Optimal. Leaf size=49 \[ -\frac {2 a x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m-1;\frac {3}{2};\frac {b x}{a}+1\right )}{b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {67, 65} \[ -\frac {2 a x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m-1;\frac {3}{2};\frac {b x}{a}+1\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rubi steps
\begin {align*} \int \frac {x^{1+m}}{\sqrt {a+b x}} \, dx &=-\frac {\left (a x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^{1+m}}{\sqrt {a+b x}} \, dx}{b}\\ &=-\frac {2 a x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-1-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 1.00 \[ -\frac {2 a x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m-1;\frac {3}{2};\frac {b x}{a}+1\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m + 1}}{\sqrt {b x + a}}, x\right ) \]
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 \[ \int \frac {x^{m + 1}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{m +1}}{\sqrt {b x +a}}\, dx \]
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 \[ \int \frac {x^{m + 1}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{m+1}}{\sqrt {a+b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.47, size = 37, normalized size = 0.76 \[ \frac {x^{2} x^{m} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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