Integrand size = 15, antiderivative size = 101 \[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\frac {2 x^{1-2 n} \sqrt {1+\frac {a x^{j-n}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1-\frac {5 n}{2}}{j-n},1+\frac {1-\frac {5 n}{2}}{j-n},-\frac {a x^{j-n}}{b}\right )}{b^2 (2-5 n) \sqrt {a x^j+b x^n}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2036, 372, 371} \[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\frac {2 x^{1-2 n} \sqrt {\frac {a x^{j-n}}{b}+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1-\frac {5 n}{2}}{j-n},\frac {1-\frac {5 n}{2}}{j-n}+1,-\frac {a x^{j-n}}{b}\right )}{b^2 (2-5 n) \sqrt {a x^j+b x^n}} \]
[In]
[Out]
Rule 371
Rule 372
Rule 2036
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{n/2} \sqrt {b+a x^{j-n}}\right ) \int \frac {x^{-5 n/2}}{\left (b+a x^{j-n}\right )^{5/2}} \, dx}{\sqrt {a x^j+b x^n}} \\ & = \frac {\left (x^{n/2} \sqrt {1+\frac {a x^{j-n}}{b}}\right ) \int \frac {x^{-5 n/2}}{\left (1+\frac {a x^{j-n}}{b}\right )^{5/2}} \, dx}{b^2 \sqrt {a x^j+b x^n}} \\ & = \frac {2 x^{1-2 n} \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {5}{2},\frac {1-\frac {5 n}{2}}{j-n};1+\frac {1-\frac {5 n}{2}}{j-n};-\frac {a x^{j-n}}{b}\right )}{b^2 (2-5 n) \sqrt {a x^j+b x^n}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\frac {2 x^{1-2 j} \left (-\left ((-2+4 j+n) \left (a (-2+j+4 n) x^j+b (-2+2 j+3 n) x^n\right )\right )+\left (4+8 j^2-8 n+3 n^2+2 j (-6+7 n)\right ) \sqrt {1+\frac {a x^{j-n}}{b}} \left (a x^j+b x^n\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {-2+4 j+n}{2 (j-n)},\frac {2-2 j-3 n}{2 j-2 n},-\frac {a x^{j-n}}{b}\right )\right )}{3 a^2 (2-4 j-n) (j-n)^2 \left (a x^j+b x^n\right )^{3/2}} \]
[In]
[Out]
\[\int \frac {1}{\left (a \,x^{j}+b \,x^{n}\right )^{\frac {5}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\int \frac {1}{\left (a x^{j} + b x^{n}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\int { \frac {1}{{\left (a x^{j} + b x^{n}\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=\int { \frac {1}{{\left (a x^{j} + b x^{n}\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Time = 9.51 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a x^j+b x^n\right )^{5/2}} \, dx=-\frac {x\,{\left (\frac {b\,x^{n-j}}{a}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {\frac {5\,j}{2}-1}{j-n};\ \frac {\frac {5\,j}{2}-1}{j-n}+1;\ -\frac {b\,x^{n-j}}{a}\right )}{\left (\frac {5\,j}{2}-1\right )\,{\left (a\,x^j+b\,x^n\right )}^{5/2}} \]
[In]
[Out]