3.26.30 \(\int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx\)

Optimal. Leaf size=207 \[ \frac {(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}+\frac {(c+d x)^{n-2} (e+f x)^{1-n} (2 a d f+b (c f (1-n)-d e (3-n)))}{d (2-n) (3-n) (d e-c f)^2}-\frac {f (c+d x)^{n-1} (e+f x)^{1-n} (2 a d f+b (c f (1-n)-d e (3-n)))}{d (1-n) (2-n) (3-n) (d e-c f)^3} \]

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Rubi [A]  time = 0.12, antiderivative size = 205, normalized size of antiderivative = 0.99, 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, 45, 37} \begin {gather*} \frac {(b c-a d) (c+d x)^{n-3} (e+f x)^{1-n}}{d (3-n) (d e-c f)}+\frac {(c+d x)^{n-2} (e+f x)^{1-n} (2 a d f+b c f (1-n)-b d e (3-n))}{d (2-n) (3-n) (d e-c f)^2}-\frac {f (c+d x)^{n-1} (e+f x)^{1-n} (2 a d f+b c f (1-n)-b d e (3-n))}{d (1-n) (2-n) (3-n) (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]

[Out]

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

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Mathematica [A]  time = 0.33, size = 112, normalized size = 0.54 \begin {gather*} \frac {(c+d x)^{n-3} (e+f x)^{1-n} \left (\frac {(c+d x) (-c f (n-2)+d e (n-1)+d f x) (2 a d f-b c f (n-1)+b d e (n-3))}{(n-2) (n-1) (d e-c f)^2}+a d-b c\right )}{d (n-3) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (c+d x)^{-4+n} (e+f x)^{-n} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n,x]

[Out]

Defer[IntegrateAlgebraic][((a + b*x)*(c + d*x)^(-4 + n))/(e + f*x)^n, x]

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fricas [B]  time = 1.91, size = 884, normalized size = 4.27 \begin {gather*} -\frac {{\left (6 \, a c^{3} e f^{2} - {\left (3 \, b d^{3} e f^{2} - {\left (b c d^{2} + 2 \, a d^{3}\right )} f^{3} - {\left (b d^{3} e f^{2} - b c d^{2} f^{3}\right )} n\right )} x^{4} + {\left (b c^{2} d + 2 \, a c d^{2}\right )} e^{3} - 3 \, {\left (b c^{3} + 2 \, a c^{2} d\right )} e^{2} f - {\left (12 \, b c d^{2} e f^{2} - 4 \, {\left (b c^{2} d + 2 \, a c d^{2}\right )} f^{3} - {\left (b d^{3} e^{2} f - 2 \, b c d^{2} e f^{2} + b c^{2} d f^{3}\right )} n^{2} + {\left (3 \, b d^{3} e^{2} f - 2 \, {\left (4 \, b c d^{2} + a d^{3}\right )} e f^{2} + {\left (5 \, b c^{2} d + 2 \, a c d^{2}\right )} f^{3}\right )} n\right )} x^{3} + {\left (a c d^{2} e^{3} - 2 \, a c^{2} d e^{2} f + a c^{3} e f^{2}\right )} n^{2} + {\left (3 \, b d^{3} e^{3} - 9 \, b c d^{2} e^{2} f - 9 \, b c^{2} d e f^{2} + 3 \, {\left (b c^{3} + 4 \, a c^{2} d\right )} f^{3} + {\left (b d^{3} e^{3} - {\left (b c d^{2} - a d^{3}\right )} e^{2} f - {\left (b c^{2} d + 2 \, a c d^{2}\right )} e f^{2} + {\left (b c^{3} + a c^{2} d\right )} f^{3}\right )} n^{2} - {\left (4 \, b d^{3} e^{3} - {\left (4 \, b c d^{2} - a d^{3}\right )} e^{2} f - 4 \, {\left (b c^{2} d + 2 \, a c d^{2}\right )} e f^{2} + {\left (4 \, b c^{3} + 7 \, a c^{2} d\right )} f^{3}\right )} n\right )} x^{2} - {\left (5 \, a c^{3} e f^{2} + {\left (b c^{2} d + 3 \, a c d^{2}\right )} e^{3} - {\left (b c^{3} + 8 \, a c^{2} d\right )} e^{2} f\right )} n + {\left (6 \, a c^{2} d e f^{2} + 6 \, a c^{3} f^{3} + 2 \, {\left (2 \, b c d^{2} + a d^{3}\right )} e^{3} - 6 \, {\left (2 \, b c^{2} d + a c d^{2}\right )} e^{2} f + {\left (a c^{3} f^{3} + {\left (b c d^{2} + a d^{3}\right )} e^{3} - {\left (2 \, b c^{2} d + a c d^{2}\right )} e^{2} f + {\left (b c^{3} - a c^{2} d\right )} e f^{2}\right )} n^{2} - {\left (5 \, a c^{3} f^{3} + {\left (5 \, b c d^{2} + 3 \, a d^{3}\right )} e^{3} - {\left (8 \, b c^{2} d + 7 \, a c d^{2}\right )} e^{2} f + {\left (3 \, b c^{3} - a c^{2} d\right )} e f^{2}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n - 4}}{{\left (6 \, d^{3} e^{3} - 18 \, c d^{2} e^{2} f + 18 \, c^{2} d e f^{2} - 6 \, c^{3} f^{3} - {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} n^{3} + 6 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} n^{2} - 11 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} n\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)^(-4+n)/((f*x+e)^n),x, algorithm="fricas")

