3.2553 \(\int (d+e x)^3 \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=327 \[ -\frac{2^{p-1} (2 c d-b e) \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{e \left (a+b x+c x^2\right )^{p+1} \left (-2 c (2 p+3) \left (c d^2 (2 p+5)-e (a e+b d (p+2))\right )-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right )}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \]

[Out]

(e*(d + e*x)^2*(a + b*x + c*x^2)^(1 + p))/(2*c*(2 + p)) - (e*(b*e*(2*c*d - b*e)*
(2 + p)*(3 + p) - 2*c*(3 + 2*p)*(c*d^2*(5 + 2*p) - e*(a*e + b*d*(2 + p))) - 2*c*
e*(2*c*d - b*e)*(1 + p)*(3 + p)*x)*(a + b*x + c*x^2)^(1 + p))/(4*c^3*(1 + p)*(2
+ p)*(3 + 2*p)) - (2^(-1 + p)*(2*c*d - b*e)*(b^2*e^2*(3 + p) + 2*c^2*d^2*(3 + 2*
p) - 2*c*e*(3*a*e + b*d*(3 + 2*p)))*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2
- 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p
, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c^3*Sqrt[b^2 - 4*a*c]
*(1 + p)*(3 + 2*p))

_______________________________________________________________________________________

Rubi [A]  time = 1.01697, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{2^{p-1} (2 c d-b e) \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-2 c e (3 a e+b d (2 p+3))+b^2 e^2 (p+3)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^3 (p+1) (2 p+3) \sqrt{b^2-4 a c}}-\frac{e \left (a+b x+c x^2\right )^{p+1} \left (-2 c (2 p+3) \left (c d^2 (2 p+5)-e (a e+b d (p+2))\right )-2 c e (p+1) (p+3) x (2 c d-b e)+b e (p+2) (p+3) (2 c d-b e)\right )}{4 c^3 (p+1) (p+2) (2 p+3)}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{p+1}}{2 c (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

