3.3058 \(\int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx\)

Optimal. Leaf size=300 \[ \frac{2 b^2 (a+b x)^{1-n} (e+f x)^{n-1} (b (3 c f+d e (1-n))-a d f (4-n))}{f (1-n) (2-n) (3-n) (4-n) (b e-a f)^4}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-3} (b (3 c f+d e (1-n))-a d f (4-n))}{f (3-n) (4-n) (b e-a f)^2}+\frac{2 b (a+b x)^{1-n} (e+f x)^{n-2} (b (3 c f+d e (1-n))-a d f (4-n))}{f (2-n) (3-n) (4-n) (b e-a f)^3} \]

[Out]

-(((d*e - c*f)*(a + b*x)^(1 - n)*(e + f*x)^(-4 + n))/(f*(b*e - a*f)*(4 - n))) + ((b*(3*c*f + d*e*(1 - n)) - a*
d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-3 + n))/(f*(b*e - a*f)^2*(3 - n)*(4 - n)) + (2*b*(b*(3*c*f + d*e*(1
- n)) - a*d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-2 + n))/(f*(b*e - a*f)^3*(2 - n)*(3 - n)*(4 - n)) + (2*b
^2*(b*(3*c*f + d*e*(1 - n)) - a*d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-1 + n))/(f*(b*e - a*f)^4*(1 - n)*(2
- n)*(3 - n)*(4 - n))

________________________________________________________________________________________

Rubi [A]  time = 0.192607, antiderivative size = 297, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {79, 45, 37} \[ \frac{2 b^2 (a+b x)^{1-n} (e+f x)^{n-1} (-a d f (4-n)+3 b c f+b d e (1-n))}{f (1-n) (2-n) (3-n) (4-n) (b e-a f)^4}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-4}}{f (4-n) (b e-a f)}+\frac{(a+b x)^{1-n} (e+f x)^{n-3} (-a d f (4-n)+3 b c f+b d e (1-n))}{f (3-n) (4-n) (b e-a f)^2}+\frac{2 b (a+b x)^{1-n} (e+f x)^{n-2} (-a d f (4-n)+3 b c f+b d e (1-n))}{f (2-n) (3-n) (4-n) (b e-a f)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*e - c*f)*(a + b*x)^(1 - n)*(e + f*x)^(-4 + n))/(f*(b*e - a*f)*(4 - n))) + ((3*b*c*f + b*d*e*(1 - n) - a*
d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-3 + n))/(f*(b*e - a*f)^2*(3 - n)*(4 - n)) + (2*b*(3*b*c*f + b*d*e*(
1 - n) - a*d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-2 + n))/(f*(b*e - a*f)^3*(2 - n)*(3 - n)*(4 - n)) + (2*b
^2*(3*b*c*f + b*d*e*(1 - n) - a*d*f*(4 - n))*(a + b*x)^(1 - n)*(e + f*x)^(-1 + n))/(f*(b*e - a*f)^4*(1 - n)*(2
- n)*(3 - n)*(4 - n))

