3.31.52 \(\int (a+b x) (c+d x)^{-5+n} (e+f x)^{-n} \, dx\) [3052]

Optimal. Leaf size=299 \[ \frac {(b c-a d) (c+d x)^{-4+n} (e+f x)^{1-n}}{d (d e-c f) (4-n)}+\frac {(3 a d f+b (c f (1-n)-d e (4-n))) (c+d x)^{-3+n} (e+f x)^{1-n}}{d (d e-c f)^2 (3-n) (4-n)}-\frac {2 f (3 a d f+b (c f (1-n)-d e (4-n))) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f)^3 (2-n) (3-n) (4-n)}+\frac {2 f^2 (3 a d f+b (c f (1-n)-d e (4-n))) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f)^4 (1-n) (2-n) (3-n) (4-n)} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 296, 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 = {80, 47, 37} \begin {gather*} \frac {2 f^2 (c+d x)^{n-1} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (1-n) (2-n) (3-n) (4-n) (d e-c f)^4}+\frac {(b c-a d) (c+d x)^{n-4} (e+f x)^{1-n}}{d (4-n) (d e-c f)}+\frac {(c+d x)^{n-3} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (3-n) (4-n) (d e-c f)^2}-\frac {2 f (c+d x)^{n-2} (e+f x)^{1-n} (3 a d f+b c f (1-n)-b d e (4-n))}{d (2-n) (3-n) (4-n) (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 47

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

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Mathematica [A]
time = 0.27, size = 165, normalized size = 0.55 \begin {gather*} \frac {(c+d x)^{-4+n} (e+f x)^{1-n} \left (-b c+a d+\frac {(3 a d f+b d e (-4+n)-b c f (-1+n)) (c+d x) \left (c^2 f^2 \left (6-5 n+n^2\right )-2 c d f (-3+n) (e (-1+n)+f x)+d^2 \left (e^2 \left (2-3 n+n^2\right )+2 e f (-1+n) x+2 f^2 x^2\right )\right )}{(d e-c f)^3 (-3+n) (-2+n) (-1+n)}\right )}{d (d e-c f) (-4+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^(-4 + n)*(e + f*x)^(1 - n)*(-(b*c) + a*d + ((3*a*d*f + b*d*e*(-4 + n) - b*c*f*(-1 + n))*(c + d*x)*(
c^2*f^2*(6 - 5*n + n^2) - 2*c*d*f*(-3 + n)*(e*(-1 + n) + f*x) + d^2*(e^2*(2 - 3*n + n^2) + 2*e*f*(-1 + n)*x +
2*f^2*x^2)))/((d*e - c*f)^3*(-3 + n)*(-2 + n)*(-1 + n))))/(d*(d*e - c*f)*(-4 + n))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1186\) vs. \(2(299)=598\).
time = 0.10, size = 1187, normalized size = 3.97

method	result	size