[Out]

-(6*a*c^3*e*f^2 - (3*b*d^3*e*f^2 - (b*c*d^2 + 2*a*d^3)*f^3 - (b*d^3*e*f^2 - b*c*d^2*f^3)*n)*x^4 + (b*c^2*d + 2
*a*c*d^2)*e^3 - 3*(b*c^3 + 2*a*c^2*d)*e^2*f - (12*b*c*d^2*e*f^2 - 4*(b*c^2*d + 2*a*c*d^2)*f^3 - (b*d^3*e^2*f -
2*b*c*d^2*e*f^2 + b*c^2*d*f^3)*n^2 + (3*b*d^3*e^2*f - 2*(4*b*c*d^2 + a*d^3)*e*f^2 + (5*b*c^2*d + 2*a*c*d^2)*f
^3)*n)*x^3 + (a*c*d^2*e^3 - 2*a*c^2*d*e^2*f + a*c^3*e*f^2)*n^2 + (3*b*d^3*e^3 - 9*b*c*d^2*e^2*f - 9*b*c^2*d*e*
f^2 + 3*(b*c^3 + 4*a*c^2*d)*f^3 + (b*d^3*e^3 - (b*c*d^2 - a*d^3)*e^2*f - (b*c^2*d + 2*a*c*d^2)*e*f^2 + (b*c^3
+ a*c^2*d)*f^3)*n^2 - (4*b*d^3*e^3 - (4*b*c*d^2 - a*d^3)*e^2*f - 4*(b*c^2*d + 2*a*c*d^2)*e*f^2 + (4*b*c^3 + 7*
a*c^2*d)*f^3)*n)*x^2 - (5*a*c^3*e*f^2 + (b*c^2*d + 3*a*c*d^2)*e^3 - (b*c^3 + 8*a*c^2*d)*e^2*f)*n + (6*a*c^2*d*
e*f^2 + 6*a*c^3*f^3 + 2*(2*b*c*d^2 + a*d^3)*e^3 - 6*(2*b*c^2*d + a*c*d^2)*e^2*f + (a*c^3*f^3 + (b*c*d^2 + a*d^
3)*e^3 - (2*b*c^2*d + a*c*d^2)*e^2*f + (b*c^3 - a*c^2*d)*e*f^2)*n^2 - (5*a*c^3*f^3 + (5*b*c*d^2 + 3*a*d^3)*e^3
- (8*b*c^2*d + 7*a*c*d^2)*e^2*f + (3*b*c^3 - a*c^2*d)*e*f^2)*n)*x)*(d*x + c)^(n - 4)/((6*d^3*e^3 - 18*c*d^2*e
^2*f + 18*c^2*d*e*f^2 - 6*c^3*f^3 - (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*n^3 + 6*(d^3*e^3 - 3*c
*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*n^2 - 11*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*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 - 4}}{{\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)^(-4+n)/((f*x+e)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n, x)