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

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

Rubi steps

\begin{align*} \int (a+b x)^{-n} (c+d x) (e+f x)^{-5+n} \, dx &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-4+n}}{f (b e-a f) (4-n)}-\frac{(-3 b c f-d (b e (1-n)+a f (-4+n))) \int (a+b x)^{-n} (e+f x)^{-4+n} \, dx}{f (-b e+a f) (-4+n)}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-4+n}}{f (b e-a f) (4-n)}+\frac{(3 b c f+b d e (1-n)-a d f (4-n)) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f)^2 (3-n) (4-n)}-\frac{(2 b (-3 b c f-d (b e (1-n)+a f (-4+n)))) \int (a+b x)^{-n} (e+f x)^{-3+n} \, dx}{f (b e-a f) (-b e+a f) (3-n) (-4+n)}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-4+n}}{f (b e-a f) (4-n)}+\frac{(3 b c f+b d e (1-n)-a d f (4-n)) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f)^2 (3-n) (4-n)}+\frac{2 b (3 b c f+b d e (1-n)-a d f (4-n)) (a+b x)^{1-n} (e+f x)^{-2+n}}{f (b e-a f)^3 (2-n) (3-n) (4-n)}-\frac{\left (2 b^2 (-3 b c f-d (b e (1-n)+a f (-4+n)))\right ) \int (a+b x)^{-n} (e+f x)^{-2+n} \, dx}{f (b e-a f)^2 (-b e+a f) (2-n) (3-n) (-4+n)}\\ &=-\frac{(d e-c f) (a+b x)^{1-n} (e+f x)^{-4+n}}{f (b e-a f) (4-n)}+\frac{(3 b c f+b d e (1-n)-a d f (4-n)) (a+b x)^{1-n} (e+f x)^{-3+n}}{f (b e-a f)^2 (3-n) (4-n)}+\frac{2 b (3 b c f+b d e (1-n)-a d f (4-n)) (a+b x)^{1-n} (e+f x)^{-2+n}}{f (b e-a f)^3 (2-n) (3-n) (4-n)}+\frac{2 b^2 (3 b c f+b d e (1-n)-a d f (4-n)) (a+b x)^{1-n} (e+f x)^{-1+n}}{f (b e-a f)^4 (1-n) (2-n) (3-n) (4-n)}\\ \end{align*}

Mathematica [A]  time = 0.41755, size = 145, normalized size = 0.48 \[ \frac{(a+b x)^{1-n} (e+f x)^{n-4} \left (-\frac{(e+f x) \left ((n-2) (n-1) (b e-a f)^2-2 b (e+f x) (-a f (n-1)+b e (n-2)-b f x)\right ) (a d f (n-4)+3 b c f-b d e (n-1))}{(n-3) (n-2) (n-1) (b e-a f)^3}+c f-d e\right )}{f (n-4) (a f-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 - n)*(e + f*x)^(-4 + n)*(-(d*e) + c*f - ((3*b*c*f + a*d*f*(-4 + n) - b*d*e*(-1 + n))*(e + f*x)*(
(b*e - a*f)^2*(-2 + n)*(-1 + n) - 2*b*(e + f*x)*(b*e*(-2 + n) - a*f*(-1 + n) - b*f*x)))/((b*e - a*f)^3*(-3 + n
)*(-2 + n)*(-1 + n))))/(f*(-(b*e) + a*f)*(-4 + n))

