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

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 = 0.462599, 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, Rules used = {6725, 135, 133} \[ \frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},\frac{\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac{x^{m+1} (e+f x)^n \left (\frac{f x}{e}+1\right )^{-n} F_1\left (m+1;-n,1;m+2;-\frac{f x}{e},-\frac{(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)} \]

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)

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 135

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c^IntPart[n]*(c +
d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int \frac{x^m (e+f x)^n}{a+b x^3} \, dx &=\int \left (-\frac{x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{x^m (e+f x)^n}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac{\int \frac{x^m (e+f x)^n}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac{\int \frac{x^m (e+f x)^n}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=-\frac{\left ((e+f x)^n \left (1+\frac{f x}{e}\right )^{-n}\right ) \int \frac{x^m \left (1+\frac{f x}{e}\right )^n}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac{\left ((e+f x)^n \left (1+\frac{f x}{e}\right )^{-n}\right ) \int \frac{x^m \left (1+\frac{f x}{e}\right )^n}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}-\frac{\left ((e+f x)^n \left (1+\frac{f x}{e}\right )^{-n}\right ) \int \frac{x^m \left (1+\frac{f x}{e}\right )^n}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{2/3}}\\ &=\frac{x^{1+m} (e+f x)^n \left (1+\frac{f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac{f x}{e},-\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}+\frac{x^{1+m} (e+f x)^n \left (1+\frac{f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac{f x}{e},\frac{\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}+\frac{x^{1+m} (e+f x)^n \left (1+\frac{f x}{e}\right )^{-n} F_1\left (1+m;-n,1;2+m;-\frac{f x}{e},-\frac{(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}\\ \end{align*}

Mathematica [F]  time = 0.168223, 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 = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( fx+e \right ) ^{n}}{b{x}^{3}+a}}\, dx \end{align*}

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. \begin{align*} \int \frac{{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(f*x+e)^n/(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. \begin{align*}{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(f*x+e)^n/(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. \begin{align*} \text{Timed out} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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