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

Optimal. Leaf size=263 \[ -\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a^{2/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a^{2/3} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 a^{2/3} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )} \]

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

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Rubi [A]  time = 0.471876, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a^{2/3} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a^{2/3} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 a^{2/3} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 60.8382, size = 223, normalized size = 0.85 \[ \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} e}} \right )}}{3 a^{\frac{2}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f - \left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} e\right )} + \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{-1} \sqrt [3]{b} \left (e + f x\right )}{\sqrt [3]{a} f + \sqrt [3]{-1} \sqrt [3]{b} e}} \right )}}{3 a^{\frac{2}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f + \sqrt [3]{-1} \sqrt [3]{b} e\right )} + \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 a^{\frac{2}{3}} \left (n + 1\right ) \left (\sqrt [3]{a} f - \sqrt [3]{b} e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.051229, size = 122, normalized size = 0.46 \[ \frac{f^2 (e+f x)^n \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]}{3 b n} \]

Warning: Unable to verify antiderivative.

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

[Out]

(f^2*(e + f*x)^n*RootSum[b*e^3 - a*f^3 - 3*b*e^2*#1 + 3*b*e*#1^2 - b*#1^3 & , Hy
pergeometric2F1[-n, -n, 1 - n, -(#1/(e + f*x - #1))]/(((e + f*x)/(e + f*x - #1))
^n*(e^2 - 2*e*#1 + #1^2)) & ])/(3*b*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^n/(b*x^3+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^n/(b*x^3 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^n/(b*x^3 + a), x)