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 1188, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)*(f*x+e)^(-4+n)*(a^3*d*f^3*n^3*x-3*a^2*b*d*e*f^2*n^3*x+2*a^2*b*d*f^3*n^2*x^2+3*a*b^2*d*e^2*f*n^3*x-4*a*
b^2*d*e*f^2*n^2*x^2+2*a*b^2*d*f^3*n*x^3-b^3*d*e^3*n^3*x+2*b^3*d*e^2*f*n^2*x^2-2*b^3*d*e*f^2*n*x^3+a^3*c*f^3*n^
3-7*a^3*d*f^3*n^2*x-3*a^2*b*c*e*f^2*n^3+3*a^2*b*c*f^3*n^2*x+22*a^2*b*d*e*f^2*n^2*x-10*a^2*b*d*f^3*n*x^2+3*a*b^
2*c*e^2*f*n^3-6*a*b^2*c*e*f^2*n^2*x+6*a*b^2*c*f^3*n*x^2-23*a*b^2*d*e^2*f*n^2*x+20*a*b^2*d*e*f^2*n*x^2-8*a*b^2*
d*f^3*x^3-b^3*c*e^3*n^3+3*b^3*c*e^2*f*n^2*x-6*b^3*c*e*f^2*n*x^2+6*b^3*c*f^3*x^3+8*b^3*d*e^3*n^2*x-10*b^3*d*e^2
*f*n*x^2+2*b^3*d*e*f^2*x^3-6*a^3*c*f^3*n^2-a^3*d*e*f^2*n^2+14*a^3*d*f^3*n*x+21*a^2*b*c*e*f^2*n^2-9*a^2*b*c*f^3
*n*x+2*a^2*b*d*e^2*f*n^2-53*a^2*b*d*e*f^2*n*x+8*a^2*b*d*f^3*x^2-24*a*b^2*c*e^2*f*n^2+30*a*b^2*c*e*f^2*n*x-6*a*
b^2*c*f^3*x^2-a*b^2*d*e^3*n^2+58*a*b^2*d*e^2*f*n*x-34*a*b^2*d*e*f^2*x^2+9*b^3*c*e^3*n^2-21*b^3*c*e^2*f*n*x+24*
b^3*c*e*f^2*x^2-19*b^3*d*e^3*n*x+8*b^3*d*e^2*f*x^2+11*a^3*c*f^3*n+3*a^3*d*e*f^2*n-8*a^3*d*f^3*x-42*a^2*b*c*e*f
^2*n+6*a^2*b*c*f^3*x-10*a^2*b*d*e^2*f*n+34*a^2*b*d*e*f^2*x+57*a*b^2*c*e^2*f*n-24*a*b^2*c*e*f^2*x+7*a*b^2*d*e^3
*n-56*a*b^2*d*e^2*f*x-26*b^3*c*e^3*n+36*b^3*c*e^2*f*x+12*b^3*d*e^3*x-6*a^3*c*f^3-2*a^3*d*e*f^2+24*a^2*b*c*e*f^
2+8*a^2*b*d*e^2*f-36*a*b^2*c*e^2*f-12*a*b^2*d*e^3+24*b^3*c*e^3)/(a^4*f^4*n^4-4*a^3*b*e*f^3*n^4+6*a^2*b^2*e^2*f
^2*n^4-4*a*b^3*e^3*f*n^4+b^4*e^4*n^4-10*a^4*f^4*n^3+40*a^3*b*e*f^3*n^3-60*a^2*b^2*e^2*f^2*n^3+40*a*b^3*e^3*f*n
^3-10*b^4*e^4*n^3+35*a^4*f^4*n^2-140*a^3*b*e*f^3*n^2+210*a^2*b^2*e^2*f^2*n^2-140*a*b^3*e^3*f*n^2+35*b^4*e^4*n^
2-50*a^4*f^4*n+200*a^3*b*e*f^3*n-300*a^2*b^2*e^2*f^2*n+200*a*b^3*e^3*f*n-50*b^4*e^4*n+24*a^4*f^4-96*a^3*b*e*f^
3+144*a^2*b^2*e^2*f^2-96*a*b^3*e^3*f+24*b^4*e^4)/((b*x+a)^n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.93825, size = 3572, normalized size = 11.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*a^4*c*e*f^3 - 2*(b^4*d*e*f^3 + (3*b^4*c - 4*a*b^3*d)*f^4 - (b^4*d*e*f^3 - a*b^3*d*f^4)*n)*x^5 - 12*(2*a*b^
3*c - a^2*b^2*d)*e^4 + 4*(9*a^2*b^2*c - 2*a^3*b*d)*e^3*f - 2*(12*a^3*b*c - a^4*d)*e^2*f^2 - 2*(5*b^4*d*e^2*f^2
+ 5*(3*b^4*c - 4*a*b^3*d)*e*f^3 + (b^4*d*e^2*f^2 - 2*a*b^3*d*e*f^3 + a^2*b^2*d*f^4)*n^2 - (6*b^4*d*e^2*f^2 +
(3*b^4*c - 10*a*b^3*d)*e*f^3 - (3*a*b^3*c - 4*a^2*b^2*d)*f^4)*n)*x^4 + (a*b^3*c*e^4 - 3*a^2*b^2*c*e^3*f + 3*a^
