Optimal. Leaf size=40 \[ \frac{2}{3 a c (c-a c x)^{3/2}}-\frac{4}{5 a (c-a c x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0886908, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6167, 6130, 21, 43} \[ \frac{2}{3 a c (c-a c x)^{3/2}}-\frac{4}{5 a (c-a c x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6130
Rule 21
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx\\ &=-\int \frac{1+a x}{(1-a x) (c-a c x)^{5/2}} \, dx\\ &=-\left (c \int \frac{1+a x}{(c-a c x)^{7/2}} \, dx\right )\\ &=-\left (c \int \left (\frac{2}{(c-a c x)^{7/2}}-\frac{1}{c (c-a c x)^{5/2}}\right ) \, dx\right )\\ &=-\frac{4}{5 a (c-a c x)^{5/2}}+\frac{2}{3 a c (c-a c x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0585227, size = 34, normalized size = 0.85 \[ \frac{2 (5 a x+1) \sqrt{c-a c x}}{15 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 21, normalized size = 0.5 \begin{align*} -{\frac{10\,ax+2}{15\,a} \left ( -acx+c \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0351, size = 32, normalized size = 0.8 \begin{align*} -\frac{2 \,{\left (5 \, a c x + c\right )}}{15 \,{\left (-a c x + c\right )}^{\frac{5}{2}} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.48353, size = 117, normalized size = 2.92 \begin{align*} \frac{2 \, \sqrt{-a c x + c}{\left (5 \, a x + 1\right )}}{15 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 25.0109, size = 31, normalized size = 0.78 \begin{align*} - \frac{4}{5 a \left (- a c x + c\right )^{\frac{5}{2}}} + \frac{2}{3 a c \left (- a c x + c\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12408, size = 46, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (5 \, a c x + c\right )}}{15 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]