3.1851 \(\int \frac{(a+b x)^m}{(c+d x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]

[Out]

(b^2*(a + b*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*(1 + m
))

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Rubi [A]  time = 0.0131582, antiderivative size = 54, 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} \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m/(c + d*x)^3,x]

[Out]

(b^2*(a + b*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*(1 + m
))

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)^m}{(c+d x)^3} \, dx &=\frac{b^2 (a+b x)^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0138603, size = 54, normalized size = 1. \[ \frac{b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m/(c + d*x)^3,x]

[Out]

(b^2*(a + b*x)^(1 + m)*Hypergeometric2F1[3, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*(1 + m
))

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m/(d*x+c)^3,x)

[Out]

int((b*x+a)^m/(d*x+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m/(d*x+c)^3,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m/(d*x + c)^3, 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 )}^{m}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m/(d*x+c)^3,x, algorithm="fricas")

[Out]

integral((b*x + a)^m/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right )^{m}}{\left (c + d x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m/(d*x+c)**3,x)

[Out]

Integral((a + b*x)**m/(c + d*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m/(d*x+c)^3,x, algorithm="giac")

[Out]

integrate((b*x + a)^m/(d*x + c)^3, x)