x+71\right ) \sqrt {c-a c x}}{105 a \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 83, normalized size = 0.72 \[ \frac {2 \, {\left (15 \, a^{4} c^{2} x^{4} - 24 \, a^{3} c^{2} x^{3} - 22 \, a^{2} c^{2} x^{2} + 88 \, a c^{2} x + 71 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 99, normalized size = 0.86 \[ \frac {2 \, {\left (\frac {64 \, \sqrt {2} \sqrt {-c} c^{2}}{\mathrm {sgn}\relax (c)} + \frac {15 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} - 84 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c - 140 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{2}}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{105 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 56, normalized size = 0.49 \[ \frac {2 \left (a x +1\right ) \left (15 a^{2} x^{2}-54 a x +71\right ) \left (-a c x +c \right )^{\frac {5}{2}}}{105 a \left (a x -1\right )^{2} \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 67, normalized size = 0.58 \[ \frac {2 \, {\left (15 \, a^{3} \sqrt {-c} c^{2} x^{3} - 39 \, a^{2} \sqrt {-c} c^{2} x^{2} + 17 \, a \sqrt {-c} c^{2} x + 71 \, \sqrt {-c} c^{2}\right )} \sqrt {a x + 1}}{105 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.41, size = 60, normalized size = 0.52 \[ \frac {2\,c^2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (15\,a^2\,x^2-54\,a\,x+71\right )}{105\,a\,\left (a\,x-1\right )} \]
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]
________________________________________________________________________________________