3.544 \(\int \frac{x (e+f x)^n}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=198 \[ -\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{(n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{(n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )} \]

[Out]

-(((1 - b/Sqrt[b^2 - 4*a*c])*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b
- Sqrt[b^2 - 4*a*c])*f)])/((2*c*e - (b - Sqrt[b^2 - 4*a*c])*f)*(1 + n))) - ((1 + b/Sqrt[b^2 - 4*a*c])*(e + f*x
)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/((2*c*e - (
b + Sqrt[b^2 - 4*a*c])*f)*(1 + n))

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Rubi [A]  time = 0.189035, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {830, 68} \[ -\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{(n+1) \left (2 c e-f \left (b-\sqrt{b^2-4 a c}\right )\right )}-\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{(n+1) \left (2 c e-f \left (\sqrt{b^2-4 a c}+b\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x*(e + f*x)^n)/(a + b*x + c*x^2),x]

[Out]

-(((1 - b/Sqrt[b^2 - 4*a*c])*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b
- Sqrt[b^2 - 4*a*c])*f)])/((2*c*e - (b - Sqrt[b^2 - 4*a*c])*f)*(1 + n))) - ((1 + b/Sqrt[b^2 - 4*a*c])*(e + f*x
)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/((2*c*e - (
b + Sqrt[b^2 - 4*a*c])*f)*(1 + n))

Rule 830

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m, (f + g*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !RationalQ[m]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{x (e+f x)^n}{a+b x+c x^2} \, dx &=\int \left (\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) (e+f x)^n}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) (e+f x)^n}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx\\ &=\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{(e+f x)^n}{b-\sqrt{b^2-4 a c}+2 c x} \, dx+\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{(e+f x)^n}{b+\sqrt{b^2-4 a c}+2 c x} \, dx\\ &=-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{2 c (e+f x)}{2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f}\right )}{\left (2 c e-\left (b-\sqrt{b^2-4 a c}\right ) f\right ) (1+n)}-\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{\left (2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f\right ) (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.275087, size = 183, normalized size = 0.92 \[ \frac{(e+f x)^{n+1} \left (-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e+\left (\sqrt{b^2-4 a c}-b\right ) f}\right )}{f \left (\sqrt{b^2-4 a c}-b\right )+2 c e}-\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \, _2F_1\left (1,n+1;n+2;\frac{2 c (e+f x)}{2 c e-\left (b+\sqrt{b^2-4 a c}\right ) f}\right )}{2 c e-f \left (\sqrt{b^2-4 a c}+b\right )}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(e + f*x)^n)/(a + b*x + c*x^2),x]

[Out]

((e + f*x)^(1 + n)*(-(((1 - b/Sqrt[b^2 - 4*a*c])*Hypergeometric2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e + (
-b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e + (-b + Sqrt[b^2 - 4*a*c])*f)) - ((1 + b/Sqrt[b^2 - 4*a*c])*Hypergeometric
2F1[1, 1 + n, 2 + n, (2*c*(e + f*x))/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)])/(2*c*e - (b + Sqrt[b^2 - 4*a*c])*f)
))/(1 + n)

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Maple [F]  time = 1.308, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{n}x}{c{x}^{2}+bx+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(f*x+e)^n/(c*x^2+b*x+a),x)

[Out]

int(x*(f*x+e)^n/(c*x^2+b*x+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}{c x^{2} + b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(f*x+e)^n/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((f*x + e)^n*x/(c*x^2 + b*x + 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}{c x^{2} + b x + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(f*x+e)^n/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

integral((f*x + e)^n*x/(c*x^2 + b*x + 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*(f*x+e)**n/(c*x**2+b*x+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}{c x^{2} + b x + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(f*x+e)^n/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

integrate((f*x + e)^n*x/(c*x^2 + b*x + a), x)