3.2570 \(\int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=809 \[ -\frac{e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac{e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(5
+ 2*p))) - (e*(b^2*e^2*(12 + 7*p + p^2) + 2*c^2*d^2*(18 + 11*p + 2*p^2) - 2*c*e
*(3*a*e*(2 + p) + b*d*(18 + 11*p + 2*p^2)))*(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^
2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) - (e*(2*c*
d - b*e)*(3 + p)*(b^2*e^2*(8 + 6*p + p^2) + 2*c^2*d^2*(8 + 7*p + 2*p^2) - 2*c*e*
(a*e*(8 + 5*p) + b*d*(8 + 7*p + 2*p^2)))*(a + b*x + c*x^2)^(1 + p))/(4*(c*d^2 -
b*d*e + a*e^2)^4*(1 + p)*(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 + p))) - (e
*(2*c*d - b*e)*(4 + p)*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)^2*(
2 + p)*(5 + 2*p)*(d + e*x)^(2*(2 + p))) + ((b^4*e^4*(12 + 7*p + p^2) + 4*c^4*d^4
*(15 + 16*p + 4*p^2) - 8*c^3*d^2*e*(5 + 2*p)*(3*a*e + b*d*(3 + 2*p)) - 4*b^2*c*e
^3*(3 + p)*(3*a*e + b*d*(5 + 2*p)) + 12*c^2*e^2*(a^2*e^2 + 2*a*b*d*e*(5 + 2*p) +
b^2*d^2*(10 + 9*p + 2*p^2)))*(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))])/(4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^4*(1 + 2*
p)*(3 + 2*p)*(5 + 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 = 4.34609, antiderivative size = 809, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac{e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac{\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right ) \left (\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt{b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac{e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(5
+ 2*p))) - (e*(b^2*e^2*(12 + 7*p + p^2) + 2*c^2*d^2*(18 + 11*p + 2*p^2) - 2*c*e
*(3*a*e*(2 + p) + b*d*(18 + 11*p + 2*p^2)))*(d + e*x)^(-3 - 2*p)*(a + b*x + c*x^
2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) - (e*(2*c*
d - b*e)*(3 + p)*(b^2*e^2*(8 + 6*p + p^2) + 2*c^2*d^2*(8 + 7*p + 2*p^2) - 2*c*e*
(a*e*(8 + 5*p) + b*d*(8 + 7*p + 2*p^2)))*(a + b*x + c*x^2)^(1 + p))/(4*(c*d^2 -
b*d*e + a*e^2)^4*(1 + p)*(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 + p))) - (e
*(2*c*d - b*e)*(4 + p)*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)^2*(
2 + p)*(5 + 2*p)*(d + e*x)^(2*(2 + p))) + ((b^4*e^4*(12 + 7*p + p^2) + 4*c^4*d^4
*(15 + 16*p + 4*p^2) - 8*c^3*d^2*e*(5 + 2*p)*(3*a*e + b*d*(3 + 2*p)) - 4*b^2*c*e
^3*(3 + p)*(3*a*e + b*d*(5 + 2*p)) + 12*c^2*e^2*(a^2*e^2 + 2*a*b*d*e*(5 + 2*p) +
b^2*d^2*(10 + 9*p + 2*p^2)))*(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))])/(4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^4*(1 + 2*
p)*(3 + 2*p)*(5 + 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 [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)**(-6-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

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Mathematica [F]  time = 180., size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

$Aborted

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Maple [F]  time = 0.229, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-6-2\,p} \left ( c{x}^{2}+bx+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)

[Out]

int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), 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)**(-6-2*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. \[ \int{\left (c x^{2} + b x + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)