Optimal. Leaf size=48 \[ \frac {2 (-x)^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, 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 (-x)^m \sqrt {a+b x} \left (-\frac {b x}{a}\right )^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {b x}{a}+1\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 69
Rubi steps
\begin {align*} \int \frac {(-x)^m}{\sqrt {a+b x}} \, dx &=\left ((-x)^m \left (-\frac {b x}{a}\right )^{-m}\right ) \int \frac {\left (-\frac {b x}{a}\right )^m}{\sqrt {a+b x}} \, dx\\ &=\frac {2 (-x)^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 (-x)^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (-x \right )^{m}}{\sqrt {b x +a}}\, dx\]
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 = 0.70, size = 42, normalized size = 0.88 \begin {gather*} \frac {x x^{m} e^{i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (m + 2\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 {{\left (-x\right )}^m}{\sqrt {a+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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