Optimal. Leaf size=77 \[ \frac {2 \sqrt {b x} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {135, 133} \[ \frac {2 \sqrt {b x} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 135
Rubi steps
\begin {align*} \int \frac {(c+d x)^n (e+f x)^p}{\sqrt {b x}} \, dx &=\left ((c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int \frac {\left (1+\frac {d x}{c}\right )^n (e+f x)^p}{\sqrt {b x}} \, dx\\ &=\left ((c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p}\right ) \int \frac {\left (1+\frac {d x}{c}\right )^n \left (1+\frac {f x}{e}\right )^p}{\sqrt {b x}} \, dx\\ &=\frac {2 \sqrt {b x} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 77, normalized size = 1.00 \[ \frac {2 x (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d x}{c},-\frac {f x}{e}\right )}{\sqrt {b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{b x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{\sqrt {b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{n} \left (f x +e \right )^{p}}{\sqrt {b x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{\sqrt {b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e+f\,x\right )}^p\,{\left (c+d\,x\right )}^n}{\sqrt {b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________