3.2554 \(\int (d+e x)^2 \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=248 \[ -\frac{2^p \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{e (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)} \]

[Out]

(e*(2*c*d - b*e)*(2 + p)*(a + b*x + c*x^2)^(1 + p))/(2*c^2*(1 + p)*(3 + 2*p)) +
(e*(d + e*x)*(a + b*x + c*x^2)^(1 + p))/(c*(3 + 2*p)) - (2^p*(b^2*e^2*(2 + p) +
2*c^2*d^2*(3 + 2*p) - 2*c*e*(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^2*Sq
rt[b^2 - 4*a*c]*(1 + p)*(3 + 2*p))

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Rubi [A]  time = 0.49422, antiderivative size = 248, 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 \[ -\frac{2^p \left (a+b x+c x^2\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (-2 c e (a e+b d (2 p+3))+b^2 e^2 (p+2)+2 c^2 d^2 (2 p+3)\right ) \, _2F_1\left (-p,p+1;p+2;\frac{b+2 c x+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt{b^2-4 a c}}+\frac{e (p+2) (2 c d-b e) \left (a+b x+c x^2\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)} \]

Antiderivative was successfully verified.

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

[Out]

(e*(2*c*d - b*e)*(2 + p)*(a + b*x + c*x^2)^(1 + p))/(2*c^2*(1 + p)*(3 + 2*p)) +
(e*(d + e*x)*(a + b*x + c*x^2)^(1 + p))/(c*(3 + 2*p)) - (2^p*(b^2*e^2*(2 + p) +
2*c^2*d^2*(3 + 2*p) - 2*c*e*(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^2*Sq
rt[b^2 - 4*a*c]*(1 + p)*(3 + 2*p))

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Rubi in Sympy [A]  time = 52.0288, size = 223, normalized size = 0.9 \[ \frac{e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{p + 1}}{c \left (2 p + 3\right )} - \frac{e \left (p + 2\right ) \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{p + 1}}{2 c^{2} \left (p + 1\right ) \left (2 p + 3\right )} + \frac{\left (\frac{- \frac{b}{2} - c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (- b e \left (p + 2\right ) \left (b e - 2 c d\right ) + 2 c \left (- c d^{2} \left (2 p + 3\right ) + e \left (a e + b d \left (p + 1\right )\right )\right )\right ) \left (a + b x + c x^{2}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{2} \left (p + 1\right ) \left (2 p + 3\right ) \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)*(a + b*x + c*x**2)**(p + 1)/(c*(2*p + 3)) - e*(p + 2)*(b*e - 2*c*d)*
(a + b*x + c*x**2)**(p + 1)/(2*c**2*(p + 1)*(2*p + 3)) + ((-b/2 - c*x + sqrt(-4*
a*c + b**2)/2)/sqrt(-4*a*c + b**2))**(-p - 1)*(-b*e*(p + 2)*(b*e - 2*c*d) + 2*c*
(-c*d**2*(2*p + 3) + e*(a*e + b*d*(p + 1))))*(a + b*x + c*x**2)**(p + 1)*hyper((
-p, p + 1), (p + 2,), (b/2 + c*x + sqrt(-4*a*c + b**2)/2)/sqrt(-4*a*c + b**2))/(
2*c**2*(p + 1)*(2*p + 3)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 6.19653, size = 1001, normalized size = 4.04 \[ \frac{3\ 2^{-p-1} c \left (b+\sqrt{b^2-4 a c}\right ) d e x^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{b+2 c x-\sqrt{b^2-4 a c}}{c}\right )^{p+1} \left (2 a+\left (b-\sqrt{b^2-4 a c}\right ) x\right )^2 F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right ) (a+x (b+c x))^{p-1}}{\left (\sqrt{b^2-4 a c}-b\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right ) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (3;1-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (3;-p,1-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-6 a F_1\left (2;-p,-p;3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{2 \left (b+\sqrt{b^2-4 a c}\right ) e^2 x^3 \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (2 a+\left (b-\sqrt{b^2-4 a c}\right ) x\right )^2 F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right ) (a+x (b+c x))^{p-1}}{3 \left (\sqrt{b^2-4 a c}-b\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right ) \left (p x \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (4;1-p,-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (4;-p,1-p;5;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )-8 a F_1\left (3;-p,-p;4;-\frac{2 c x}{b+\sqrt{b^2-4 a c}},\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )\right )}+\frac{d^2 \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (c x^2+b x+a\right )^p \left (\frac{x-\frac{\sqrt{b^2-4 a c}-b}{2 c}}{\frac{\sqrt{b^2-4 a c}-b}{2 c}-\frac{-b-\sqrt{b^2-4 a c}}{2 c}}+1\right )^{-p} \, _2F_1\left (-p,p+1;p+2;-\frac{x-\frac{\sqrt{b^2-4 a c}-b}{2 c}}{\frac{\sqrt{b^2-4 a c}-b}{2 c}-\frac{-b-\sqrt{b^2-4 a c}}{2 c}}\right )}{2 c (p+1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*2^(-1 - p)*c*(b + Sqrt[b^2 - 4*a*c])*d*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)*Appel
lF1[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 + Sqrt
[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])*AppellF1
[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])*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])])/(3*(-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])]))) + (d^2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(a
+ b*x + c*x^2)^p*Hypergeometric2F1[-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)

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Maple [F]  time = 0.152, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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