3.957 \(\int (b x)^{5/2} (\pi +d x)^n (e+f x)^p \, dx\)

Optimal. Leaf size=63 \[ \frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]

[Out]

(2*Pi^n*(b*x)^(7/2)*(e + f*x)^p*AppellF1[7/2, -n, -p, 9/2, -((d*x)/Pi), -((f*x)/
e)])/(7*b*(1 + (f*x)/e)^p)

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Rubi [A]  time = 0.0936117, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 \pi ^n (b x)^{7/2} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(b*x)^(5/2)*(Pi + d*x)^n*(e + f*x)^p,x]

[Out]

(2*Pi^n*(b*x)^(7/2)*(e + f*x)^p*AppellF1[7/2, -n, -p, 9/2, -((d*x)/Pi), -((f*x)/
e)])/(7*b*(1 + (f*x)/e)^p)

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Rubi in Sympy [A]  time = 11.4303, size = 49, normalized size = 0.78 \[ \frac{2 \pi ^{n} \left (b x\right )^{\frac{7}{2}} \left (1 + \frac{f x}{e}\right )^{- p} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (\frac{7}{2},- n,- p,\frac{9}{2},- \frac{d x}{\pi },- \frac{f x}{e} \right )}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x)**(5/2)*(d*x+pi)**n*(f*x+e)**p,x)

[Out]

2*pi**n*(b*x)**(7/2)*(1 + f*x/e)**(-p)*(e + f*x)**p*appellf1(7/2, -n, -p, 9/2, -
d*x/pi, -f*x/e)/(7*b)

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Mathematica [B]  time = 0.377245, size = 159, normalized size = 2.52 \[ \frac{18 \pi e x (b x)^{5/2} (d x+\pi )^n (e+f x)^p F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 \left (9 \pi e F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )+2 x \left (d e n F_1\left (\frac{9}{2};1-n,-p;\frac{11}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )+\pi f p F_1\left (\frac{9}{2};-n,1-p;\frac{11}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(b*x)^(5/2)*(Pi + d*x)^n*(e + f*x)^p,x]

[Out]

(18*e*Pi*x*(b*x)^(5/2)*(Pi + d*x)^n*(e + f*x)^p*AppellF1[7/2, -n, -p, 9/2, -((d*
x)/Pi), -((f*x)/e)])/(7*(9*e*Pi*AppellF1[7/2, -n, -p, 9/2, -((d*x)/Pi), -((f*x)/
e)] + 2*x*(d*e*n*AppellF1[9/2, 1 - n, -p, 11/2, -((d*x)/Pi), -((f*x)/e)] + f*p*P
i*AppellF1[9/2, -n, 1 - p, 11/2, -((d*x)/Pi), -((f*x)/e)])))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{{\frac{5}{2}}} \left ( dx+\pi \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x)^(5/2)*(d*x+Pi)^n*(f*x+e)^p,x)

[Out]

int((b*x)^(5/2)*(d*x+Pi)^n*(f*x+e)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p,x, algorithm="maxima")

[Out]

integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p} b^{2} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p,x, algorithm="fricas")

[Out]

integral(sqrt(b*x)*(pi + d*x)^n*(f*x + e)^p*b^2*x^2, 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)**(5/2)*(d*x+pi)**n*(f*x+e)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p,x, algorithm="giac")

[Out]

integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + e)^p, x)