3.1857 \(\int \frac{(c+d x)^n}{(a+b x)^2} \, dx\)

Optimal. Leaf size=51 \[ \frac{d (c+d x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.0111316, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {68} \[ \frac{d (c+d x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^n/(a + b*x)^2,x]

[Out]

(d*(c + d*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*(1 + 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{(c+d x)^n}{(a+b x)^2} \, dx &=\frac{d (c+d x)^{1+n} \, _2F_1\left (2,1+n;2+n;\frac{b (c+d x)}{b c-a d}\right )}{(b c-a d)^2 (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0134705, size = 52, normalized size = 1.02 \[ \frac{d (c+d x)^{n+1} \, _2F_1\left (2,n+1;n+2;-\frac{b (c+d x)}{a d-b c}\right )}{(n+1) (a d-b c)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^n/(a + b*x)^2,x]

[Out]

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

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{n}}{ \left ( bx+a \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^n/(b*x + a)^2, x)