3.3103 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^p \, dx\)

Optimal. Leaf size=133 \[ \frac{b^4 (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m+5,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b c-a d)^5} \]

[Out]

(b^4*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*(e + f*x)^p*AppellF1[1 + m, 5 + m, -p, 2 + m, -((d*(a + b
*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/((b*c - a*d)^5*(1 + m)*(c + d*x)^m*((b*(e + f*x))/(b*e - a*f
))^p)

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Rubi [A]  time = 0.11384, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {140, 139, 138} \[ \frac{b^4 (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m+5,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^p,x]

[Out]

(b^4*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*(e + f*x)^p*AppellF1[1 + m, 5 + m, -p, 2 + m, -((d*(a + b
*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/((b*c - a*d)^5*(1 + m)*(c + d*x)^m*((b*(e + f*x))/(b*e - a*f
))^p)

Rule 140

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

Rule 139

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

Rule 138

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

Rubi steps

\begin{align*} \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^p \, dx &=\frac{\left (b^5 (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-5-m} (e+f x)^p \, dx}{(b c-a d)^5}\\ &=\frac{\left (b^5 (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p}\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^{-5-m} \left (\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}\right )^p \, dx}{(b c-a d)^5}\\ &=\frac{b^4 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m (e+f x)^p \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (1+m;5+m,-p;2+m;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(b c-a d)^5 (1+m)}\\ \end{align*}

Mathematica [A]  time = 3.04906, size = 131, normalized size = 0.98 \[ \frac{b^4 (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m+5,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{(m+1) (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^p,x]

[Out]

(b^4*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*(e + f*x)^p*AppellF1[1 + m, 5 + m, -p, 2 + m, (d*(a + b*x
))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/((b*c - a*d)^5*(1 + m)*(c + d*x)^m*((b*(e + f*x))/(b*e - a*f
))^p)

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Maple [F]  time = 0.143, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-5-m} \left ( fx+e \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^p,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}{\left (f x + e\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^p,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 5)*(f*x + e)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}{\left (f x + e\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^p,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m - 5)*(f*x + e)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**(-5-m)*(f*x+e)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}{\left (f x + e\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^p,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m - 5)*(f*x + e)^p, x)