3.953 \(\int \frac{(c+d x)^n (e+f x)^p}{\sqrt{b x}} \, dx\)

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]

(2*Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p*AppellF1[1/2, -n, -p, 3/2, -((d*x)/c), -((f
*x)/e)])/(b*(1 + (d*x)/c)^n*(1 + (f*x)/e)^p)

_______________________________________________________________________________________

Rubi [A]  time = 0.135999, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 \sqrt{b x} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[((c + d*x)^n*(e + f*x)^p)/Sqrt[b*x],x]

[Out]

(2*Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p*AppellF1[1/2, -n, -p, 3/2, -((d*x)/c), -((f
*x)/e)])/(b*(1 + (d*x)/c)^n*(1 + (f*x)/e)^p)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.3417, size = 60, normalized size = 0.78 \[ \frac{2 \sqrt{b x} \left (1 + \frac{d x}{c}\right )^{- n} \left (1 + \frac{f x}{e}\right )^{- p} \left (c + d x\right )^{n} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2},- n,- p,\frac{3}{2},- \frac{d x}{c},- \frac{f x}{e} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**n*(f*x+e)**p/(b*x)**(1/2),x)

[Out]

2*sqrt(b*x)*(1 + d*x/c)**(-n)*(1 + f*x/e)**(-p)*(c + d*x)**n*(e + f*x)**p*appell
f1(1/2, -n, -p, 3/2, -d*x/c, -f*x/e)/b

_______________________________________________________________________________________

Mathematica [B]  time = 0.365056, size = 157, normalized size = 2.04 \[ \frac{6 c e x (c+d x)^n (e+f x)^p F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{\sqrt{b x} \left (3 c e F_1\left (\frac{1}{2};-n,-p;\frac{3}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+2 d e n x F_1\left (\frac{3}{2};1-n,-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+2 c f p x F_1\left (\frac{3}{2};-n,1-p;\frac{5}{2};-\frac{d x}{c},-\frac{f x}{e}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((c + d*x)^n*(e + f*x)^p)/Sqrt[b*x],x]

[Out]

(6*c*e*x*(c + d*x)^n*(e + f*x)^p*AppellF1[1/2, -n, -p, 3/2, -((d*x)/c), -((f*x)/
e)])/(Sqrt[b*x]*(3*c*e*AppellF1[1/2, -n, -p, 3/2, -((d*x)/c), -((f*x)/e)] + 2*d*
e*n*x*AppellF1[3/2, 1 - n, -p, 5/2, -((d*x)/c), -((f*x)/e)] + 2*c*f*p*x*AppellF1
[3/2, -n, 1 - p, 5/2, -((d*x)/c), -((f*x)/e)]))

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{ \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p}{\frac{1}{\sqrt{bx}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^n*(f*x+e)^p/(b*x)^(1/2),x)

[Out]

int((d*x+c)^n*(f*x+e)^p/(b*x)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}}{\sqrt{b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x),x, algorithm="maxima")

[Out]

integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}}{\sqrt{b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x),x, algorithm="fricas")

[Out]

integral((d*x + c)^n*(f*x + e)^p/sqrt(b*x), x)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**n*(f*x+e)**p/(b*x)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}}{\sqrt{b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x),x, algorithm="giac")

[Out]

integrate((d*x + c)^n*(f*x + e)^p/sqrt(b*x), x)