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

Optimal. Leaf size=211 \[ \frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},\frac{\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)} \]

[Out]

(x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((b^(1/3)*x)/a
^(1/3))])/(3*a*(1 + m)*(1 + (f*x)/e)^n) + (x^(1 + m)*(e + f*x)^n*AppellF1[1 + m,
-n, 1, 2 + m, -((f*x)/e), ((-1)^(1/3)*b^(1/3)*x)/a^(1/3)])/(3*a*(1 + m)*(1 + (f
*x)/e)^n) + (x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((
(-1)^(2/3)*b^(1/3)*x)/a^(1/3))])/(3*a*(1 + m)*(1 + (f*x)/e)^n)

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Rubi [A]  time = 0.969815, antiderivative size = 211, 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 \[ \frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},\frac{\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((b^(1/3)*x)/a
^(1/3))])/(3*a*(1 + m)*(1 + (f*x)/e)^n) + (x^(1 + m)*(e + f*x)^n*AppellF1[1 + m,
-n, 1, 2 + m, -((f*x)/e), ((-1)^(1/3)*b^(1/3)*x)/a^(1/3)])/(3*a*(1 + m)*(1 + (f
*x)/e)^n) + (x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((
(-1)^(2/3)*b^(1/3)*x)/a^(1/3))])/(3*a*(1 + m)*(1 + (f*x)/e)^n)

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Rubi in Sympy [A]  time = 88.7438, size = 168, normalized size = 0.8 \[ \frac{x^{m + 1} \left (1 + \frac{f x}{e}\right )^{- n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{\sqrt [3]{b} x}{\sqrt [3]{a}},- \frac{f x}{e} \right )}}{3 a \left (m + 1\right )} + \frac{x^{m + 1} \left (1 + \frac{f x}{e}\right )^{- n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,\frac{\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}},- \frac{f x}{e} \right )}}{3 a \left (m + 1\right )} + \frac{x^{m + 1} \left (1 + \frac{f x}{e}\right )^{- n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,1,- n,m + 2,- \frac{\left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} x}{\sqrt [3]{a}},- \frac{f x}{e} \right )}}{3 a \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(m + 1)*(1 + f*x/e)**(-n)*(e + f*x)**n*appellf1(m + 1, 1, -n, m + 2, -b**(1/3
)*x/a**(1/3), -f*x/e)/(3*a*(m + 1)) + x**(m + 1)*(1 + f*x/e)**(-n)*(e + f*x)**n*
appellf1(m + 1, 1, -n, m + 2, (-1)**(1/3)*b**(1/3)*x/a**(1/3), -f*x/e)/(3*a*(m +
1)) + x**(m + 1)*(1 + f*x/e)**(-n)*(e + f*x)**n*appellf1(m + 1, 1, -n, m + 2, -
(-1)**(2/3)*b**(1/3)*x/a**(1/3), -f*x/e)/(3*a*(m + 1))

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Mathematica [A]  time = 0.0833978, size = 0, normalized size = 0. \[ \int \frac{x^m (e+f x)^n}{a+b x^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( fx+e \right ) ^{n}}{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(f*x+e)^n/(b*x^3+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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