3.2567 \(\int (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=332 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.357759, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (2 c d-b e) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )}-\frac{e (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 66.6399, size = 296, normalized size = 0.89 \[ - \frac{e \left (d + e x\right )^{- 2 p - 2} \left (a + b x + c x^{2}\right )^{p + 1}}{2 \left (p + 1\right ) \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (\frac{\left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}\right )^{- p} \left (d + e x\right )^{- 2 p - 1} \left (\frac{b e}{2} - c d\right ) \left (a + b x + c x^{2}\right )^{p} \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{4 c \left (d + e x\right ) \sqrt{- 4 a c + b^{2}}}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}} \right )}}{\left (2 p + 1\right ) \left (a e^{2} - b d e + c d^{2}\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 3.10786, size = 535, normalized size = 1.61 \[ \frac{2^{-3 (p+1)} \Gamma \left (-p-\frac{1}{2}\right ) (d+e x)^{-2 (p+1)} \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{c}\right )^p (a+x (b+c x))^p \left (\frac{e \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}\right )^{-p} \left (1-\frac{2 c (d+e x)}{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}\right )^{p+1} \left (\Gamma (1-2 p) \Gamma (-p) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right ) \left (e (2 p+1) \left (\sqrt{b^2-4 a c}-b\right )+4 c d (p+1)+2 c e x\right ) \, _2F_1\left (1,-p;-2 p;\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}\right )+\frac{4 c e \Gamma (1-p) \Gamma (-2 p) (d+e x) \left (2 c x \sqrt{b^2-4 a c}+b \sqrt{b^2-4 a c}+4 a c-b^2\right ) \, _2F_1\left (2,1-p;1-2 p;\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{b^2-4 a c}+b+2 c x}\right )}{\sqrt{\pi } e (p+1) \Gamma (1-2 p) \Gamma (-2 p) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.221, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-3-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 3), 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)**(-3-2*p)*(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. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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