3.938 \(\int \frac{x (a+b x)^n}{c+d x} \, dx\)

Optimal. Leaf size=78 \[ \frac{(a+b x)^{n+1}}{b d (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0203725, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 68} \[ \frac{(a+b x)^{n+1}}{b d (n+1)}-\frac{c (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^n)/(c + d*x),x]

[Out]

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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{x (a+b x)^n}{c+d x} \, dx &=\frac{(a+b x)^{1+n}}{b d (1+n)}-\frac{c \int \frac{(a+b x)^n}{c+d x} \, dx}{d}\\ &=\frac{(a+b x)^{1+n}}{b d (1+n)}-\frac{c (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac{d (a+b x)}{b c-a d}\right )}{d (b c-a d) (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0153543, size = 68, normalized size = 0.87 \[ -\frac{(a+b x)^{n+1} \left (b c \, _2F_1\left (1,n+1;n+2;\frac{d (a+b x)}{a d-b c}\right )+a d-b c\right )}{b d (n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*x)^n)/(c + d*x),x]

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}x}{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^n/(d*x+c),x)

[Out]

int(x*(b*x+a)^n/(d*x+c),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n*x/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^n*x/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**n/(d*x+c),x)

[Out]

Integral(x*(a + b*x)**n/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n*x/(d*x + c), x)