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

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Rubi [A]  time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x + c\right )}^{n}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \]

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)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{n}}{\left (b x +a \right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c+d\,x\right )}^n}{{\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^n/(a + b*x)^2,x)

[Out]

int((c + d*x)^n/(a + b*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{n}}{\left (a + b x\right )^{2}}\, dx \]

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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