3.136 \(\int \frac{(A+B x) (c+d x)^n (e+f x)^p}{a+b x} \, dx\)

Optimal. Leaf size=177 \[ -\frac{(A b-a B) (c+d x)^{n+1} (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p} F_1\left (n+1;1,-p;n+2;\frac{b (c+d x)}{b c-a d},-\frac{f (c+d x)}{d e-c f}\right )}{b (n+1) (b c-a d)}-\frac{B (c+d x)^{n+1} (e+f x)^{p+1} \, _2F_1\left (1,n+p+2;p+2;\frac{d (e+f x)}{d e-c f}\right )}{b (p+1) (d e-c f)} \]

[Out]

-(((A*b - a*B)*(c + d*x)^(1 + n)*(e + f*x)^p*AppellF1[1 + n, 1, -p, 2 + n, (b*(c + d*x))/(b*c - a*d), -((f*(c
+ d*x))/(d*e - c*f))])/(b*(b*c - a*d)*(1 + n)*((d*(e + f*x))/(d*e - c*f))^p)) - (B*(c + d*x)^(1 + n)*(e + f*x)
^(1 + p)*Hypergeometric2F1[1, 2 + n + p, 2 + p, (d*(e + f*x))/(d*e - c*f)])/(b*(d*e - c*f)*(1 + p))

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Rubi [A]  time = 0.118519, antiderivative size = 190, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {157, 70, 69, 137, 136} \[ \frac{B (c+d x)^{n+1} (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{b d (n+1)}-\frac{(A b-a B) (c+d x)^{n+1} (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p} F_1\left (n+1;-p,1;n+2;-\frac{f (c+d x)}{d e-c f},\frac{b (c+d x)}{b c-a d}\right )}{b (n+1) (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

Int[((A + B*x)*(c + d*x)^n*(e + f*x)^p)/(a + b*x),x]

[Out]

-(((A*b - a*B)*(c + d*x)^(1 + n)*(e + f*x)^p*AppellF1[1 + n, -p, 1, 2 + n, -((f*(c + d*x))/(d*e - c*f)), (b*(c
+ d*x))/(b*c - a*d)])/(b*(b*c - a*d)*(1 + n)*((d*(e + f*x))/(d*e - c*f))^p)) + (B*(c + d*x)^(1 + n)*(e + f*x)
^p*Hypergeometric2F1[1 + n, -p, 2 + n, -((f*(c + d*x))/(d*e - c*f))])/(b*d*(1 + n)*((d*(e + f*x))/(d*e - c*f))
^p)

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), 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*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 137

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}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rule 136

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

Rubi steps

\begin{align*} \int \frac{(A+B x) (c+d x)^n (e+f x)^p}{a+b x} \, dx &=\frac{B \int (c+d x)^n (e+f x)^p \, dx}{b}+\frac{(A b-a B) \int \frac{(c+d x)^n (e+f x)^p}{a+b x} \, dx}{b}\\ &=\frac{\left (B (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p}\right ) \int (c+d x)^n \left (\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}\right )^p \, dx}{b}+\frac{\left ((A b-a B) (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p}\right ) \int \frac{(c+d x)^n \left (\frac{d e}{d e-c f}+\frac{d f x}{d e-c f}\right )^p}{a+b x} \, dx}{b}\\ &=-\frac{(A b-a B) (c+d x)^{1+n} (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p} F_1\left (1+n;-p,1;2+n;-\frac{f (c+d x)}{d e-c f},\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) (1+n)}+\frac{B (c+d x)^{1+n} (e+f x)^p \left (\frac{d (e+f x)}{d e-c f}\right )^{-p} \, _2F_1\left (1+n,-p;2+n;-\frac{f (c+d x)}{d e-c f}\right )}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.398779, size = 199, normalized size = 1.12 \[ \frac{(c+d x)^n (e+f x)^p \left (\frac{(A b-a B) \left (\frac{b (c+d x)}{d (a+b x)}\right )^{-n} \left (\frac{b (e+f x)}{f (a+b x)}\right )^{-p} F_1\left (-n-p;-n,-p;-n-p+1;\frac{a d-b c}{d (a+b x)},\frac{a f-b e}{f (a+b x)}\right )}{n+p}+\frac{b B (e+f x) \left (\frac{f (c+d x)}{c f-d e}\right )^{-n} \, _2F_1\left (-n,p+1;p+2;\frac{d (e+f x)}{d e-c f}\right )}{f (p+1)}\right )}{b^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((A + B*x)*(c + d*x)^n*(e + f*x)^p)/(a + b*x),x]

[Out]

((c + d*x)^n*(e + f*x)^p*(((A*b - a*B)*AppellF1[-n - p, -n, -p, 1 - n - p, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*
e) + a*f)/(f*(a + b*x))])/((n + p)*((b*(c + d*x))/(d*(a + b*x)))^n*((b*(e + f*x))/(f*(a + b*x)))^p) + (b*B*(e
+ f*x)*Hypergeometric2F1[-n, 1 + p, 2 + p, (d*(e + f*x))/(d*e - c*f)])/(f*(1 + p)*((f*(c + d*x))/(-(d*e) + c*f
))^n)))/b^2

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{p} \left ( dx+c \right ) ^{n} \left ( Bx+A \right ) }{bx+a}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(d*x+c)^n*(f*x+e)^p/(b*x+a),x)

[Out]

int((B*x+A)*(d*x+c)^n*(f*x+e)^p/(b*x+a),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(d*x+c)^n*(f*x+e)^p/(b*x+a),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(d*x + c)^n*(f*x + e)^p/(b*x + a), 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 )}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{p}}{b x + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(d*x+c)^n*(f*x+e)^p/(b*x+a),x, algorithm="fricas")

[Out]

integral((B*x + A)*(d*x + c)^n*(f*x + e)^p/(b*x + a), 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)*(d*x+c)**n*(f*x+e)**p/(b*x+a),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(d*x+c)^n*(f*x+e)^p/(b*x+a),x, algorithm="giac")

[Out]

integrate((B*x + A)*(d*x + c)^n*(f*x + e)^p/(b*x + a), x)