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maple [B]  time = 0.01, size = 506, normalized size = 2.44 \begin {gather*} -\frac {\left (f x +e \right ) \left (b \,c^{2} f^{2} n^{2} x -2 b c d e f \,n^{2} x -b c d \,f^{2} n \,x^{2}+b \,d^{2} e^{2} n^{2} x +b \,d^{2} e f n \,x^{2}+a \,c^{2} f^{2} n^{2}-2 a c d e f \,n^{2}-2 a c d \,f^{2} n x +a \,d^{2} e^{2} n^{2}+2 a \,d^{2} e f n x +2 a \,d^{2} f^{2} x^{2}-4 b \,c^{2} f^{2} n x +8 b c d e f n x +b c d \,f^{2} x^{2}-4 b \,d^{2} e^{2} n x -3 b \,d^{2} e f \,x^{2}-5 a \,c^{2} f^{2} n +8 a c d e f n +6 a c d \,f^{2} x -3 a \,d^{2} e^{2} n -2 a \,d^{2} e f x +b \,c^{2} e f n +3 b \,c^{2} f^{2} x -b c d \,e^{2} n -10 b c d e f x +3 b \,d^{2} e^{2} x +6 a \,c^{2} f^{2}-6 a c d e f +2 a \,d^{2} e^{2}-3 b \,c^{2} e f +b c d \,e^{2}\right ) \left (d x +c \right )^{n -3} \left (f x +e \right )^{-n}}{c^{3} f^{3} n^{3}-3 c^{2} d e \,f^{2} n^{3}+3 c \,d^{2} e^{2} f \,n^{3}-d^{3} e^{3} n^{3}-6 c^{3} f^{3} n^{2}+18 c^{2} d e \,f^{2} n^{2}-18 c \,d^{2} e^{2} f \,n^{2}+6 d^{3} e^{3} n^{2}+11 c^{3} f^{3} n -33 c^{2} d e \,f^{2} n +33 c \,d^{2} e^{2} f n -11 d^{3} e^{3} n -6 c^{3} f^{3}+18 c^{2} d e \,f^{2}-18 c \,d^{2} e^{2} f +6 d^{3} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)^(-4+n)/((f*x+e)^n),x)

[Out]

-(d*x+c)^(n-3)*(f*x+e)*(b*c^2*f^2*n^2*x-2*b*c*d*e*f*n^2*x-b*c*d*f^2*n*x^2+b*d^2*e^2*n^2*x+b*d^2*e*f*n*x^2+a*c^
2*f^2*n^2-2*a*c*d*e*f*n^2-2*a*c*d*f^2*n*x+a*d^2*e^2*n^2+2*a*d^2*e*f*n*x+2*a*d^2*f^2*x^2-4*b*c^2*f^2*n*x+8*b*c*
d*e*f*n*x+b*c*d*f^2*x^2-4*b*d^2*e^2*n*x-3*b*d^2*e*f*x^2-5*a*c^2*f^2*n+8*a*c*d*e*f*n+6*a*c*d*f^2*x-3*a*d^2*e^2*
n-2*a*d^2*e*f*x+b*c^2*e*f*n+3*b*c^2*f^2*x-b*c*d*e^2*n-10*b*c*d*e*f*x+3*b*d^2*e^2*x+6*a*c^2*f^2-6*a*c*d*e*f+2*a
*d^2*e^2-3*b*c^2*e*f+b*c*d*e^2)/(c^3*f^3*n^3-3*c^2*d*e*f^2*n^3+3*c*d^2*e^2*f*n^3-d^3*e^3*n^3-6*c^3*f^3*n^2+18*
c^2*d*e*f^2*n^2-18*c*d^2*e^2*f*n^2+6*d^3*e^3*n^2+11*c^3*f^3*n-33*c^2*d*e*f^2*n+33*c*d^2*e^2*f*n-11*d^3*e^3*n-6
*c^3*f^3+18*c^2*d*e*f^2-18*c*d^2*e^2*f+6*d^3*e^3)/((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 - 4}}{{\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)^(-4+n)/((f*x+e)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)*(d*x + c)^(n - 4)/(f*x + e)^n, x)

