3.164 \(\int \frac{x^4 (e+f x)^n}{a+b x^3} \, dx\)

Optimal. Leaf size=332 \[ -\frac{a^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{4/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{\sqrt [3]{-1} a^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{4/3} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{(-1)^{2/3} a^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 b^{4/3} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )}-\frac{e (e+f x)^{n+1}}{b f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b f^2 (n+2)} \]

[Out]

-((e*(e + f*x)^(1 + n))/(b*f^2*(1 + n))) + (e + f*x)^(2 + n)/(b*f^2*(2 + n)) - (
a^(2/3)*(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^(4/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(
1/3)*a^(2/3)*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^
(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)])/(3*b^(4/3)*((-1)^(2/3)*b^(
1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(2/3)*a^(2/3)*(e + f*x)^(1 + n)*Hypergeomet
ric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a
^(1/3)*f)])/(3*b^(4/3)*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))

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Rubi [A]  time = 1.60278, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{4/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{\sqrt [3]{-1} a^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{4/3} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{(-1)^{2/3} a^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 b^{4/3} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )}-\frac{e (e+f x)^{n+1}}{b f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b f^2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(e + f*x)^n)/(a + b*x^3),x]

[Out]

-((e*(e + f*x)^(1 + n))/(b*f^2*(1 + n))) + (e + f*x)^(2 + n)/(b*f^2*(2 + n)) - (
a^(2/3)*(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^(4/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(
1/3)*a^(2/3)*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^
(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)])/(3*b^(4/3)*((-1)^(2/3)*b^(
1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(2/3)*a^(2/3)*(e + f*x)^(1 + n)*Hypergeomet
ric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a
^(1/3)*f)])/(3*b^(4/3)*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))

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Rubi in Sympy [A]  time = 167.999, size = 287, normalized size = 0.86 \[ - \frac{\sqrt [3]{-1} a^{\frac{2}{3}} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} e}} \right )}}{3 b^{\frac{4}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f - \left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} e\right )} + \frac{\left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{-1} \sqrt [3]{b} \left (e + f x\right )}{\sqrt [3]{a} f + \sqrt [3]{-1} \sqrt [3]{b} e}} \right )}}{3 b^{\frac{4}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f + \sqrt [3]{-1} \sqrt [3]{b} e\right )} + \frac{a^{\frac{2}{3}} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{4}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f - \sqrt [3]{b} e\right )} - \frac{e \left (e + f x\right )^{n + 1}}{b f^{2} \left (n + 1\right )} + \frac{\left (e + f x\right )^{n + 2}}{b f^{2} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(f*x+e)**n/(b*x**3+a),x)

[Out]

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

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Mathematica [C]  time = 0.691891, size = 298, normalized size = 0.9 \[ \frac{(e+f x)^n \left (\frac{a e f^3 \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]}{b n}-\frac{a f^3 \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1} \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]}{b n}-\frac{3 \left (e^2 \left (1-\left (\frac{f x}{e}+1\right )^{-n}\right )-e f n x-f^2 (n+1) x^2\right )}{n^2+3 n+2}\right )}{3 b f^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^4*(e + f*x)^n)/(a + b*x^3),x]

[Out]

((e + f*x)^n*((-3*(-(e*f*n*x) - f^2*(1 + n)*x^2 + e^2*(1 - (1 + (f*x)/e)^(-n))))
/(2 + 3*n + n^2) + (a*e*f^3*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))]/(((e + f*x)/(e +
f*x - #1))^n*(e^2 - 2*e*#1 + #1^2)) & ])/(b*n) - (a*f^3*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)) & ])/(b*n
)))/(3*b*f^2)

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( fx+e \right ) ^{n}}{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(f*x+e)^n/(b*x^3+a),x)

[Out]

int(x^4*(f*x+e)^n/(b*x^3+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x^{4}}{b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^4/(b*x^3 + a),x, algorithm="maxima")

[Out]

integrate((f*x + e)^n*x^4/(b*x^3 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{4}}{b x^{3} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^4/(b*x^3 + a),x, algorithm="fricas")

[Out]

integral((f*x + e)^n*x^4/(b*x^3 + 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**4*(f*x+e)**n/(b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x^{4}}{b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^4/(b*x^3 + a),x, algorithm="giac")

[Out]

integrate((f*x + e)^n*x^4/(b*x^3 + a), x)