gosper	\(-\frac {\left (d x +c \right )^{-4+n} \left (f x +e \right ) \left (b \,c^{3} f^{3} n^{3} x -3 b \,c^{2} d e \,f^{2} n^{3} x -2 b \,c^{2} d \,f^{3} n^{2} x^{2}+3 b c \,d^{2} e^{2} f \,n^{3} x +4 b c \,d^{2} e \,f^{2} n^{2} x^{2}+2 b c \,d^{2} f^{3} n \,x^{3}-b \,d^{3} e^{3} n^{3} x -2 b \,d^{3} e^{2} f \,n^{2} x^{2}-2 b \,d^{3} e \,f^{2} n \,x^{3}+a \,c^{3} f^{3} n^{3}-3 a \,c^{2} d e \,f^{2} n^{3}-3 a \,c^{2} d \,f^{3} n^{2} x +3 a c \,d^{2} e^{2} f \,n^{3}+6 a c \,d^{2} e \,f^{2} n^{2} x +6 a c \,d^{2} f^{3} n \,x^{2}-a \,d^{3} e^{3} n^{3}-3 a \,d^{3} e^{2} f \,n^{2} x -6 a \,d^{3} e \,f^{2} n \,x^{2}-6 a \,d^{3} f^{3} x^{3}-8 b \,c^{3} f^{3} n^{2} x +23 b \,c^{2} d e \,f^{2} n^{2} x +10 b \,c^{2} d \,f^{3} n \,x^{2}-22 b c \,d^{2} e^{2} f \,n^{2} x -20 b c \,d^{2} e \,f^{2} n \,x^{2}-2 b c \,d^{2} f^{3} x^{3}+7 b \,d^{3} e^{3} n^{2} x +10 b \,d^{3} e^{2} f n \,x^{2}+8 b \,d^{3} e \,f^{2} x^{3}-9 a \,c^{3} f^{3} n^{2}+24 a \,c^{2} d e \,f^{2} n^{2}+21 a \,c^{2} d \,f^{3} n x -21 a c \,d^{2} e^{2} f \,n^{2}-30 a c \,d^{2} e \,f^{2} n x -24 a c \,d^{2} f^{3} x^{2}+6 a \,d^{3} e^{3} n^{2}+9 a \,d^{3} e^{2} f n x +6 a \,d^{3} e \,f^{2} x^{2}+b \,c^{3} e \,f^{2} n^{2}+19 b \,c^{3} f^{3} n x -2 b \,c^{2} d \,e^{2} f \,n^{2}-58 b \,c^{2} d e \,f^{2} n x -8 b \,c^{2} d \,f^{3} x^{2}+b c \,d^{2} e^{3} n^{2}+53 b c \,d^{2} e^{2} f n x +34 b c \,d^{2} e \,f^{2} x^{2}-14 b \,d^{3} e^{3} n x -8 b \,d^{3} e^{2} f \,x^{2}+26 a \,c^{3} f^{3} n -57 a \,c^{2} d e \,f^{2} n -36 a \,c^{2} d \,f^{3} x +42 a c \,d^{2} e^{2} f n +24 a c \,d^{2} e \,f^{2} x -11 a \,d^{3} e^{3} n -6 a \,d^{3} e^{2} f x -7 b \,c^{3} e \,f^{2} n -12 b \,c^{3} f^{3} x +10 b \,c^{2} d \,e^{2} f n +56 b \,c^{2} d e \,f^{2} x -3 b c \,d^{2} e^{3} n -34 b c \,d^{2} e^{2} f x +8 b \,d^{3} e^{3} x -24 a \,c^{3} f^{3}+36 a \,c^{2} d e \,f^{2}-24 a c \,d^{2} e^{2} f +6 a \,d^{3} e^{3}+12 b \,c^{3} e \,f^{2}-8 b \,c^{2} d \,e^{2} f +2 b c \,d^{2} e^{3}\right ) \left (f x +e \right )^{-n}}{c^{4} f^{4} n^{4}-4 c^{3} d e \,f^{3} n^{4}+6 c^{2} d^{2} e^{2} f^{2} n^{4}-4 c \,d^{3} e^{3} f \,n^{4}+d^{4} e^{4} n^{4}-10 c^{4} f^{4} n^{3}+40 c^{3} d e \,f^{3} n^{3}-60 c^{2} d^{2} e^{2} f^{2} n^{3}+40 c \,d^{3} e^{3} f \,n^{3}-10 d^{4} e^{4} n^{3}+35 c^{4} f^{4} n^{2}-140 c^{3} d e \,f^{3} n^{2}+210 c^{2} d^{2} e^{2} f^{2} n^{2}-140 c \,d^{3} e^{3} f \,n^{2}+35 d^{4} e^{4} n^{2}-50 c^{4} f^{4} n +200 c^{3} d e \,f^{3} n -300 c^{2} d^{2} e^{2} f^{2} n +200 c \,d^{3} e^{3} f n -50 d^{4} e^{4} n +24 c^{4} f^{4}-96 c^{3} d e \,f^{3}+144 c^{2} d^{2} e^{2} f^{2}-96 c \,d^{3} e^{3} f +24 d^{4} e^{4}}\)	\(1187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1734 vs. \(2 (281) = 562\).
