Optimal. Leaf size=65 \[ -\frac {\left (a+b x^n\right ) \, _2F_1\left (3,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^3 x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1369, 371}
\begin {gather*} -\frac {\left (a+b x^n\right ) \, _2F_1\left (3,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^3 x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 1369
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=-\frac {\left (a+b x^n\right ) \, _2F_1\left (3,-\frac {1}{n};-\frac {1-n}{n};-\frac {b x^n}{a}\right )}{a^3 x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 53, normalized size = 0.82 \begin {gather*} -\frac {\left (a+b x^n\right )^3 \, _2F_1\left (3,-\frac {1}{n};1-\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 x \left (\left (a+b x^n\right )^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^2\,{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________