Optimal. Leaf size=77 \[ \frac {8 a^2 c^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 (c-a c x)^{3/2}}+\frac {2 a^2 c^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 \sqrt {c-a c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 89, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6176, 6181, 78, 37} \[ \frac {2 x \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 37
Rule 78
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac {(c-a c x)^{3/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\left (7 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {14 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{3/2}}{15 a \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{3/2}}{5 \left (1-\frac {1}{a x}\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 57, normalized size = 0.74 \[ -\frac {2 c \sqrt {\frac {1}{a x}+1} (a x+1) (3 a x-7) \sqrt {c-a c x}}{15 a \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 64, normalized size = 0.83 \[ -\frac {2 \, {\left (3 \, a^{3} c x^{3} - a^{2} c x^{2} - 11 \, a c x - 7 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{2} x - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 74, normalized size = 0.96 \[ \frac {2 \, {\left (\frac {8 \, \sqrt {2} \sqrt {-c} c}{\mathrm {sgn}\relax (c)} - \frac {3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} + 10 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c}{c \mathrm {sgn}\left (-a c x - c\right )}\right )}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 48, normalized size = 0.62 \[ \frac {2 \left (a x +1\right ) \left (3 a x -7\right ) \left (-a c x +c \right )^{\frac {3}{2}}}{15 a \left (a x -1\right ) \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 45, normalized size = 0.58 \[ -\frac {2 \, {\left (3 \, a^{2} \sqrt {-c} c x^{2} - 4 \, a \sqrt {-c} c x - 7 \, \sqrt {-c} c\right )} \sqrt {a x + 1}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.37, size = 50, normalized size = 0.65 \[ -\frac {2\,c\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\left (3\,a\,x-7\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{15\,a\,\left (a\,x-1\right )} \]
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]
________________________________________________________________________________________