time = 2.05, size = 1734, normalized size = 5.80 \begin {gather*} -\frac {{\left (2 \, {\left (b c d^{3} f^{4} n - {\left (b c d^{3} + 3 \, a d^{4}\right )} f^{4}\right )} x^{5} - 2 \, {\left (b c^{2} d^{2} f^{4} n^{2} - 3 \, {\left (2 \, b c^{2} d^{2} + a c d^{3}\right )} f^{4} n + 5 \, {\left (b c^{2} d^{2} + 3 \, a c d^{3}\right )} f^{4}\right )} x^{4} + {\left (b c^{3} d f^{4} n^{3} - {\left (10 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{4} n^{2} + {\left (29 \, b c^{3} d + 27 \, a c^{2} d^{2}\right )} f^{4} n - 20 \, {\left (b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{4}\right )} x^{3} + {\left ({\left (b c^{4} + a c^{3} d\right )} f^{4} n^{3} - 4 \, {\left (2 \, b c^{4} + 3 \, a c^{3} d\right )} f^{4} n^{2} + {\left (19 \, b c^{4} + 47 \, a c^{3} d\right )} f^{4} n - 12 \, {\left (b c^{4} + 5 \, a c^{3} d\right )} f^{4}\right )} x^{2} + {\left (a c^{4} f^{4} n^{3} - 9 \, a c^{4} f^{4} n^{2} + 26 \, a c^{4} f^{4} n - 24 \, a c^{4} f^{4}\right )} x - {\left (a c d^{3} n^{3} - 2 \, b c^{2} d^{2} - 6 \, a c d^{3} - {\left (b c^{2} d^{2} + 6 \, a c d^{3}\right )} n^{2} + {\left (b d^{4} n^{3} - 7 \, b d^{4} n^{2} + 14 \, b d^{4} n - 8 \, b d^{4}\right )} x^{2} + {\left (3 \, b c^{2} d^{2} + 11 \, a c d^{3}\right )} n - {\left (10 \, b c d^{3} + 6 \, a d^{4} - {\left (b c d^{3} + a d^{4}\right )} n^{3} + 2 \, {\left (4 \, b c d^{3} + 3 \, a d^{4}\right )} n^{2} - {\left (17 \, b c d^{3} + 11 \, a d^{4}\right )} n\right )} x\right )} e^{4} + {\left (3 \, a c^{2} d^{2} f n^{3} - {\left (2 \, b c^{3} d + 21 \, a c^{2} d^{2}\right )} f n^{2} - {\left (b d^{4} f n^{3} - 5 \, b d^{4} f n^{2} + 4 \, b d^{4} f n\right )} x^{3} + 2 \, {\left (5 \, b c^{3} d + 21 \, a c^{2} d^{2}\right )} f n - {\left (32 \, b c d^{3} f - {\left (2 \, b c d^{3} - a d^{4}\right )} f n^{3} + {\left (16 \, b c d^{3} - 3 \, a d^{4}\right )} f n^{2} - 2 \, {\left (23 \, b c d^{3} - a d^{4}\right )} f n\right )} x^{2} - 8 \, {\left (b c^{3} d + 3 \, a c^{2} d^{2}\right )} f + {\left ({\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} f n^{3} - {\left (23 \, b c^{2} d^{2} + 18 \, a c d^{3}\right )} f n^{2} + 20 \, {\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} f n - 8 \, {\left (5 \, b c^{2} d^{2} + 3 \, a c d^{3}\right )} f\right )} x\right )} e^{3} - {\left (3 \, a c^{3} d f^{2} n^{3} - {\left (b c^{4} + 24 \, a c^{3} d\right )} f^{2} n^{2} + 2 \, {\left (b d^{4} f^{2} n^{2} - 4 \, b d^{4} f^{2} n\right )} x^{4} + {\left (7 \, b c^{4} + 57 \, a c^{3} d\right )} f^{2} n - {\left (3 \, b c d^{3} f^{2} n^{3} - {\left (20 \, b c d^{3} + 3 \, a d^{4}\right )} f^{2} n^{2} + {\left (41 \, b c d^{3} + 3 \, a d^{4}\right )} f^{2} n\right )} x^{3} - 12 \, {\left (b c^{4} + 3 \, a c^{3} d\right )} f^{2} - 3 \, {\left (a c d^{3} f^{2} n^{3} + 16 \, b c^{2} d^{2} f^{2} + {\left (b c^{2} d^{2} - 6 \, a c d^{3}\right )} f^{2} n^{2} - 5 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} f^{2} n\right )} x^{2} + {\left (3 \, b c^{3} d f^{2} n^{3} - {\left (22 \, b c^{3} d + 9 \, a c^{2} d^{2}\right )} f^{2} n^{2} + 5 \, {\left (11 \, b c^{3} d + 9 \, a c^{2} d^{2}\right )} f^{2} n - 12 \, {\left (5 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{2}\right )} x\right )} e^{2} + {\left (a c^{4} f^{3} n^{3} - 9 \, a c^{4} f^{3} n^{2} + 26 \, a c^{4} f^{3} n - 24 \, a c^{4} f^{3} - 2 \, {\left (b d^{4} f^{3} n - 4 \, b d^{4} f^{3}\right )} x^{5} + 2 \, {\left (2 \, b c d^{3} f^{3} n^{2} + 20 \, b c d^{3} f^{3} - {\left (10 \, b c d^{3} + 3 \, a d^{4}\right )} f^{3} n\right )} x^{4} - {\left (3 \, b c^{2} d^{2} f^{3} n^{3} - 80 \, b c^{2} d^{2} f^{3} - {\left (25 \, b c^{2} d^{2} + 6 \, a c d^{3}\right )} f^{3} n^{2} + 6 \, {\left (11 \, b c^{2} d^{2} + 5 \, a c d^{3}\right )} f^{3} n\right )} x^{3} + {\left (48 \, b c^{3} d f^{3} - {\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} f^{3} n^{3} + {\left (14 \, b c^{3} d + 27 \, a c^{2} d^{2}\right )} f^{3} n^{2} - 12 \, {\left (3 \, b c^{3} d + 5 \, a c^{2} d^{2}\right )} f^{3} n\right )} x^{2} - {\left (24 \, a c^{3} d f^{3} - {\left (b c^{4} - 2 \, a c^{3} d\right )} f^{3} n^{3} + {\left (7 \, b c^{4} - 12 \, a c^{3} d\right )} f^{3} n^{2} - 2 \, {\left (6 \, b c^{4} - 5 \, a c^{3} d\right )} f^{3} n\right )} x\right )} e\right )} {\left (d x + c\right )}^{n - 5}}{{\left (c^{4} f^{4} n^{4} - 10 \, c^{4} f^{4} n^{3} + 35 \, c^{4} f^{4} n^{2} - 50 \, c^{4} f^{4} n + 24 \, c^{4} f^{4} + {\left (d^{4} n^{4} - 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} - 50 \, d^{4} n + 24 \, d^{4}\right )} e^{4} - 4 \, {\left (c d^{3} f n^{4} - 10 \, c d^{3} f n^{3} + 