3.542 \(\int \frac{x^3 (e+f x)^n}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=289 \[ \frac{\left (\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-b f+\sqrt{b^2-4 a c} f}\right )}{c (n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{\left (-\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{c (n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{(b f+c e) (e+f x)^{n+1}}{c^2 f^2 (n+1)}+\frac{(e+f x)^{n+2}}{c f^2 (n+2)} \]

[Out]

-(((c*e + b*f)*(e + f*x)^(1 + n))/(c^2*f^2*(1 + n))) + (e + f*x)^(2 + n)/(c*f^2*
(2 + n)) + ((a - b^2/c + (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(e + f*x)^(1 +
n)*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - b*f + Sqrt[b^2 -
4*a*c]*f)])/(c*(2*c*e - (b - Sqrt[b^2 - 4*a*c])*f)*(1 + n)) + ((a - b^2/c - (b*
(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 +
n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(c*(2*c*e - (b
+ Sqrt[b^2 - 4*a*c])*f)*(1 + n))

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Rubi [A]  time = 1.24327, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\left (\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-b f+\sqrt{b^2-4 a c} f}\right )}{c (n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}+\frac{\left (-\frac{b \left (b^2-3 a c\right )}{c \sqrt{b^2-4 a c}}+a-\frac{b^2}{c}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{c (n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )}-\frac{(b f+c e) (e+f x)^{n+1}}{c^2 f^2 (n+1)}+\frac{(e+f x)^{n+2}}{c f^2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(e + f*x)^n)/(a + b*x + c*x^2),x]

[Out]

-(((c*e + b*f)*(e + f*x)^(1 + n))/(c^2*f^2*(1 + n))) + (e + f*x)^(2 + n)/(c*f^2*
(2 + n)) + ((a - b^2/c + (b*(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(e + f*x)^(1 +
n)*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - b*f + Sqrt[b^2 -
4*a*c]*f)])/(c*(2*c*e - (b - Sqrt[b^2 - 4*a*c])*f)*(1 + n)) + ((a - b^2/c - (b*
(b^2 - 3*a*c))/(c*Sqrt[b^2 - 4*a*c]))*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 +
n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(c*(2*c*e - (b
+ Sqrt[b^2 - 4*a*c])*f)*(1 + n))

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Rubi in Sympy [A]  time = 72.3735, size = 272, normalized size = 0.94 \[ \frac{\left (e + f x\right )^{n + 2}}{c f^{2} \left (n + 2\right )} + \frac{\left (e + f x\right )^{n + 1} \left (b \left (- 3 a c + b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (- 2 e - 2 f x\right )}{b f - 2 c e - f \sqrt{- 4 a c + b^{2}}}} \right )}}{c^{2} \left (n + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c e - f \left (b - \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{\left (e + f x\right )^{n + 1} \left (b \left (- 3 a c + b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (- 2 e - 2 f x\right )}{b f - 2 c e + f \sqrt{- 4 a c + b^{2}}}} \right )}}{c^{2} \left (n + 1\right ) \sqrt{- 4 a c + b^{2}} \left (2 c e - f \left (b + \sqrt{- 4 a c + b^{2}}\right )\right )} - \frac{\left (e + f x\right )^{n + 1} \left (b f + c e\right )}{c^{2} f^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(f*x+e)**n/(c*x**2+b*x+a),x)

[Out]

(e + f*x)**(n + 2)/(c*f**2*(n + 2)) + (e + f*x)**(n + 1)*(b*(-3*a*c + b**2) - sq
rt(-4*a*c + b**2)*(-a*c + b**2))*hyper((1, n + 1), (n + 2,), c*(-2*e - 2*f*x)/(b
*f - 2*c*e - f*sqrt(-4*a*c + b**2)))/(c**2*(n + 1)*sqrt(-4*a*c + b**2)*(2*c*e -
f*(b - sqrt(-4*a*c + b**2)))) - (e + f*x)**(n + 1)*(b*(-3*a*c + b**2) + sqrt(-4*
a*c + b**2)*(-a*c + b**2))*hyper((1, n + 1), (n + 2,), c*(-2*e - 2*f*x)/(b*f - 2
*c*e + f*sqrt(-4*a*c + b**2)))/(c**2*(n + 1)*sqrt(-4*a*c + b**2)*(2*c*e - f*(b +
sqrt(-4*a*c + b**2)))) - (e + f*x)**(n + 1)*(b*f + c*e)/(c**2*f**2*(n + 1))

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Mathematica [A]  time = 1.18899, size = 353, normalized size = 1.22 \[ \frac{2^{-n-1} (e+f x)^n \left (\left (b^2 \sqrt{f^2 \left (b^2-4 a c\right )}-a c \sqrt{f^2 \left (b^2-4 a c\right )}+3 a b c f+b^3 (-f)\right ) \left (\frac{c (e+f x)}{-\sqrt{f^2 \left (b^2-4 a c\right )}+b f+2 c f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{2 c e-b f+\sqrt{\left (b^2-4 a c\right ) f^2}}{-b f-2 c x f+\sqrt{\left (b^2-4 a c\right ) f^2}}\right )+\left (b^2 \sqrt{f^2 \left (b^2-4 a c\right )}-a c \sqrt{f^2 \left (b^2-4 a c\right )}-3 a b c f+b^3 f\right ) \left (\frac{c (e+f x)}{\sqrt{f^2 \left (b^2-4 a c\right )}+b f+2 c f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{-2 c e+b f+\sqrt{\left (b^2-4 a c\right ) f^2}}{b f+2 c x f+\sqrt{\left (b^2-4 a c\right ) f^2}}\right )\right )}{c^3 n \sqrt{f^2 \left (b^2-4 a c\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^3*(e + f*x)^n)/(a + b*x + c*x^2),x]

[Out]

(2^(-1 - n)*(e + f*x)^n*(((-(b^3*f) + 3*a*b*c*f + b^2*Sqrt[(b^2 - 4*a*c)*f^2] -
a*c*Sqrt[(b^2 - 4*a*c)*f^2])*Hypergeometric2F1[-n, -n, 1 - n, (2*c*e - b*f + Sqr
t[(b^2 - 4*a*c)*f^2])/(-(b*f) + Sqrt[(b^2 - 4*a*c)*f^2] - 2*c*f*x)])/((c*(e + f*
x))/(b*f - Sqrt[(b^2 - 4*a*c)*f^2] + 2*c*f*x))^n + ((b^3*f - 3*a*b*c*f + b^2*Sqr
t[(b^2 - 4*a*c)*f^2] - a*c*Sqrt[(b^2 - 4*a*c)*f^2])*Hypergeometric2F1[-n, -n, 1
- n, (-2*c*e + b*f + Sqrt[(b^2 - 4*a*c)*f^2])/(b*f + Sqrt[(b^2 - 4*a*c)*f^2] + 2
*c*f*x)])/((c*(e + f*x))/(b*f + Sqrt[(b^2 - 4*a*c)*f^2] + 2*c*f*x))^n))/(c^3*Sqr
t[(b^2 - 4*a*c)*f^2]*n)

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3} \left ( fx+e \right ) ^{n}}{c{x}^{2}+bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(f*x+e)^n/(c*x^2+b*x+a),x)

[Out]

int(x^3*(f*x+e)^n/(c*x^2+b*x+a),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^3/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

integrate((f*x + e)^n*x^3/(c*x^2 + b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^3/(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

integral((f*x + e)^n*x^3/(c*x^2 + b*x + a), 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(x**3*(f*x+e)**n/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x^{3}}{c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^3/(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

integrate((f*x + e)^n*x^3/(c*x^2 + b*x + a), x)