3.955 \(\int (b x)^m (\pi +d x)^n (e+f x)^p \, dx\)

Optimal. Leaf size=65 \[ \frac{\pi ^n (b x)^{m+1} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d x}{\pi },-\frac{f x}{e}\right )}{b (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.101886, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\pi ^n (b x)^{m+1} (e+f x)^p \left (\frac{f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac{d x}{\pi },-\frac{f x}{e}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.4436, size = 49, normalized size = 0.75 \[ \frac{\pi ^{n} \left (b x\right )^{m + 1} \left (1 + \frac{f x}{e}\right )^{- p} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- n,- p,m + 2,- \frac{d x}{\pi },- \frac{f x}{e} \right )}}{b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.514002, size = 163, normalized size = 2.51 \[ \frac{\pi e (m+2) x (b x)^m (d x+\pi )^n (e+f x)^p F_1\left (m+1;-n,-p;m+2;-\frac{d x}{\pi },-\frac{f x}{e}\right )}{(m+1) \left (\pi e (m+2) F_1\left (m+1;-n,-p;m+2;-\frac{d x}{\pi },-\frac{f x}{e}\right )+x \left (d e n F_1\left (m+2;1-n,-p;m+3;-\frac{d x}{\pi },-\frac{f x}{e}\right )+\pi f p F_1\left (m+2;-n,1-p;m+3;-\frac{d x}{\pi },-\frac{f x}{e}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.178, size = 0, normalized size = 0. \[ \int \left ( bx \right ) ^{m} \left ( dx+\pi \right ) ^{n} \left ( fx+e \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x)^m*(d*x+Pi)^n*(f*x+e)^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((pi + d*x)^n*(b*x)^m*(f*x + e)^p, 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)**m*(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. \[ \int{\left (\pi + d x\right )}^{n} \left (b x\right )^{m}{\left (f x + e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((pi + d*x)^n*(b*x)^m*(f*x + e)^p, x)