35 \, c d^{3} f n^{2} - 50 \, c d^{3} f n + 24 \, c d^{3} f\right )} e^{3} + 6 \, {\left (c^{2} d^{2} f^{2} n^{4} - 10 \, c^{2} d^{2} f^{2} n^{3} + 35 \, c^{2} d^{2} f^{2} n^{2} - 50 \, c^{2} d^{2} f^{2} n + 24 \, c^{2} d^{2} f^{2}\right )} e^{2} - 4 \, {\left (c^{3} d f^{3} n^{4} - 10 \, c^{3} d f^{3} n^{3} + 35 \, c^{3} d f^{3} n^{2} - 50 \, c^{3} d f^{3} n + 24 \, c^{3} d f^{3}\right )} e\right )} {\left (f x + e\right )}^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(b*c*d^3*f^4*n - (b*c*d^3 + 3*a*d^4)*f^4)*x^5 - 2*(b*c^2*d^2*f^4*n^2 - 3*(2*b*c^2*d^2 + a*c*d^3)*f^4*n + 5
*(b*c^2*d^2 + 3*a*c*d^3)*f^4)*x^4 + (b*c^3*d*f^4*n^3 - (10*b*c^3*d + 3*a*c^2*d^2)*f^4*n^2 + (29*b*c^3*d + 27*a
*c^2*d^2)*f^4*n - 20*(b*c^3*d + 3*a*c^2*d^2)*f^4)*x^3 + ((b*c^4 + a*c^3*d)*f^4*n^3 - 4*(2*b*c^4 + 3*a*c^3*d)*f
^4*n^2 + (19*b*c^4 + 47*a*c^3*d)*f^4*n - 12*(b*c^4 + 5*a*c^3*d)*f^4)*x^2 + (a*c^4*f^4*n^3 - 9*a*c^4*f^4*n^2 +
26*a*c^4*f^4*n - 24*a*c^4*f^4)*x - (a*c*d^3*n^3 - 2*b*c^2*d^2 - 6*a*c*d^3 - (b*c^2*d^2 + 6*a*c*d^3)*n^2 + (b*d
^4*n^3 - 7*b*d^4*n^2 + 14*b*d^4*n - 8*b*d^4)*x^2 + (3*b*c^2*d^2 + 11*a*c*d^3)*n - (10*b*c*d^3 + 6*a*d^4 - (b*c
*d^3 + a*d^4)*n^3 + 2*(4*b*c*d^3 + 3*a*d^4)*n^2 - (17*b*c*d^3 + 11*a*d^4)*n)*x)*e^4 + (3*a*c^2*d^2*f*n^3 - (2*
b*c^3*d + 21*a*c^2*d^2)*f*n^2 - (b*d^4*f*n^3 - 5*b*d^4*f*n^2 + 4*b*d^4*f*n)*x^3 + 2*(5*b*c^3*d + 21*a*c^2*d^2)
*f*n - (32*b*c*d^3*f - (2*b*c*d^3 - a*d^4)*f*n^3 + (16*b*c*d^3 - 3*a*d^4)*f*n^2 - 2*(23*b*c*d^3 - a*d^4)*f*n)*
x^2 - 8*(b*c^3*d + 3*a*c^2*d^2)*f + ((3*b*c^2*d^2 + 2*a*c*d^3)*f*n^3 - (23*b*c^2*d^2 + 18*a*c*d^3)*f*n^2 + 20*
(3*b*c^2*d^2 + 2*a*c*d^3)*f*n - 8*(5*b*c^2*d^2 + 3*a*c*d^3)*f)*x)*e^3 - (3*a*c^3*d*f^2*n^3 - (b*c^4 + 24*a*c^3
*d)*f^2*n^2 + 2*(b*d^4*f^2*n^2 - 4*b*d^4*f^2*n)*x^4 + (7*b*c^4 + 57*a*c^3*d)*f^2*n - (3*b*c*d^3*f^2*n^3 - (20*
b*c*d^3 + 3*a*d^4)*f^2*n^2 + (41*b*c*d^3 + 3*a*d^4)*f^2*n)*x^3 - 12*(b*c^4 + 3*a*c^3*d)*f^2 - 3*(a*c*d^3*f^2*n
^3 + 16*b*c^2*d^2*f^2 + (b*c^2*d^2 - 6*a*c*d^3)*f^2*n^2 - 5*(b*c^2*d^2 - a*c*d^3)*f^2*n)*x^2 + (3*b*c^3*d*f^2*
n^3 - (22*b*c^3*d + 9*a*c^2*d^2)*f^2*n^2 + 5*(11*b*c^3*d + 9*a*c^2*d^2)*f^2*n - 12*(5*b*c^3*d + 3*a*c^2*d^2)*f
^2)*x)*e^2 + (a*c^4*f^3*n^3 - 9*a*c^4*f^3*n^2 + 26*a*c^4*f^3*n - 24*a*c^4*f^3 - 2*(b*d^4*f^3*n - 4*b*d^4*f^3)*
