3.450 \(\int (d+e x)^m \left (b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=103 \[ \frac{\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac{e x}{d}\right )^{-p} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.147408, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac{e x}{d}\right )^{-p} \left (1-\frac{c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{d+e x}{d},\frac{c (d+e x)}{c d-b e}\right )}{e (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m*(b*x + c*x^2)^p,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.5791, size = 76, normalized size = 0.74 \[ \frac{\left (- \frac{e x}{d}\right )^{- p} \left (d + e x\right )^{m + 1} \left (b x + c x^{2}\right )^{p} \left (\frac{c \left (d + e x\right )}{b e - c d} + 1\right )^{- p} \operatorname{appellf_{1}}{\left (m + 1,- p,- p,m + 2,\frac{d + e x}{d},\frac{c \left (- d - e x\right )}{b e - c d} \right )}}{e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+b*x)**p,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.538833, size = 160, normalized size = 1.55 \[ \frac{b d (p+2) x (x (b+c x))^p (d+e x)^m F_1\left (p+1;-p,-m;p+2;-\frac{c x}{b},-\frac{e x}{d}\right )}{(p+1) \left (b d (p+2) F_1\left (p+1;-p,-m;p+2;-\frac{c x}{b},-\frac{e x}{d}\right )+x \left (c d p F_1\left (p+2;1-p,-m;p+3;-\frac{c x}{b},-\frac{e x}{d}\right )+b e m F_1\left (p+2;-p,1-m;p+3;-\frac{c x}{b},-\frac{e x}{d}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^m*(b*x + c*x^2)^p,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.128, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m*(c*x^2+b*x)^p,x)

[Out]

int((e*x+d)^m*(c*x^2+b*x)^p,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p*(e*x + d)^m, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x)^p*(e*x + d)^m, 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((e*x+d)**m*(c*x**2+b*x)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^p*(e*x + d)^m, x)