3.955 \(\int \frac{(b x)^m (c+d x)^n}{(e+f x)^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e^2 (m+1)} \]

[Out]

((b*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((d*x)/c), -((f*x)/e)])/(b*e^2*(1 + m)*(1 + (d*x)/c)
^n)

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Rubi [A]  time = 0.0307787, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {135, 133} \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[((b*x)^m*(c + d*x)^n)/(e + f*x)^2,x]

[Out]

((b*x)^(1 + m)*(c + d*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((d*x)/c), -((f*x)/e)])/(b*e^2*(1 + m)*(1 + (d*x)/c)
^n)

Rule 135

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c^IntPart[n]*(c +
d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int \frac{(b x)^m (c+d x)^n}{(e+f x)^2} \, dx &=\left ((c+d x)^n \left (1+\frac{d x}{c}\right )^{-n}\right ) \int \frac{(b x)^m \left (1+\frac{d x}{c}\right )^n}{(e+f x)^2} \, dx\\ &=\frac{(b x)^{1+m} (c+d x)^n \left (1+\frac{d x}{c}\right )^{-n} F_1\left (1+m;-n,2;2+m;-\frac{d x}{c},-\frac{f x}{e}\right )}{b e^2 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0601175, size = 60, normalized size = 0.95 \[ \frac{x (b x)^m (c+d x)^n \left (\frac{c+d x}{c}\right )^{-n} F_1\left (m+1;-n,2;m+2;-\frac{d x}{c},-\frac{f x}{e}\right )}{e^2 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*x)^m*(c + d*x)^n*AppellF1[1 + m, -n, 2, 2 + m, -((d*x)/c), -((f*x)/e)])/(e^2*(1 + m)*((c + d*x)/c)^n)

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n}}{ \left ( fx+e \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(d*x+c)^n/(f*x+e)^2,x)

[Out]

int((b*x)^m*(d*x+c)^n/(f*x+e)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x)^m*(d*x + c)^n/(f*x + e)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x)^m*(d*x + c)^n/(f^2*x^2 + 2*e*f*x + e^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*(d*x+c)**n/(f*x+e)**2,x)

[Out]

Integral((b*x)**m*(c + d*x)**n/(e + f*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x)^m*(d*x + c)^n/(f*x + e)^2, x)