Optimal. Leaf size=51 \[ -\frac {\left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac {5}{2}-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 a x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac {a \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {\frac {b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^{3/2}}{x^3} \, dx &=\frac {\left (a \sqrt {a+b x^n}\right ) \int \frac {\left (1+\frac {b x^n}{a}\right )^{3/2}}{x^3} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=-\frac {a \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {1+\frac {b x^n}{a}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 58, normalized size = 1.14 \[ -\frac {a \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},-\frac {2}{n};1-\frac {2}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {\frac {b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{n}+a \right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x^n\right )}^{3/2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.99, size = 46, normalized size = 0.90 \[ \frac {a^{\frac {3}{2}} \Gamma \left (- \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {2}{n} \\ 1 - \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x^{2} \Gamma \left (1 - \frac {2}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________