Optimal. Leaf size=102 \[ \frac {2 x (c x)^m \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {m-\frac {n}{2}+1}{j-n};\frac {m-\frac {n}{2}+1}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{(2 m-n+2) \sqrt {a x^j+b x^n}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2032, 365, 364} \[ \frac {2 x (c x)^m \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {m-\frac {n}{2}+1}{j-n};\frac {m-\frac {n}{2}+1}{j-n}+1;-\frac {a x^{j-n}}{b}\right )}{(2 m-n+2) \sqrt {a x^j+b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 365
Rule 2032
Rubi steps
\begin {align*} \int \frac {(c x)^m}{\sqrt {a x^j+b x^n}} \, dx &=\frac {\left (x^{-m+\frac {n}{2}} (c x)^m \sqrt {b+a x^{j-n}}\right ) \int \frac {x^{m-\frac {n}{2}}}{\sqrt {b+a x^{j-n}}} \, dx}{\sqrt {a x^j+b x^n}}\\ &=\frac {\left (x^{-m+\frac {n}{2}} (c x)^m \sqrt {1+\frac {a x^{j-n}}{b}}\right ) \int \frac {x^{m-\frac {n}{2}}}{\sqrt {1+\frac {a x^{j-n}}{b}}} \, dx}{\sqrt {a x^j+b x^n}}\\ &=\frac {2 x (c x)^m \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {1}{2},\frac {1+m-\frac {n}{2}}{j-n};1+\frac {1+m-\frac {n}{2}}{j-n};-\frac {a x^{j-n}}{b}\right )}{(2+2 m-n) \sqrt {a x^j+b x^n}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 106, normalized size = 1.04 \[ \frac {2 x (c x)^m \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {2 m-n+2}{2 j-2 n};\frac {2 m-n+2}{2 j-2 n}+1;-\frac {a x^{j-n}}{b}\right )}{(2 m-n+2) \sqrt {a x^j+b x^n}} \]
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 \[ \int \frac {\left (c x\right )^{m}}{\sqrt {a x^{j} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{m}}{\sqrt {a \,x^{j}+b \,x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{\sqrt {a x^{j} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x\right )}^m}{\sqrt {a\,x^j+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x\right )^{m}}{\sqrt {a x^{j} + b x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________