3.810 \(\int \frac{(f+g x)^n (a+2 c d x+c e x^2)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=88 \[ \frac{c (f+g x)^{n+1}}{e g (n+1)}-\frac{g \left (c d^2-a e\right ) (f+g x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{e (n+1) (e f-d g)^2} \]

[Out]

(c*(f + g*x)^(1 + n))/(e*g*(1 + n)) - ((c*d^2 - a*e)*g*(f + g*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (e
*(f + g*x))/(e*f - d*g)])/(e*(e*f - d*g)^2*(1 + n))

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Rubi [A]  time = 0.0830026, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {947, 68} \[ \frac{c (f+g x)^{n+1}}{e g (n+1)}-\frac{g \left (c d^2-a e\right ) (f+g x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{e (n+1) (e f-d g)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(f + g*x)^(1 + n))/(e*g*(1 + n)) - ((c*d^2 - a*e)*g*(f + g*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (e
*(f + g*x))/(e*f - d*g)])/(e*(e*f - d*g)^2*(1 + n))

Rule 947

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (E
qQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0]))

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

Mathematica [A]  time = 0.0926034, size = 83, normalized size = 0.94 \[ \frac{(f+g x)^{n+1} \left (g^2 \left (a e-c d^2\right ) \, _2F_1\left (2,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )+c (e f-d g)^2\right )}{e g (n+1) (e f-d g)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((f + g*x)^(1 + n)*(c*(e*f - d*g)^2 + (-(c*d^2) + a*e)*g^2*Hypergeometric2F1[2, 1 + n, 2 + n, (e*(f + g*x))/(e
*f - d*g)]))/(e*g*(e*f - d*g)^2*(1 + n))

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Maple [F]  time = 0.692, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{n} \left ( ce{x}^{2}+2\,cdx+a \right ) }{ \left ( ex+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c e x^{2} + 2 \, c d x + a\right )}{\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c e x^{2} + 2 \, c d x + a\right )}{\left (g x + f\right )}^{n}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*e*x^2 + 2*c*d*x + a)*(g*x + f)^n/(e^2*x^2 + 2*d*e*x + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**n*(c*e*x**2+2*c*d*x+a)/(e*x+d)**2,x)

[Out]

Integral((f + g*x)**n*(a + 2*c*d*x + c*e*x**2)/(d + e*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c e x^{2} + 2 \, c d x + a\right )}{\left (g x + f\right )}^{n}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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