3.944 \(\int \frac{x (a+b x)^n}{(c+d x)^2} \, dx\)

Optimal. Leaf size=99 \[ -\frac{(a+b x)^{n+1} (a d-b c (n+1)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)^2}-\frac{c (a+b x)^{n+1}}{d (c+d x) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0330376, antiderivative size = 99, 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 = {78, 68} \[ -\frac{(a+b x)^{n+1} (a d-b c (n+1)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{d (n+1) (b c-a d)^2}-\frac{c (a+b x)^{n+1}}{d (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x)^n)/(c + d*x)^2,x]

[Out]

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

Rule 78

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

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)^2} \, dx &=-\frac{c (a+b x)^{1+n}}{d (b c-a d) (c+d x)}+\frac{(a d-b c (1+n)) \int \frac{(a+b x)^n}{c+d x} \, dx}{d (-b c+a d)}\\ &=-\frac{c (a+b x)^{1+n}}{d (b c-a d) (c+d x)}-\frac{(a d-b c (1+n)) (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)^2 (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0357209, size = 83, normalized size = 0.84 \[ \frac{(a+b x)^{n+1} \left (\frac{(b c (n+1)-a d) \, _2F_1\left (1,n+1;n+2;\frac{d (a+b x)}{a d-b c}\right )}{n+1}+\frac{c (a d-b c)}{c+d x}\right )}{d (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}x}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**n/(d*x+c)**2,x)

[Out]

Integral(x*(a + b*x)**n/(c + d*x)**2, 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}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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