3.2645 \(\int (A+B x) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=349 \[ \frac{(B d-A e) (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )}-\frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p (-2 a B e+A b e-2 A c d+b B d) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

((B*d - A*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))) - ((b*B*d - 2*A*c*d + A*b*e - 2*a*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 = 0.56996, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(B d-A e) (d+e x)^{-2 (p+1)} \left (a+b x+c x^2\right )^{p+1}}{2 (p+1) \left (a e^2-b d e+c d^2\right )}-\frac{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) (d+e x)^{-2 p-1} \left (a+b x+c x^2\right )^p (-2 a B e+A b e-2 A c d+b B d) \left (\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{2 (2 p+1) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((B*d - A*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))) - ((b*B*d - 2*A*c*d + A*b*e - 2*a*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 = 96.1494, size = 313, normalized size = 0.9 \[ - \frac{\left (d + e x\right )^{- 2 p - 2} \left (A e - B d\right ) \left (a + b x + c x^{2}\right )^{p + 1}}{2 \left (p + 1\right ) \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (\frac{\left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}\right )^{- p} \left (d + e x\right )^{- 2 p - 1} \left (a + b x + c x^{2}\right )^{p} \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (- A c d - B a e + \frac{b \left (A e + B d\right )}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{4 c \left (d + e x\right ) \sqrt{- 4 a c + b^{2}}}{\left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right ) \left (b e - 2 c d + e \sqrt{- 4 a c + b^{2}}\right )}} \right )}}{\left (2 p + 1\right ) \left (a e^{2} - b d e + c d^{2}\right ) \left (b e - 2 c d - e \sqrt{- 4 a c + b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)**(-2*p - 2)*(A*e - B*d)*(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)*(a + b*x + c*x**2)**p*(b + 2*c*x - sqrt(-
4*a*c + b**2))*(-A*c*d - B*a*e + b*(A*e + B*d)/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 [B]  time = 7.89225, size = 1158, normalized size = 3.32 \[ \frac{2^{-3 p-2} \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x\right )^{-p} \left (\frac{e \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )^{-p} \left (\frac{b+2 c x-\sqrt{b^2-4 a c}}{c}\right )^p \left (\frac{e \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d}\right )^{-p} (d+e x)^{-2 (p+1)} (a+x (b+c x))^p \left (1-\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )^{p+1} \left (-\frac{4^{p+1} B (d+e x) \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}\right ) \left (1-\frac{2 c (d+e x)}{2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e}\right )^p}{2 p+1}-\frac{B d \left (1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^p \Gamma \left (-p-\frac{1}{2}\right ) \left (\left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \left (4 c d (p+1)+\left (\sqrt{b^2-4 a c}-b\right ) e (2 p+1)+2 c e x\right ) \Gamma (1-2 p) \Gamma (-p) \, _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 \left (-b^2+\sqrt{b^2-4 a c} b+4 a c+2 c \sqrt{b^2-4 a c} x\right ) (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _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 )}{b+2 c x+\sqrt{b^2-4 a c}}\right )}{2 \left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right )^2 (p+1) \sqrt{\pi } \Gamma (1-2 p) \Gamma (-2 p)}+\frac{A e \left (1-\frac{2 c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )^p \Gamma \left (-p-\frac{1}{2}\right ) \left (\left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right ) \left (4 c d (p+1)+\left (\sqrt{b^2-4 a c}-b\right ) e (2 p+1)+2 c e x\right ) \Gamma (1-2 p) \Gamma (-p) \, _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 \left (-b^2+\sqrt{b^2-4 a c} b+4 a c+2 c \sqrt{b^2-4 a c} x\right ) (d+e x) \Gamma (1-p) \Gamma (-2 p) \, _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 )}{b+2 c x+\sqrt{b^2-4 a c}}\right )}{2 \left (2 c d+\left (\sqrt{b^2-4 a c}-b\right ) e\right )^2 (p+1) \sqrt{\pi } \Gamma (1-2 p) \Gamma (-2 p)}\right )}{e^2} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2^(-2 - 3*p)*((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)*(-(B*d*(1 - (2*c*(d +
e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^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*(2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)^2*(1 + p)*Sqrt[Pi]*Gamma[1 - 2*p]*Gamma[-2*p]) + (A*e*(1 - (2*c*(d +
e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^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*(2*c*d + (-b + Sqrt[b^2 -
4*a*c])*e)^2*(1 + p)*Sqrt[Pi]*Gamma[1 - 2*p]*Gamma[-2*p]) - (4^(1 + p)*B*(d + e
*x)*(1 - (2*c*(d + e*x))/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e))^p*Hypergeometric2
F1[-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))])/(1 + 2*p)))/(e^2*((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*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*d + (b + Sqr
t[b^2 - 4*a*c])*e))^p*(d + e*x)^(2*(1 + p)))

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Maple [F]  time = 0.249, size = 0, normalized size = 0. \[ \int \left ( Bx+A \right ) \left ( ex+d \right ) ^{-3-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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