Optimal. Leaf size=359 \[ -\frac {2 n x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \, _2F_1\left (1,\frac {n-1}{2};\frac {n+1}{2};\frac {a+\frac {1}{x}}{a-\frac {1}{x}}\right )}{a (1-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (n^2+2 n+2\right ) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {(n+2) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 363, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6192, 6194, 129, 155, 12, 131} \[ \frac {2 n x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-3}{2}} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (3-n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (n^2+2 n+2\right ) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}}}{a (1-n) (n+1) \left (c-a^2 c x^2\right )^{3/2}}+\frac {x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {(n+2) x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{\frac {n-1}{2}} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-n-1)}}{a (n+1) \left (c-a^2 c x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 129
Rule 131
Rule 155
Rule 6192
Rule 6194
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{n \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \, dx}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{x^2} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {n}{a}-\frac {2 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{-\frac {3}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{\left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {(2+n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (a \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}} \left (\frac {n (1+n)}{a^2}+\frac {(2+n) x}{a^3}\right )}{x} \, dx,x,\frac {1}{x}\right )}{(1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {(2+n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {n \left (1-n^2\right ) \left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{a^3 x} \, dx,x,\frac {1}{x}\right )}{(1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {(2+n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}-\frac {\left (n \left (1-n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{\frac {1}{2}-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+\frac {n}{2}}}{x} \, dx,x,\frac {1}{x}\right )}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}\\ &=-\frac {(2+n) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (2+2 n+n^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^3}{a (1-n) (1+n) \left (c-a^2 c x^2\right )^{3/2}}+\frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {1}{2} (-1-n)} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-1+n)} x^4}{\left (c-a^2 c x^2\right )^{3/2}}+\frac {2 n \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (1-\frac {1}{a x}\right )^{\frac {3-n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-3+n)} x^3 \, _2F_1\left (1,\frac {3-n}{2};\frac {5-n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (3-n) \left (c-a^2 c x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.61, size = 133, normalized size = 0.37 \[ \frac {\frac {c (a n x-1) e^{n \coth ^{-1}(a x)}}{n^2-1}-\frac {c \left (a^2 x^2-1\right ) \left (\frac {2 n e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};e^{2 \coth ^{-1}(a x)}\right )}{a x \sqrt {1-\frac {1}{a^2 x^2}}}+(n+1) e^{n \coth ^{-1}(a x)}\right )}{n+1}}{a^4 c^2 \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} x^{3}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________