3*b*c*e^2*f^2 - a^4*c*e*f^3)*n^3 - (20*b^4*d*e^3*f + 20*(3*b^4*c - 4*a*b^3*d)*e^2*f^2 - (b^4*d*e^3*f - 3*a*b^3
*d*e^2*f^2 + 3*a^2*b^2*d*e*f^3 - a^3*b*d*f^4)*n^3 + (10*b^4*d*e^3*f + (3*b^4*c - 25*a*b^3*d)*e^2*f^2 - 2*(3*a*
b^3*c - 10*a^2*b^2*d)*e*f^3 + (3*a^2*b^2*c - 5*a^3*b*d)*f^4)*n^2 - (29*b^4*d*e^3*f + 3*(9*b^4*c - 22*a*b^3*d)*
e^2*f^2 - (30*a*b^3*c - 41*a^2*b^2*d)*e*f^3 + (3*a^2*b^2*c - 4*a^3*b*d)*f^4)*n)*x^3 + (6*a^4*c*e*f^3 - (9*a*b^
3*c - a^2*b^2*d)*e^4 + 2*(12*a^2*b^2*c - a^3*b*d)*e^3*f - (21*a^3*b*c - a^4*d)*e^2*f^2)*n^2 - (12*b^4*d*e^4 -
48*a^2*b^2*d*e^2*f^2 + 32*a^3*b*d*e*f^3 - 8*a^4*d*f^4 + 12*(5*b^4*c - 4*a*b^3*d)*e^3*f - (b^4*d*e^4 - 3*a*b^3*
c*e^2*f^2 + (b^4*c - 2*a*b^3*d)*e^3*f + (3*a^2*b^2*c + 2*a^3*b*d)*e*f^3 - (a^3*b*c + a^4*d)*f^4)*n^3 + (8*b^4*
d*e^4 + 2*(6*b^4*c - 7*a*b^3*d)*e^3*f - 3*(9*a*b^3*c + a^2*b^2*d)*e^2*f^2 + 2*(9*a^2*b^2*c + 8*a^3*b*d)*e*f^3
- (3*a^3*b*c + 7*a^4*d)*f^4)*n^2 - (19*b^4*d*e^4 + (47*b^4*c - 36*a*b^3*d)*e^3*f - 15*(4*a*b^3*c + a^2*b^2*d)*
e^2*f^2 + (15*a^2*b^2*c + 46*a^3*b*d)*e*f^3 - 2*(a^3*b*c + 7*a^4*d)*f^4)*n)*x^2 - (11*a^4*c*e*f^3 - (26*a*b^3*
c - 7*a^2*b^2*d)*e^4 + (57*a^2*b^2*c - 10*a^3*b*d)*e^3*f - 3*(14*a^3*b*c - a^4*d)*e^2*f^2)*n - (24*b^4*c*e^4 -
6*a^4*c*f^4 + 12*(2*a*b^3*c - 5*a^2*b^2*d)*e^3*f - 4*(9*a^2*b^2*c - 10*a^3*b*d)*e^2*f^2 + 2*(12*a^3*b*c - 5*a
^4*d)*e*f^3 - (3*a^3*b*d*e^2*f^2 - a^4*c*f^4 + (b^4*c + a*b^3*d)*e^4 - (2*a*b^3*c + 3*a^2*b^2*d)*e^3*f + (2*a^
3*b*c - a^4*d)*e*f^3)*n^3 - (6*a^4*c*f^4 - (9*b^4*c + 7*a*b^3*d)*e^4 + 2*(6*a*b^3*c + 11*a^2*b^2*d)*e^3*f + (9
*a^2*b^2*c - 23*a^3*b*d)*e^2*f^2 - 2*(9*a^3*b*c - 4*a^4*d)*e*f^3)*n^2 + (11*a^4*c*f^4 - 2*(13*b^4*c + 6*a*b^3*
d)*e^4 + 5*(2*a*b^3*c + 11*a^2*b^2*d)*e^3*f + 15*(3*a^2*b^2*c - 4*a^3*b*d)*e^2*f^2 - (40*a^3*b*c - 17*a^4*d)*e
*f^3)*n)*x)*(f*x + e)^(n - 5)/((24*b^4*e^4 - 96*a*b^3*e^3*f + 144*a^2*b^2*e^2*f^2 - 96*a^3*b*e*f^3 + 24*a^4*f^
4 + (b^4*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n^4 - 10*(b^4*e^4 - 4*a*b^3*e^3*f
+ 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n^3 + 35*(b^4*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b
*e*f^3 + a^4*f^4)*n^2 - 50*(b^4*e^4 - 4*a*b^3*e^3*f + 6*a^2*b^2*e^2*f^2 - 4*a^3*b*e*f^3 + a^4*f^4)*n)*(b*x + a
)^n)

________________________________________________________________________________________

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((d*x+c)*(f*x+e)**(-5+n)/((b*x+a)**n),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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