3.3046 \(\int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx\)

Optimal. Leaf size=125 \[ -\frac{(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac{B (a+b x)^p (d+e x)^{-p} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b (d+e x)}{b d-a e}\right )}{e^2 p} \]

[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)

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Rubi [A]  time = 0.0686309, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {79, 70, 69} \[ -\frac{(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac{B (a+b x)^p (d+e x)^{-p} \left (-\frac{e (a+b x)}{b d-a e}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b (d+e x)}{b d-a e}\right )}{e^2 p} \]

Antiderivative was successfully verified.

[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 79

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]

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]))

Rubi steps

\begin{align*} \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 \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]  time = 0.191395, size = 114, normalized size = 0.91 \[ \frac{(a+b x)^p (d+e x)^{-p} \left (\frac{e (a+b x) (A e-B d)}{(p+1) (d+e x) (b d-a e)}-\frac{B \left (\frac{e (a+b x)}{a e-b d}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac{b (d+e x)}{b d-a e}\right )}{p}\right )}{e^2} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{p} \left ( Bx+A \right ) \left ( ex+d \right ) ^{-2-p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x + A\right )}{\left (b x + a\right )}^{p}{\left (e x + d\right )}^{-p - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B x + A\right )}{\left (b x + a\right )}^{p}{\left (e x + d\right )}^{-p - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)**p*(B*x+A)*(e*x+d)**(-2-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 )}{\left (b x + a\right )}^{p}{\left (e x + d\right )}^{-p - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)