3.163 \(\int \frac{x^5 (e+f x)^n}{a+b x^3} \, dx\)

Optimal. Leaf size=324 \[ \frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{5/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 b^{5/3} (n+1) \left (\sqrt [3]{-1} \sqrt [3]{a} f+\sqrt [3]{b} e\right )}+\frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 b^{5/3} (n+1) \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right )}+\frac{e^2 (e+f x)^{n+1}}{b f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b f^3 (n+3)} \]

[Out]

(e^2*(e + f*x)^(1 + n))/(b*f^3*(1 + n)) - (2*e*(e + f*x)^(2 + n))/(b*f^3*(2 + n)
) + (e + f*x)^(3 + n)/(b*f^3*(3 + n)) + (a*(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^(5/3)*(b^(1/3
)*e - a^(1/3)*f)*(1 + n)) + (a*(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^(5/3)*(b^(1/3)
*e + (-1)^(1/3)*a^(1/3)*f)*(1 + n)) + (a*(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^(5/3
)*(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n))

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Rubi [A]  time = 1.47782, antiderivative size = 324, 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 \[ \frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{5/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 b^{5/3} (n+1) \left (\sqrt [3]{-1} \sqrt [3]{a} f+\sqrt [3]{b} e\right )}+\frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 b^{5/3} (n+1) \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right )}+\frac{e^2 (e+f x)^{n+1}}{b f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b f^3 (n+3)} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(e + f*x)^n)/(a + b*x^3),x]

[Out]

(e^2*(e + f*x)^(1 + n))/(b*f^3*(1 + n)) - (2*e*(e + f*x)^(2 + n))/(b*f^3*(2 + n)
) + (e + f*x)^(3 + n)/(b*f^3*(3 + n)) + (a*(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^(5/3)*(b^(1/3
)*e - a^(1/3)*f)*(1 + n)) + (a*(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^(5/3)*(b^(1/3)
*e + (-1)^(1/3)*a^(1/3)*f)*(1 + n)) + (a*(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^(5/3
)*(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n))

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Rubi in Sympy [A]  time = 143.417, size = 272, normalized size = 0.84 \[ \frac{a \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{5}{3}} \left (n + 1\right ) \left (- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} f + \sqrt [3]{b} e\right )} + \frac{a \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{\sqrt [3]{-1} \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{5}{3}} \left (n + 1\right ) \left (\sqrt [3]{-1} \sqrt [3]{a} f + \sqrt [3]{b} e\right )} - \frac{a \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 b^{\frac{5}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f - \sqrt [3]{b} e\right )} + \frac{e^{2} \left (e + f x\right )^{n + 1}}{b f^{3} \left (n + 1\right )} - \frac{2 e \left (e + f x\right )^{n + 2}}{b f^{3} \left (n + 2\right )} + \frac{\left (e + f x\right )^{n + 3}}{b f^{3} \left (n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(f*x+e)**n/(b*x**3+a),x)

[Out]

a*(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**(5/3)*(n + 1)*(-(-1)**(2/3)*a**(1/3)*f + b**(1
/3)*e)) + a*(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**(5/3)*(n + 1)*((-1)**(1/3)*a**(1/3)*f
+ b**(1/3)*e)) - a*(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**(5/3)*(n + 1)*(a**(1/3)*f - b**(1/3)*e))
+ e**2*(e + f*x)**(n + 1)/(b*f**3*(n + 1)) - 2*e*(e + f*x)**(n + 2)/(b*f**3*(n
+ 2)) + (e + f*x)**(n + 3)/(b*f**3*(n + 3))

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Mathematica [C]  time = 0.590475, size = 423, normalized size = 1.31 \[ \frac{(e+f x)^n \left (\frac{3 \left (e^3 \left (2-2 \left (\frac{f x}{e}+1\right )^{-n}\right )-2 e^2 f n x+e f^2 n (n+1) x^2+f^3 \left (n^2+3 n+2\right ) x^3\right )}{n^3+6 n^2+11 n+6}-\frac{a f^3 \left (e^2 \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]-2 e \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1} \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]+\text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1}^2 \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]\right )}{b n}\right )}{3 b f^3} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^5*(e + f*x)^n)/(a + b*x^3),x]

[Out]

((e + f*x)^n*((3*(-2*e^2*f*n*x + e*f^2*n*(1 + n)*x^2 + f^3*(2 + 3*n + n^2)*x^3 +
e^3*(2 - 2/(1 + (f*x)/e)^n)))/(6 + 11*n + 6*n^2 + n^3) - (a*f^3*(e^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))]/(((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 & , (Hypergeom
etric2F1[-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)) & ]))/(b*n)))/(3*b*f^3)

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5} \left ( fx+e \right ) ^{n}}{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(f*x+e)^n/(b*x^3+a),x)

[Out]

int(x^5*(f*x+e)^n/(b*x^3+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x^{5}}{b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^5/(b*x^3 + a),x, algorithm="maxima")

[Out]

integrate((f*x + e)^n*x^5/(b*x^3 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{5}}{b x^{3} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^5/(b*x^3 + a),x, algorithm="fricas")

[Out]

integral((f*x + e)^n*x^5/(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**5*(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. \[ \int \frac{{\left (f x + e\right )}^{n} x^{5}}{b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^n*x^5/(b*x^3 + a),x, algorithm="giac")

[Out]

integrate((f*x + e)^n*x^5/(b*x^3 + a), x)