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

Optimal. Leaf size=253 \[ -\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{2/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 b^{2/3} (n+1) \left (\sqrt [3]{-1} \sqrt [3]{a} f+\sqrt [3]{b} e\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 b^{2/3} (n+1) \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.689617, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{2/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 b^{2/3} (n+1) \left (\sqrt [3]{-1} \sqrt [3]{a} f+\sqrt [3]{b} e\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 b^{2/3} (n+1) \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 73.0247, size = 209, normalized size = 0.83 \[ - \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{2}{3}} \left (n + 1\right ) \left (- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} f + \sqrt [3]{b} e\right )} - \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{\sqrt [3]{-1} \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{2}{3}} \left (n + 1\right ) \left (\sqrt [3]{-1} \sqrt [3]{a} f + \sqrt [3]{b} e\right )} + \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{2}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f - \sqrt [3]{b} e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(e + f*x)**(n + 1)*hyper((1, n + 1), (n + 2,), b**(1/3)*(e + f*x)/(-(-1)**(2/3)
*a**(1/3)*f + b**(1/3)*e))/(3*b**(2/3)*(n + 1)*(-(-1)**(2/3)*a**(1/3)*f + b**(1/
3)*e)) - (e + f*x)**(n + 1)*hyper((1, n + 1), (n + 2,), b**(1/3)*(e + f*x)/((-1)
**(1/3)*a**(1/3)*f + b**(1/3)*e))/(3*b**(2/3)*(n + 1)*((-1)**(1/3)*a**(1/3)*f +
b**(1/3)*e)) + (e + f*x)**(n + 1)*hyper((1, n + 1), (n + 2,), b**(1/3)*(e + f*x)
/(-a**(1/3)*f + b**(1/3)*e))/(3*b**(2/3)*(n + 1)*(a**(1/3)*f - b**(1/3)*e))

_______________________________________________________________________________________

Mathematica [C]  time = 0.122362, size = 337, normalized size = 1.33 \[ \frac{(e+f x)^n \left (e^2 \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]-2 e \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1} \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]+\text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1}^2 \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]\right )}{3 b n} \]

Warning: Unable to verify antiderivative.

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

[Out]

((e + f*x)^n*(e^2*RootSum[b*e^3 - a*f^3 - 3*b*e^2*#1 + 3*b*e*#1^2 - b*#1^3 & , H
ypergeometric2F1[-n, -n, 1 - n, -(#1/(e + f*x - #1))]/(((e + f*x)/(e + f*x - #1)
)^n*(e^2 - 2*e*#1 + #1^2)) & ] - 2*e*RootSum[b*e^3 - a*f^3 - 3*b*e^2*#1 + 3*b*e*
#1^2 - b*#1^3 & , (Hypergeometric2F1[-n, -n, 1 - n, -(#1/(e + f*x - #1))]*#1)/((
(e + f*x)/(e + f*x - #1))^n*(e^2 - 2*e*#1 + #1^2)) & ] + RootSum[b*e^3 - a*f^3 -
3*b*e^2*#1 + 3*b*e*#1^2 - b*#1^3 & , (Hypergeometric2F1[-n, -n, 1 - n, -(#1/(e
+ f*x - #1))]*#1^2)/(((e + f*x)/(e + f*x - #1))^n*(e^2 - 2*e*#1 + #1^2)) & ]))/(
3*b*n)

_______________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( fx+e \right ) ^{n}}{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(f*x+e)^n/(b*x^3+a),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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