Optimal. Leaf size=100 \[ \frac {2 x (c x)^m \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {m+\frac {n}{2}+1}{j-n};\frac {2 m+n+2}{2 j-2 n}+1;-\frac {a x^{j-n}}{b}\right )}{(2 m+n+2) \sqrt {\frac {a x^{j-n}}{b}+1}} \]
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Rubi [A] time = 0.09, antiderivative size = 100, 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} \[ \frac {2 x (c x)^m \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {m+\frac {n}{2}+1}{j-n};\frac {2 m+n+2}{2 j-2 n}+1;-\frac {a x^{j-n}}{b}\right )}{(2 m+n+2) \sqrt {\frac {a x^{j-n}}{b}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2032
Rubi steps
\begin {align*} \int (c x)^m \sqrt {a x^j+b x^n} \, dx &=\frac {\left (x^{-m-\frac {n}{2}} (c x)^m \sqrt {a x^j+b x^n}\right ) \int x^{m+\frac {n}{2}} \sqrt {b+a x^{j-n}} \, dx}{\sqrt {b+a x^{j-n}}}\\ &=\frac {\left (x^{-m-\frac {n}{2}} (c x)^m \sqrt {a x^j+b x^n}\right ) \int x^{m+\frac {n}{2}} \sqrt {1+\frac {a x^{j-n}}{b}} \, dx}{\sqrt {1+\frac {a x^{j-n}}{b}}}\\ &=\frac {2 x (c x)^m \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {1+m+\frac {n}{2}}{j-n};1+\frac {2+2 m+n}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )}{(2+2 m+n) \sqrt {1+\frac {a x^{j-n}}{b}}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 156, normalized size = 1.56 \[ \frac {2 x (c x)^m \left ((2 j+2 m-n+2) \left (a x^j+b x^n\right )-a (j-n) x^j \sqrt {\frac {a x^{j-n}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {2 j+2 m-n+2}{2 j-2 n};\frac {4 j+2 m-3 n+2}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )\right )}{(2 m+n+2) (2 j+2 m-n+2) \sqrt {a x^j+b x^n}} \]
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x^{j} + b x^{n}} \left (c x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.84, size = 0, normalized size = 0.00 \[ \int \sqrt {a \,x^{j}+b \,x^{n}}\, \left (c x \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a x^{j} + b x^{n}} \left (c x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,x\right )}^m\,\sqrt {a\,x^j+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x\right )^{m} \sqrt {a x^{j} + b x^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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