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

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

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Rubi [A]  time = 0.20, 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 = {79, 45, 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 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 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]

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 = 0.29, size = 165, normalized size = 0.55 \begin {gather*} \frac {(c+d x)^{n-4} (e+f x)^{1-n} \left (\frac {(c+d x) \left (c^2 f^2 \left (n^2-5 n+6\right )-2 c d f (n-3) (e (n-1)+f x)+d^2 \left (e^2 \left (n^2-3 n+2\right )+2 e f (n-1) x+2 f^2 x^2\right )\right ) (3 a d f-b c f (n-1)+b d e (n-4))}{(n-3) (n-2) (n-1) (d e-c f)^3}+a d-b c\right )}{d (n-4) (d e-c f)} \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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IntegrateAlgebraic [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (c+d x)^{-5+n} (e+f x)^{-n} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.35, size = 1741, normalized size = 5.82

result too large to display

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]

(24*a*c^4*e*f^3 - 2*(4*b*d^4*e*f^3 - (b*c*d^3 + 3*a*d^4)*f^4 - (b*d^4*e*f^3 - b*c*d^3*f^4)*n)*x^5 - 2*(b*c^2*d
^2 + 3*a*c*d^3)*e^4 + 8*(b*c^3*d + 3*a*c^2*d^2)*e^3*f - 12*(b*c^4 + 3*a*c^3*d)*e^2*f^2 - 2*(20*b*c*d^3*e*f^3 -
5*(b*c^2*d^2 + 3*a*c*d^3)*f^4 - (b*d^4*e^2*f^2 - 2*b*c*d^3*e*f^3 + b*c^2*d^2*f^4)*n^2 + (4*b*d^4*e^2*f^2 - (1
0*b*c*d^3 + 3*a*d^4)*e*f^3 + 3*(2*b*c^2*d^2 + a*c*d^3)*f^4)*n)*x^4 + (a*c*d^3*e^4 - 3*a*c^2*d^2*e^3*f + 3*a*c^
3*d*e^2*f^2 - a*c^4*e*f^3)*n^3 - (80*b*c^2*d^2*e*f^3 - 20*(b*c^3*d + 3*a*c^2*d^2)*f^4 - (b*d^4*e^3*f - 3*b*c*d
^3*e^2*f^2 + 3*b*c^2*d^2*e*f^3 - b*c^3*d*f^4)*n^3 + (5*b*d^4*e^3*f - (20*b*c*d^3 + 3*a*d^4)*e^2*f^2 + (25*b*c^
2*d^2 + 6*a*c*d^3)*e*f^3 - (10*b*c^3*d + 3*a*c^2*d^2)*f^4)*n^2 - (4*b*d^4*e^3*f - (41*b*c*d^3 + 3*a*d^4)*e^2*f
^2 + 6*(11*b*c^2*d^2 + 5*a*c*d^3)*e*f^3 - (29*b*c^3*d + 27*a*c^2*d^2)*f^4)*n)*x^3 + (9*a*c^4*e*f^3 - (b*c^2*d^
2 + 6*a*c*d^3)*e^4 + (2*b*c^3*d + 21*a*c^2*d^2)*e^3*f - (b*c^4 + 24*a*c^3*d)*e^2*f^2)*n^2 - (8*b*d^4*e^4 - 32*
b*c*d^3*e^3*f + 48*b*c^2*d^2*e^2*f^2 + 48*b*c^3*d*e*f^3 - 12*(b*c^4 + 5*a*c^3*d)*f^4 - (b*d^4*e^4 - 3*a*c*d^3*
e^2*f^2 - (2*b*c*d^3 - a*d^4)*e^3*f + (2*b*c^3*d + 3*a*c^2*d^2)*e*f^3 - (b*c^4 + a*c^3*d)*f^4)*n^3 + (7*b*d^4*
e^4 - (16*b*c*d^3 - 3*a*d^4)*e^3*f + 3*(b*c^2*d^2 - 6*a*c*d^3)*e^2*f^2 + (14*b*c^3*d + 27*a*c^2*d^2)*e*f^3 - 4
*(2*b*c^4 + 3*a*c^3*d)*f^4)*n^2 - (14*b*d^4*e^4 - 2*(23*b*c*d^3 - a*d^4)*e^3*f + 15*(b*c^2*d^2 - a*c*d^3)*e^2*
f^2 + 12*(3*b*c^3*d + 5*a*c^2*d^2)*e*f^3 - (19*b*c^4 + 47*a*c^3*d)*f^4)*n)*x^2 - (26*a*c^4*e*f^3 - (3*b*c^2*d^
2 + 11*a*c*d^3)*e^4 + 2*(5*b*c^3*d + 21*a*c^2*d^2)*e^3*f - (7*b*c^4 + 57*a*c^3*d)*e^2*f^2)*n + (24*a*c^3*d*e*f
^3 + 24*a*c^4*f^4 - 2*(5*b*c*d^3 + 3*a*d^4)*e^4 + 8*(5*b*c^2*d^2 + 3*a*c*d^3)*e^3*f - 12*(5*b*c^3*d + 3*a*c^2*
d^2)*e^2*f^2 + (3*b*c^3*d*e^2*f^2 - a*c^4*f^4 + (b*c*d^3 + a*d^4)*e^4 - (3*b*c^2*d^2 + 2*a*c*d^3)*e^3*f - (b*c
^4 - 2*a*c^3*d)*e*f^3)*n^3 + (9*a*c^4*f^4 - 2*(4*b*c*d^3 + 3*a*d^4)*e^4 + (23*b*c^2*d^2 + 18*a*c*d^3)*e^3*f -
(22*b*c^3*d + 9*a*c^2*d^2)*e^2*f^2 + (7*b*c^4 - 12*a*c^3*d)*e*f^3)*n^2 - (26*a*c^4*f^4 - (17*b*c*d^3 + 11*a*d^
4)*e^4 + 20*(3*b*c^2*d^2 + 2*a*c*d^3)*e^3*f - 5*(11*b*c^3*d + 9*a*c^2*d^2)*e^2*f^2 + 2*(6*b*c^4 - 5*a*c^3*d)*e
*f^3)*n)*x)*(d*x + c)^(n - 5)/((24*d^4*e^4 - 96*c*d^3*e^3*f + 144*c^2*d^2*e^2*f^2 - 96*c^3*d*e*f^3 + 24*c^4*f^
4 + (d^4*e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n^4 - 10*(d^4*e^4 - 4*c*d^3*e^3*f
+ 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n^3 + 35*(d^4*e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d
*e*f^3 + c^4*f^4)*n^2 - 50*(d^4*e^4 - 4*c*d^3*e^3*f + 6*c^2*d^2*e^2*f^2 - 4*c^3*d*e*f^3 + c^4*f^4)*n)*(f*x + e
)^n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n - 5}}{{\left (f x + e\right )}^{n}}\,{d x} \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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maple [B]  time = 0.01, size = 1187, normalized size = 3.97 \begin {gather*} -\frac {\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 (d x +c \right )^{n -4} \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}} \end {gather*}

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)

[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*} \int \frac {{\left (b x + a\right )} {\left (d x + c\right )}^{n - 5}}{{\left (f x + e\right )}^{n}}\,{d x} \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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mupad [B]  time = 4.67, size = 1657, normalized size = 5.54

result too large to display

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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