3.2558 \(\int \frac{\left (a+b x+c x^2\right )^p}{(d+e x)^2} \, dx\)

Optimal. Leaf size=196 \[ -\frac{4^p \left (a+b x+c x^2\right )^p \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} F_1\left (1-2 p;-p,-p;2 (1-p);\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{e (1-2 p) (d+e x)} \]

[Out]

-((4^p*(a + b*x + c*x^2)^p*AppellF1[1 - 2*p, -p, -p, 2*(1 - p), (2*c*d - (b - Sq
rt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c)/(2*(d
+ e*x))])/(e*(1 - 2*p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((
e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*(d + e*x)))

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Rubi [A]  time = 0.476011, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{4^p \left (a+b x+c x^2\right )^p \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} F_1\left (1-2 p;-p,-p;2 (1-p);\frac{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 d-\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{e (1-2 p) (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

-((4^p*(a + b*x + c*x^2)^p*AppellF1[1 - 2*p, -p, -p, 2*(1 - p), (2*c*d - (b - Sq
rt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c)/(2*(d
+ e*x))])/(e*(1 - 2*p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((
e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*(d + e*x)))

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Rubi in Sympy [A]  time = 31.9402, size = 167, normalized size = 0.85 \[ - \frac{\left (\frac{e \left (b + 2 c x - \sqrt{- 4 a c + b^{2}}\right )}{2 c \left (d + e x\right )}\right )^{- p} \left (\frac{e \left (b + 2 c x + \sqrt{- 4 a c + b^{2}}\right )}{2 c \left (d + e x\right )}\right )^{- p} \left (a + b x + c x^{2}\right )^{p} \left (\frac{1}{d + e x}\right )^{2 p} \left (\frac{1}{d + e x}\right )^{- 2 p + 1} \operatorname{appellf_{1}}{\left (- 2 p + 1,- p,- p,- 2 p + 2,\frac{c d - \frac{e \left (b - \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )},\frac{c d - \frac{e \left (b + \sqrt{- 4 a c + b^{2}}\right )}{2}}{c \left (d + e x\right )} \right )}}{e \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(e*(b + 2*c*x - sqrt(-4*a*c + b**2))/(2*c*(d + e*x)))**(-p)*(e*(b + 2*c*x + sqr
t(-4*a*c + b**2))/(2*c*(d + e*x)))**(-p)*(a + b*x + c*x**2)**p*(1/(d + e*x))**(2
*p)*(1/(d + e*x))**(-2*p + 1)*appellf1(-2*p + 1, -p, -p, -2*p + 2, (c*d - e*(b -
sqrt(-4*a*c + b**2))/2)/(c*(d + e*x)), (c*d - e*(b + sqrt(-4*a*c + b**2))/2)/(c
*(d + e*x)))/(e*(-2*p + 1))

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Mathematica [A]  time = 0.583298, size = 191, normalized size = 0.97 \[ \frac{4^p (a+x (b+c x))^p \left (\frac{e \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac{e \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c (d+e x)},\frac{2 c d-b e+\sqrt{b^2-4 a c} e}{2 c d+2 c e x}\right )}{e (2 p-1) (d+e x)} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4^p*(a + x*(b + c*x))^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, (2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*
c*e*x)])/(e*(-1 + 2*p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((e
*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*(d + e*x))

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Maple [F]  time = 0.157, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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