Optimal. Leaf size=95 \[ \frac {c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6268,
25, 528, 382, 79, 52, 65, 214} \begin {gather*} -\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+x \left (c-\frac {c}{a x}\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 25
Rule 52
Rule 65
Rule 79
Rule 214
Rule 382
Rule 528
Rule 6268
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx\\ &=-\int \frac {\left (c-\frac {c}{a x}\right )^{5/2} (1+a x)}{1-a x} \, dx\\ &=\frac {c \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} (1+a x)}{x} \, dx}{a}\\ &=\frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{3/2} \, dx}{a}\\ &=-\frac {c \text {Subst}\left (\int \frac {(a+x) \left (c-\frac {c x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {c \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {c^2 \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x+\frac {c^3 \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x-c^2 \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=\frac {c^2 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\left (c-\frac {c}{a x}\right )^{5/2} x-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 75, normalized size = 0.79 \begin {gather*} \frac {c^2 \sqrt {c-\frac {c}{a x}} \left (2-2 a x+3 a^2 x^2\right )-3 a c^{5/2} x \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{3 a^2 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 108, normalized size = 1.14
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (-6 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} x^{3}+3 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{2} x^{3}+4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{6 x^{2} \sqrt {\left (a x -1\right ) x}\, a^{\frac {5}{2}}}\) | \(108\) |
risch | \(\frac {\left (3 a^{3} x^{3}-5 a^{2} x^{2}+4 a x -2\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \,a^{2} \left (a x -1\right )}-\frac {\ln \left (\frac {-\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 182, normalized size = 1.92 \begin {gather*} \left [\frac {3 \, a c^{\frac {5}{2}} x \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (3 \, a^{2} c^{2} x^{2} - 2 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, a^{2} x}, \frac {3 \, a \sqrt {-c} c^{2} x \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (3 \, a^{2} c^{2} x^{2} - 2 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________