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

Optimal. Leaf size=300 \[ \frac{\sqrt [3]{b} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{\sqrt [3]{b} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 a (n+1) \left (\sqrt [3]{-1} \sqrt [3]{a} f+\sqrt [3]{b} e\right )}+\frac{\sqrt [3]{b} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 a (n+1) \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{a e (n+1)} \]

[Out]

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

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Rubi [A]  time = 1.24279, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt [3]{b} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}+\frac{\sqrt [3]{b} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 a (n+1) \left (\sqrt [3]{-1} \sqrt [3]{a} f+\sqrt [3]{b} e\right )}+\frac{\sqrt [3]{b} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 a (n+1) \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right )}-\frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{f x}{e}+1\right )}{a e (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 129.179, size = 241, normalized size = 0.8 \[ \frac{\sqrt [3]{b} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 a \left (n + 1\right ) \left (- \left (-1\right )^{\frac{2}{3}} \sqrt [3]{a} f + \sqrt [3]{b} e\right )} + \frac{\sqrt [3]{b} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{\sqrt [3]{-1} \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 a \left (n + 1\right ) \left (\sqrt [3]{-1} \sqrt [3]{a} f + \sqrt [3]{b} e\right )} - \frac{\sqrt [3]{b} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 a \left (n + 1\right ) \left (\sqrt [3]{a} f - \sqrt [3]{b} e\right )} - \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{f x}{e}} \right )}}{a e \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.263433, size = 377, normalized size = 1.26 \[ \frac{(e+f x)^n \left (-e^2 \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]+2 e \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1} \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]-\text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1}^2 \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]+3 \left (\frac{e}{f x}+1\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{e}{f x}\right )\right )}{3 a n} \]

Warning: Unable to verify antiderivative.

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

[Out]

((e + f*x)^n*((3*Hypergeometric2F1[-n, -n, 1 - n, -(e/(f*x))])/(1 + 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 & , Hypergeometric
2F1[-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*a*n)

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{ \left ( fx+e \right ) ^{n}}{x \left ( b{x}^{3}+a \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^n/x/(b*x^3+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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