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

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

[Out]

(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],-d*(b*x+a)/(-a*d+b*c))/(-a*d+b*c)/(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} \[ \frac {(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/(c + d*x),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 51, normalized size = 1.00 \[ -\frac {(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )}{(n+1) (a d-b c)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/(c + d*x),x]

[Out]

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

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fricas [F]  time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^n/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n/(d*x + c), x)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{n}}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/(c + d*x),x)

[Out]

int((a + b*x)^n/(c + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*x)**n/(c + d*x), x)

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