(e*(d + e*x)^2*(a + b*x + c*x^2)^(1 + p))/(2*c*(2 + p)) - (e*(b*e*(2*c*d - b*e)*
(2 + p)*(3 + p) - 2*c*(3 + 2*p)*(c*d^2*(5 + 2*p) - e*(a*e + b*d*(2 + p))) - 2*c*
e*(2*c*d - b*e)*(1 + p)*(3 + p)*x)*(a + b*x + c*x^2)^(1 + p))/(4*c^3*(1 + p)*(2
+ p)*(3 + 2*p)) - (2^(-1 + p)*(2*c*d - b*e)*(b^2*e^2*(3 + p) + 2*c^2*d^2*(3 + 2*
p) - 2*c*e*(3*a*e + b*d*(3 + 2*p)))*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2
- 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p
, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(c^3*Sqrt[b^2 - 4*a*c]
*(1 + p)*(3 + 2*p))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 14.6624, size = 1415, normalized size = 4.33 \[ \frac{9\ 2^{-p-2} c \left (b+\sqrt{b^2-4 a c}\right ) d^2 e x^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{b+2 c x-\sqrt{b^2-4 a c}}{c}\right )^{p+1} \left (2 a+\left (b-\sqrt{b^2-4 a c}\right ) x\right )^2 F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right ) (a+x (b+c x))^{p-1}}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right ) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (3;1-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (3;-p,1-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-6 a F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{2 \left (b+\sqrt{b^2-4 a c}\right ) d e^2 x^3 \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (2 a+\left (b-\sqrt{b^2-4 a c}\right ) x\right )^2 F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right ) (a+x (b+c x))^{p-1}}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right ) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (4;1-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (4;-p,1-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-8 a F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{5\ 2^{-p-3} c \left (b+\sqrt{b^2-4 a c}\right ) e^3 x^4 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{b+2 c x-\sqrt{b^2-4 a c}}{c}\right )^{p+1} \left (2 a+\left (b-\sqrt{b^2-4 a c}\right ) x\right )^2 F_1\left (4;-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right ) (a+x (b+c x))^{p-1}}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right ) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (5;1-p,-p;6;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (5;-p,1-p;6;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-10 a F_1\left (4;-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{d^3 \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (c x^2+b x+a\right )^p \left (\frac{x-\frac{\sqrt{b^2-4 a c}-b}{2 c}}{\frac{\sqrt{b^2-4 a c}-b}{2 c}-\frac{-b-\sqrt{b^2-4 a c}}{2 c}}+1\right )^{-p} \, _2F_1\left (-p,p+1;p+2;-\frac{x-\frac{\sqrt{b^2-4 a c}-b}{2 c}}{\frac{\sqrt{b^2-4 a c}-b}{2 c}-\frac{-b-\sqrt{b^2-4 a c}}{2 c}}\right )}{2 c (p+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(9*2^(-2 - p)*c*(b + Sqrt[b^2 - 4*a*c])*d^2*e*x^2*((b - Sqrt[b^2 - 4*a*c] + 2*c*
x)/c)^(1 + p)*(2*a + (b - Sqrt[b^2 - 4*a*c])*x)^2*(a + x*(b + c*x))^(-1 + p)*App
ellF1[2, -p, -p, 3, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4
*a*c])])/((-b + Sqrt[b^2 - 4*a*c])*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x)^p*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x)*(-6*a*AppellF1[2, -p, -p, 3, (-2*c*x)/(b + Sqrt[b^2 - 4
*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + p*x*((-b + Sqrt[b^2 - 4*a*c])*Appell
F1[3, 1 - p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4
*a*c])] - (b + Sqrt[b^2 - 4*a*c])*AppellF1[3, -p, 1 - p, 4, (-2*c*x)/(b + Sqrt[b
^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])]))) + (2*(b + Sqrt[b^2 - 4*a*c])*
d*e^2*x^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(2*a + (b - Sqrt[b^2 - 4*a*c])*x)^2*(a
+ x*(b + c*x))^(-1 + p)*AppellF1[3, -p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])
, (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4
*a*c] + 2*c*x)*(-8*a*AppellF1[3, -p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2
*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + p*x*((-b + Sqrt[b^2 - 4*a*c])*AppellF1[4, 1 -
p, -p, 5, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] -
(b + Sqrt[b^2 - 4*a*c])*AppellF1[4, -p, 1 - p, 5, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c
]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])]))) + (5*2^(-3 - p)*c*(b + Sqrt[b^2 - 4*a*c
])*e^3*x^4*((b - Sqrt[b^2 - 4*a*c] + 2*c*x)/c)^(1 + p)*(2*a + (b - Sqrt[b^2 - 4*
a*c])*x)^2*(a + x*(b + c*x))^(-1 + p)*AppellF1[4, -p, -p, 5, (-2*c*x)/(b + Sqrt[
b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 - 4*a*c])*((b
- Sqrt[b^2 - 4*a*c])/(2*c) + x)^p*(b + Sqrt[b^2 - 4*a*c] + 2*c*x)*(-10*a*AppellF
1[4, -p, -p, 5, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c
])] + p*x*((-b + Sqrt[b^2 - 4*a*c])*AppellF1[5, 1 - p, -p, 6, (-2*c*x)/(b + Sqrt
[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] - (b + Sqrt[b^2 - 4*a*c])*Appe
llF1[5, -p, 1 - p, 6, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 -
4*a*c])]))) + (d^3*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + b*x + c*x^2)^p*Hypergeo
metric2F1[-p, 1 + p, 2 + p, -((-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x)/(-(-b - Sqrt
[b^2 - 4*a*c])/(2*c) + (-b + Sqrt[b^2 - 4*a*c])/(2*c)))])/(2*c*(1 + p)*(1 + (-(-
b + Sqrt[b^2 - 4*a*c])/(2*c) + x)/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + (-b + Sqrt[
b^2 - 4*a*c])/(2*c)))^p)

_______________________________________________________________________________________

Maple [F]  time = 0.164, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+d)^3*(c*x^2+b*x+a)^p,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^3*(c*x^2 + b*x + a)^p, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (c x^{2} + b x + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(c*x^2 + b*x + a)^p, 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)**3*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^3*(c*x^2 + b*x + a)^p, x)