x^5 + 2*(2*b*c*d^3*f^3*n^2 + 20*b*c*d^3*f^3 - (10*b*c*d^3 + 3*a*d^4)*f^3*n)*x^4 - (3*b*c^2*d^2*f^3*n^3 - 80*b*
c^2*d^2*f^3 - (25*b*c^2*d^2 + 6*a*c*d^3)*f^3*n^2 + 6*(11*b*c^2*d^2 + 5*a*c*d^3)*f^3*n)*x^3 + (48*b*c^3*d*f^3 -
(2*b*c^3*d + 3*a*c^2*d^2)*f^3*n^3 + (14*b*c^3*d + 27*a*c^2*d^2)*f^3*n^2 - 12*(3*b*c^3*d + 5*a*c^2*d^2)*f^3*n)
*x^2 - (24*a*c^3*d*f^3 - (b*c^4 - 2*a*c^3*d)*f^3*n^3 + (7*b*c^4 - 12*a*c^3*d)*f^3*n^2 - 2*(6*b*c^4 - 5*a*c^3*d
)*f^3*n)*x)*e)*(d*x + c)^(n - 5)/((c^4*f^4*n^4 - 10*c^4*f^4*n^3 + 35*c^4*f^4*n^2 - 50*c^4*f^4*n + 24*c^4*f^4 +
(d^4*n^4 - 10*d^4*n^3 + 35*d^4*n^2 - 50*d^4*n + 24*d^4)*e^4 - 4*(c*d^3*f*n^4 - 10*c*d^3*f*n^3 + 35*c*d^3*f*n^
2 - 50*c*d^3*f*n + 24*c*d^3*f)*e^3 + 6*(c^2*d^2*f^2*n^4 - 10*c^2*d^2*f^2*n^3 + 35*c^2*d^2*f^2*n^2 - 50*c^2*d^2
*f^2*n + 24*c^2*d^2*f^2)*e^2 - 4*(c^3*d*f^3*n^4 - 10*c^3*d*f^3*n^3 + 35*c^3*d*f^3*n^2 - 50*c^3*d*f^3*n + 24*c^
3*d*f^3)*e)*(f*x + e)^n)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6191 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.67, size = 1657, normalized size = 5.54 \begin {gather*} \frac {x\,{\left (c+d\,x\right )}^{n-5}\,\left (-b\,c^4\,e\,f^3\,n^3+7\,b\,c^4\,e\,f^3\,n^2-12\,b\,c^4\,e\,f^3\,n-a\,c^4\,f^4\,n^3+9\,a\,c^4\,f^4\,n^2-26\,a\,c^4\,f^4\,n+24\,a\,c^4\,f^4+3\,b\,c^3\,d\,e^2\,f^2\,n^3-22\,b\,c^3\,d\,e^2\,f^2\,n^2+55\,b\,c^3\,d\,e^2\,f^2\,n-60\,b\,c^3\,d\,e^2\,f^2+2\,a\,c^3\,d\,e\,f^3\,n^3-12\,a\,c^3\,d\,e\,f^3\,n^2+10\,a\,c^3\,d\,e\,f^3\,n+24\,a\,c^3\,d\,e\,f^3-3\,b\,c^2\,d^2\,e^3\,f\,n^3+23\,b\,c^2\,d^2\,e^3\,f\,n^2-60\,b\,c^2\,d^2\,e^3\,f\,n+40\,b\,c^2\,d^2\,e^3\,f-9\,a\,c^2\,d^2\,e^2\,f^2\,n^2+45\,a\,c^2\,d^2\,e^2\,f^2\,n-36\,a\,c^2\,d^2\,e^2\,f^2+b\,c\,d^3\,e^4\,n^3-8\,b\,c\,d^3\,e^4\,n^2+17\,b\,c\,d^3\,e^4\,n-10\,b\,c\,d^3\,e^4-2\,a\,c\,d^3\,e^3\,f\,n^3+18\,a\,c\,d^3\,e^3\,f\,n^2-40\,a\,c\,d^3\,e^3\,f\,n+24\,a\,c\,d^3\,e^3\,f+a\,d^4\,e^4\,n^3-6\,a\,d^4\,e^4\,n^2+11\,a\,d^4\,e^4\,n-6\,a\,d^4\,e^4\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}-\frac {{\left (c+d\,x\right )}^{n-5}\,\left (b\,c^4\,e^2\,f^2\,n^2-7\,b\,c^4\,e^2\,f^2\,n+12\,b\,c^4\,e^2\,f^2+a\,c^4\,e\,f^3\,n^3-9\,a\,c^4\,e\,f^3\,n^2+26\,a\,c^4\,e\,f^3\,n-24\,a\,c^4\,e\,f^3-2\,b\,c^3\,d\,e^3\,f\,n^2+10\,b\,c^3\,d\,e^3\,f\,n-8\,b\,c^3\,d\,e^3\,f-3\,a\,c^3\,d\,e^2\,f^2\,n^3+24\,a\,c^3\,d\,e^2\,f^2\,n^2-57\,a\,c^3\,d\,e^2\,f^2\,n+36\,a\,c^3\,d\,e^2\,f^2+b\,c^2\,d^2\,e^4\,n^2-3\,b\,c^2\,d^2\,e^4\,n+2\,b\,c^2\,d^2\,e^4+3\,a\,c^2\,d^2\,e^3\,f\,n^3-21\,a\,c^2\,d^2\,e^3\,f\,n^2+42\,a\,c^2\,d^2\,e^3\,f\,n-24\,a\,c^2\,d^2\,e^3\,f-a\,c\,d^3\,e^4\,n^3+6\,a\,c\,d^3\,e^4\,n^2-11\,a\,c\,d^3\,e^4\,n+6\,a\,c\,d^3\,e^4\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {x^2\,{\left (c+d\,x\right )}^{n-5}\,\left (-b\,c^4\,f^4\,n^3+8\,b\,c^4\,f^4\,n^2-19\,b\,c^4\,f^4\,n+12\,b\,c^4\,f^4+2\,b\,c^3\,d\,e\,f^3\,n^3-14\,b\,c^3\,d\,e\,f^3\,n^2+36\,b\,c^3\,d\,e\,f^3\,n-48\,b\,c^3\,d\,e\,f^3-a\,c^3\,d\,f^4\,n^3+12\,a\,c^3\,d\,f^4\,n^2-47\,a\,c^3\,d\,f^4\,n+60\,a\,c^3\,d\,f^4-3\,b\,c^2\,d^2\,e^2\,f^2\,n^2+15\,b\,c^2\,d^2\,e^2\,f^2\,n-48\,b\,c^2\,d^2\,e^2\,f^2+3\,a\,c^2\,d^2\,e\,f^3\,n^3-27\,a\,c^2\,d^2\,e\,f^3\,n^2+60\,a\,c^2\,d^2\,e\,f^3\,n-2\,b\,c\,d^3\,e^3\,f\,n^3+16\,b\,c\,d^3\,e^3\,f\,n^2-46\,b\,c\,d^3\,e^3\,f\,n+32\,b\,c\,d^3\,e^3\,f-3\,a\,c\,d^3\,e^2\,f^2\,n^3+18\,a\,c\,d^3\,e^2\,f^2\,n^2-15\,a\,c\,d^3\,e^2\,f^2\,n+b\,d^4\,e^4\,n^3-7\,b\,d^4\,e^4\,n^2+14\,b\,d^4\,e^4\,n-8\,b\,d^4\,e^4+a\,d^4\,e^3\,f\,n^3-3\,a\,d^4\,e^3\,f\,n^2+2\,a\,d^4\,e^3\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {2\,d^3\,f^3\,x^5\,{\left (c+d\,x\right )}^{n-5}\,\left (3\,a\,d\,f+b\,c\,f-4\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {2\,d^2\,f^2\,x^4\,{\left (c+d\,x\right )}^{n-5}\,\left (5\,c\,f-c\,f\,n+d\,e\,n\right )\,\left (3\,a\,d\,f+b\,c\,f-4\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )}+\frac {d\,f\,x^3\,{\left (c+d\,x\right )}^{n-5}\,\left (3\,a\,d\,f+b\,c\,f-4\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )\,\left (c^2\,f^2\,n^2-9\,c^2\,f^2\,n+20\,c^2\,f^2-2\,c\,d\,e\,f\,n^2+10\,c\,d\,e\,f\,n+d^2\,e^2\,n^2-d^2\,e^2\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^4\,\left (n^4-10\,n^3+35\,n^2-50\,n+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(c + d*x)^(n - 5)*(24*a*c^4*f^4 - 6*a*d^4*e^4 + 9*a*c^4*f^4*n^2 - 6*a*d^4*e^4*n^2 - a*c^4*f^4*n^3 + a*d^4*e
^4*n^3 - 10*b*c*d^3*e^4 - 26*a*c^4*f^4*n + 11*a*d^4*e^4*n + 24*a*c*d^3*e^3*f + 24*a*c^3*d*e*f^3 + 17*b*c*d^3*e
^4*n - 12*b*c^4*e*f^3*n + 40*b*c^2*d^2*e^3*f - 60*b*c^3*d*e^2*f^2 - 8*b*c*d^3*e^4*n^2 + b*c*d^3*e^4*n^3 + 7*b*
c^4*e*f^3*n^2 - b*c^4*e*f^3*n^3 - 36*a*c^2*d^2*e^2*f^2 + 45*a*c^2*d^2*e^2*f^2*n + 23*b*c^2*d^2*e^3*f*n^2 - 22*
