Optimal. Leaf size=51 \[ \frac {2 a^2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-2-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, 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 = {69, 67}
\begin {gather*} \frac {2 a^2 x^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m-2;\frac {3}{2};\frac {b x}{a}+1\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 69
Rubi steps
\begin {align*} \int \frac {x^{2+m}}{\sqrt {a+b x}} \, dx &=\frac {\left (a^2 x^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^{2+m}}{\sqrt {a+b x}} \, dx}{b^2}\\ &=\frac {2 a^2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-2-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 51, normalized size = 1.00 \begin {gather*} \frac {2 a^2 x^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-2-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{2+m}}{\sqrt {b x +a}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.61, size = 37, normalized size = 0.73 \begin {gather*} \frac {x^{3} x^{m} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 4\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{m+2}}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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