\(\int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx\) [3046]

Optimal result

Integrand size = 24, antiderivative size = 125 \[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=-\frac {(B d-A e) (a+b x)^{1+p} (d+e x)^{-1-p}}{e (b d-a e) (1+p)}-\frac {B (a+b x)^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-p} (d+e x)^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {b (d+e x)}{b d-a e}\right )}{e^2 p} \]

[Out]

-(-A*e+B*d)*(b*x+a)^(p+1)*(e*x+d)^(-1-p)/e/(-a*e+b*d)/(p+1)-B*(b*x+a)^p*hypergeom([-p, -p],[1-p],b*(e*x+d)/(-a
*e+b*d))/e^2/p/((-e*(b*x+a)/(-a*e+b*d))^p)/((e*x+d)^p)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 72, 71} \[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=-\frac {(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac {B (a+b x)^p (d+e x)^{-p} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {b (d+e x)}{b d-a e}\right )}{e^2 p} \]

[In]

Int[(a + b*x)^p*(A + B*x)*(d + e*x)^(-2 - p),x]

[Out]

-(((B*d - A*e)*(a + b*x)^(1 + p)*(d + e*x)^(-1 - p))/(e*(b*d - a*e)*(1 + p))) - (B*(a + b*x)^p*Hypergeometric2
F1[-p, -p, 1 - p, (b*(d + e*x))/(b*d - a*e)])/(e^2*p*(-((e*(a + b*x))/(b*d - a*e)))^p*(d + e*x)^p)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c
, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.91 \[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=\frac {(a+b x)^p (d+e x)^{-p} \left (\frac {e (-B d+A e) (a+b x)}{(b d-a e) (1+p) (d+e x)}-\frac {B \left (\frac {e (a+b x)}{-b d+a e}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {b (d+e x)}{b d-a e}\right )}{p}\right )}{e^2} \]

[In]

Integrate[(a + b*x)^p*(A + B*x)*(d + e*x)^(-2 - p),x]

[Out]

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

Maple [F]

\[\int \left (b x +a \right )^{p} \left (B x +A \right ) \left (e x +d \right )^{-2-p}d x\]

[In]

int((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x)

[Out]

int((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x)

Fricas [F]

\[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{p} {\left (e x + d\right )}^{-p - 2} \,d x } \]

[In]

integrate((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x, algorithm="fricas")

[Out]

integral((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2), x)

Sympy [F]

\[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=\int \left (A + B x\right ) \left (a + b x\right )^{p} \left (d + e x\right )^{- p - 2}\, dx \]

[In]

integrate((b*x+a)**p*(B*x+A)*(e*x+d)**(-2-p),x)

[Out]

Integral((A + B*x)*(a + b*x)**p*(d + e*x)**(-p - 2), x)

Maxima [F]

\[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{p} {\left (e x + d\right )}^{-p - 2} \,d x } \]

[In]

integrate((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2), x)

Giac [F]

\[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{p} {\left (e x + d\right )}^{-p - 2} \,d x } \]

[In]

integrate((b*x+a)^p*(B*x+A)*(e*x+d)^(-2-p),x, algorithm="giac")

[Out]

integrate((B*x + A)*(b*x + a)^p*(e*x + d)^(-p - 2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^p}{{\left (d+e\,x\right )}^{p+2}} \,d x \]

[In]

int(((A + B*x)*(a + b*x)^p)/(d + e*x)^(p + 2),x)

[Out]

int(((A + B*x)*(a + b*x)^p)/(d + e*x)^(p + 2), x)