b*c^3*d*e^2*f^2*n^2 - 3*b*c^2*d^2*e^3*f*n^3 + 3*b*c^3*d*e^2*f^2*n^3 - 40*a*c*d^3*e^3*f*n + 10*a*c^3*d*e*f^3*n
- 9*a*c^2*d^2*e^2*f^2*n^2 + 18*a*c*d^3*e^3*f*n^2 - 12*a*c^3*d*e*f^3*n^2 - 2*a*c*d^3*e^3*f*n^3 + 2*a*c^3*d*e*f^
3*n^3 - 60*b*c^2*d^2*e^3*f*n + 55*b*c^3*d*e^2*f^2*n))/((e + f*x)^n*(c*f - d*e)^4*(35*n^2 - 50*n - 10*n^3 + n^4
+ 24)) - ((c + d*x)^(n - 5)*(2*b*c^2*d^2*e^4 + 12*b*c^4*e^2*f^2 + 6*a*c*d^3*e^4 - 24*a*c^4*e*f^3 - 8*b*c^3*d*
e^3*f - 11*a*c*d^3*e^4*n + 26*a*c^4*e*f^3*n - 24*a*c^2*d^2*e^3*f + 36*a*c^3*d*e^2*f^2 + 6*a*c*d^3*e^4*n^2 - a*
c*d^3*e^4*n^3 - 3*b*c^2*d^2*e^4*n - 9*a*c^4*e*f^3*n^2 + a*c^4*e*f^3*n^3 - 7*b*c^4*e^2*f^2*n + b*c^2*d^2*e^4*n^
2 + b*c^4*e^2*f^2*n^2 - 21*a*c^2*d^2*e^3*f*n^2 + 24*a*c^3*d*e^2*f^2*n^2 + 3*a*c^2*d^2*e^3*f*n^3 - 3*a*c^3*d*e^
2*f^2*n^3 + 10*b*c^3*d*e^3*f*n + 42*a*c^2*d^2*e^3*f*n - 57*a*c^3*d*e^2*f^2*n - 2*b*c^3*d*e^3*f*n^2))/((e + f*x
)^n*(c*f - d*e)^4*(35*n^2 - 50*n - 10*n^3 + n^4 + 24)) + (x^2*(c + d*x)^(n - 5)*(12*b*c^4*f^4 - 8*b*d^4*e^4 +
8*b*c^4*f^4*n^2 - 7*b*d^4*e^4*n^2 - b*c^4*f^4*n^3 + b*d^4*e^4*n^3 + 60*a*c^3*d*f^4 - 19*b*c^4*f^4*n + 14*b*d^4
*e^4*n + 32*b*c*d^3*e^3*f - 48*b*c^3*d*e*f^3 - 47*a*c^3*d*f^4*n + 2*a*d^4*e^3*f*n + 12*a*c^3*d*f^4*n^2 - a*c^3
*d*f^4*n^3 - 3*a*d^4*e^3*f*n^2 + a*d^4*e^3*f*n^3 - 48*b*c^2*d^2*e^2*f^2 + 18*a*c*d^3*e^2*f^2*n^2 - 27*a*c^2*d^
2*e*f^3*n^2 - 3*a*c*d^3*e^2*f^2*n^3 + 3*a*c^2*d^2*e*f^3*n^3 + 15*b*c^2*d^2*e^2*f^2*n - 46*b*c*d^3*e^3*f*n + 36
*b*c^3*d*e*f^3*n - 3*b*c^2*d^2*e^2*f^2*n^2 - 15*a*c*d^3*e^2*f^2*n + 60*a*c^2*d^2*e*f^3*n + 16*b*c*d^3*e^3*f*n^
2 - 14*b*c^3*d*e*f^3*n^2 - 2*b*c*d^3*e^3*f*n^3 + 2*b*c^3*d*e*f^3*n^3))/((e + f*x)^n*(c*f - d*e)^4*(35*n^2 - 50
*n - 10*n^3 + n^4 + 24)) + (2*d^3*f^3*x^5*(c + d*x)^(n - 5)*(3*a*d*f + b*c*f - 4*b*d*e - b*c*f*n + b*d*e*n))/(
(e + f*x)^n*(c*f - d*e)^4*(35*n^2 - 50*n - 10*n^3 + n^4 + 24)) + (2*d^2*f^2*x^4*(c + d*x)^(n - 5)*(5*c*f - c*f
*n + d*e*n)*(3*a*d*f + b*c*f - 4*b*d*e - b*c*f*n + b*d*e*n))/((e + f*x)^n*(c*f - d*e)^4*(35*n^2 - 50*n - 10*n^
3 + n^4 + 24)) + (d*f*x^3*(c + d*x)^(n - 5)*(3*a*d*f + b*c*f - 4*b*d*e - b*c*f*n + b*d*e*n)*(20*c^2*f^2 - 9*c^
2*f^2*n - d^2*e^2*n + c^2*f^2*n^2 + d^2*e^2*n^2 + 10*c*d*e*f*n - 2*c*d*e*f*n^2))/((e + f*x)^n*(c*f - d*e)^4*(3
5*n^2 - 50*n - 10*n^3 + n^4 + 24))

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