3.951 \(\int (b x)^{3/2} (c+d x)^n (e+f x)^p \, dx\)

Optimal. Leaf size=79 \[ \frac{2 (b x)^{5/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{5}{2};-n,-p;\frac{7}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{5 b} \]

[Out]

(2*(b*x)^(5/2)*(c + d*x)^n*(e + f*x)^p*AppellF1[5/2, -n, -p, 7/2, -((d*x)/c), -(
(f*x)/e)])/(5*b*(1 + (d*x)/c)^n*(1 + (f*x)/e)^p)

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Rubi [A]  time = 0.133871, antiderivative size = 79, 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 \[ \frac{2 (b x)^{5/2} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{5}{2};-n,-p;\frac{7}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[(b*x)^(3/2)*(c + d*x)^n*(e + f*x)^p,x]

[Out]

(2*(b*x)^(5/2)*(c + d*x)^n*(e + f*x)^p*AppellF1[5/2, -n, -p, 7/2, -((d*x)/c), -(
(f*x)/e)])/(5*b*(1 + (d*x)/c)^n*(1 + (f*x)/e)^p)

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Rubi in Sympy [A]  time = 18.5378, size = 61, normalized size = 0.77 \[ \frac{2 \left (b x\right )^{\frac{5}{2}} \left (1 + \frac{d x}{c}\right )^{- n} \left (1 + \frac{f x}{e}\right )^{- p} \left (c + d x\right )^{n} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (\frac{5}{2},- n,- p,\frac{7}{2},- \frac{d x}{c},- \frac{f x}{e} \right )}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x)**(3/2)*(d*x+c)**n*(f*x+e)**p,x)

[Out]

2*(b*x)**(5/2)*(1 + d*x/c)**(-n)*(1 + f*x/e)**(-p)*(c + d*x)**n*(e + f*x)**p*app
ellf1(5/2, -n, -p, 7/2, -d*x/c, -f*x/e)/(5*b)

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Mathematica [B]  time = 0.383816, size = 159, normalized size = 2.01 \[ \frac{14 c e x (b x)^{3/2} (c+d x)^n (e+f x)^p F_1\left (\frac{5}{2};-n,-p;\frac{7}{2};-\frac{d x}{c},-\frac{f x}{e}\right )}{5 \left (7 c e F_1\left (\frac{5}{2};-n,-p;\frac{7}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+2 x \left (d e n F_1\left (\frac{7}{2};1-n,-p;\frac{9}{2};-\frac{d x}{c},-\frac{f x}{e}\right )+c f p F_1\left (\frac{7}{2};-n,1-p;\frac{9}{2};-\frac{d x}{c},-\frac{f x}{e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(b*x)^(3/2)*(c + d*x)^n*(e + f*x)^p,x]

[Out]

(14*c*e*x*(b*x)^(3/2)*(c + d*x)^n*(e + f*x)^p*AppellF1[5/2, -n, -p, 7/2, -((d*x)
/c), -((f*x)/e)])/(5*(7*c*e*AppellF1[5/2, -n, -p, 7/2, -((d*x)/c), -((f*x)/e)] +
2*x*(d*e*n*AppellF1[7/2, 1 - n, -p, 9/2, -((d*x)/c), -((f*x)/e)] + c*f*p*Appell
F1[7/2, -n, 1 - p, 9/2, -((d*x)/c), -((f*x)/e)])))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{{\frac{3}{2}}} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x)^(3/2)*(d*x+c)^n*(f*x+e)^p,x)

[Out]

int((b*x)^(3/2)*(d*x+c)^n*(f*x+e)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{\frac{3}{2}}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(3/2)*(d*x + c)^n*(f*x + e)^p,x, algorithm="maxima")

[Out]

integrate((b*x)^(3/2)*(d*x + c)^n*(f*x + e)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(3/2)*(d*x + c)^n*(f*x + e)^p,x, algorithm="fricas")

[Out]

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

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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((b*x)**(3/2)*(d*x+c)**n*(f*x+e)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{\frac{3}{2}}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(3/2)*(d*x + c)^n*(f*x + e)^p,x, algorithm="giac")

[Out]

integrate((b*x)^(3/2)*(d*x + c)^n*(f*x + e)^p, x)