Optimal. Leaf size=250 \[ \frac {2 \sqrt {a+b x} (A b-a B) (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2}+\frac {2 B (a+b x)^{3/2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac {3}{2};-n,-p;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {159, 140, 139, 138} \[ \frac {2 \sqrt {a+b x} (A b-a B) (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2}+\frac {2 B (a+b x)^{3/2} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac {3}{2};-n,-p;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 138
Rule 139
Rule 140
Rule 159
Rubi steps
\begin {align*} \int \frac {(A+B x) (c+d x)^n (e+f x)^p}{\sqrt {a+b x}} \, dx &=\frac {B \int \sqrt {a+b x} (c+d x)^n (e+f x)^p \, dx}{b}+\frac {(A b-a B) \int \frac {(c+d x)^n (e+f x)^p}{\sqrt {a+b x}} \, dx}{b}\\ &=\frac {\left (B (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int \sqrt {a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p \, dx}{b}+\frac {\left ((A b-a B) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n (e+f x)^p}{\sqrt {a+b x}} \, dx}{b}\\ &=\frac {\left (B (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int \sqrt {a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p \, dx}{b}+\frac {\left ((A b-a B) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p}\right ) \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^p}{\sqrt {a+b x}} \, dx}{b}\\ &=\frac {2 (A b-a B) \sqrt {a+b x} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2}+\frac {2 B (a+b x)^{3/2} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} (e+f x)^p \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} F_1\left (\frac {3}{2};-n,-p;\frac {5}{2};-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{3 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 184, normalized size = 0.74 \[ \frac {2 \sqrt {a+b x} (c+d x)^n (e+f x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \left (3 (A b-a B) F_1\left (\frac {1}{2};-n,-p;\frac {3}{2};\frac {d (a+b x)}{a d-b c},\frac {f (a+b x)}{a f-b e}\right )+B (a+b x) F_1\left (\frac {3}{2};-n,-p;\frac {5}{2};\frac {d (a+b x)}{a d-b c},\frac {f (a+b x)}{a f-b e}\right )\right )}{3 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{\sqrt {b x + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (d x +c \right )^{n} \left (f x +e \right )^{p}}{\sqrt {b x +a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p}}{\sqrt {b x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^p\,\left (A+B\,x\right )\,{\left (c+d\,x\right )}^n}{\sqrt {a+b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________