Optimal. Leaf size=48 \[ \frac {2 (-x)^m \left (-\frac {b x}{a}\right )^{-m} \sqrt {a+b x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {b x}{a}\right )}{b} \]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
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*}
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 2.44, size = 40, normalized size = 0.83 \begin {gather*} \frac {E^{I \text {Pi} m} x^{1+m} \text {hyper}\left [\left \{\frac {1}{2},1+m\right \},\left \{2+m\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{\sqrt {a} \left (1+m\right )} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.73, 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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] N/A
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________