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mupad [B]  time = 3.43, size = 874, normalized size = 4.22 \begin {gather*} -\frac {x\,{\left (c+d\,x\right )}^{n-4}\,\left (b\,c^3\,e\,f^2\,n^2-3\,b\,c^3\,e\,f^2\,n+a\,c^3\,f^3\,n^2-5\,a\,c^3\,f^3\,n+6\,a\,c^3\,f^3-2\,b\,c^2\,d\,e^2\,f\,n^2+8\,b\,c^2\,d\,e^2\,f\,n-12\,b\,c^2\,d\,e^2\,f-a\,c^2\,d\,e\,f^2\,n^2+a\,c^2\,d\,e\,f^2\,n+6\,a\,c^2\,d\,e\,f^2+b\,c\,d^2\,e^3\,n^2-5\,b\,c\,d^2\,e^3\,n+4\,b\,c\,d^2\,e^3-a\,c\,d^2\,e^2\,f\,n^2+7\,a\,c\,d^2\,e^2\,f\,n-6\,a\,c\,d^2\,e^2\,f+a\,d^3\,e^3\,n^2-3\,a\,d^3\,e^3\,n+2\,a\,d^3\,e^3\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {x^2\,{\left (c+d\,x\right )}^{n-4}\,\left (b\,c^3\,f^3\,n^2-4\,b\,c^3\,f^3\,n+3\,b\,c^3\,f^3-b\,c^2\,d\,e\,f^2\,n^2+4\,b\,c^2\,d\,e\,f^2\,n-9\,b\,c^2\,d\,e\,f^2+a\,c^2\,d\,f^3\,n^2-7\,a\,c^2\,d\,f^3\,n+12\,a\,c^2\,d\,f^3-b\,c\,d^2\,e^2\,f\,n^2+4\,b\,c\,d^2\,e^2\,f\,n-9\,b\,c\,d^2\,e^2\,f-2\,a\,c\,d^2\,e\,f^2\,n^2+8\,a\,c\,d^2\,e\,f^2\,n+b\,d^3\,e^3\,n^2-4\,b\,d^3\,e^3\,n+3\,b\,d^3\,e^3+a\,d^3\,e^2\,f\,n^2-a\,d^3\,e^2\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {c\,e\,{\left (c+d\,x\right )}^{n-4}\,\left (b\,c^2\,e\,f\,n-3\,b\,c^2\,e\,f+a\,c^2\,f^2\,n^2-5\,a\,c^2\,f^2\,n+6\,a\,c^2\,f^2-b\,c\,d\,e^2\,n+b\,c\,d\,e^2-2\,a\,c\,d\,e\,f\,n^2+8\,a\,c\,d\,e\,f\,n-6\,a\,c\,d\,e\,f+a\,d^2\,e^2\,n^2-3\,a\,d^2\,e^2\,n+2\,a\,d^2\,e^2\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {d^2\,f^2\,x^4\,{\left (c+d\,x\right )}^{n-4}\,\left (2\,a\,d\,f+b\,c\,f-3\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )}-\frac {d\,f\,x^3\,{\left (c+d\,x\right )}^{n-4}\,\left (4\,c\,f-c\,f\,n+d\,e\,n\right )\,\left (2\,a\,d\,f+b\,c\,f-3\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^3\,\left (n^3-6\,n^2+11\,n-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(c + d*x)^(n - 4))/(e + f*x)^n,x)

[Out]

- (x*(c + d*x)^(n - 4)*(6*a*c^3*f^3 + 2*a*d^3*e^3 + a*c^3*f^3*n^2 + a*d^3*e^3*n^2 + 4*b*c*d^2*e^3 - 5*a*c^3*f^
3*n - 3*a*d^3*e^3*n - 6*a*c*d^2*e^2*f + 6*a*c^2*d*e*f^2 - 12*b*c^2*d*e^2*f - 5*b*c*d^2*e^3*n - 3*b*c^3*e*f^2*n
+ b*c*d^2*e^3*n^2 + b*c^3*e*f^2*n^2 + 7*a*c*d^2*e^2*f*n + a*c^2*d*e*f^2*n + 8*b*c^2*d*e^2*f*n - a*c*d^2*e^2*f
*n^2 - a*c^2*d*e*f^2*n^2 - 2*b*c^2*d*e^2*f*n^2))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 6)) - (x^2*(
c + d*x)^(n - 4)*(3*b*c^3*f^3 + 3*b*d^3*e^3 + b*c^3*f^3*n^2 + b*d^3*e^3*n^2 + 12*a*c^2*d*f^3 - 4*b*c^3*f^3*n -
4*b*d^3*e^3*n - 9*b*c*d^2*e^2*f - 9*b*c^2*d*e*f^2 - 7*a*c^2*d*f^3*n - a*d^3*e^2*f*n + a*c^2*d*f^3*n^2 + a*d^3
*e^2*f*n^2 + 8*a*c*d^2*e*f^2*n + 4*b*c*d^2*e^2*f*n + 4*b*c^2*d*e*f^2*n - 2*a*c*d^2*e*f^2*n^2 - b*c*d^2*e^2*f*n
^2 - b*c^2*d*e*f^2*n^2))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 6)) - (c*e*(c + d*x)^(n - 4)*(6*a*c^
2*f^2 + 2*a*d^2*e^2 + a*c^2*f^2*n^2 + a*d^2*e^2*n^2 + b*c*d*e^2 - 3*b*c^2*e*f - 5*a*c^2*f^2*n - 3*a*d^2*e^2*n
- 6*a*c*d*e*f - b*c*d*e^2*n + b*c^2*e*f*n - 2*a*c*d*e*f*n^2 + 8*a*c*d*e*f*n))/((e + f*x)^n*(c*f - d*e)^3*(11*n
- 6*n^2 + n^3 - 6)) - (d^2*f^2*x^4*(c + d*x)^(n - 4)*(2*a*d*f + b*c*f - 3*b*d*e - b*c*f*n + b*d*e*n))/((e + f
*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 6)) - (d*f*x^3*(c + d*x)^(n - 4)*(4*c*f - c*f*n + d*e*n)*(2*a*d*f +
b*c*f - 3*b*d*e - b*c*f*n + b*d*e*n))/((e + f*x)^n*(c*f - d*e)^3*(11*n - 6*n^2 + n^3 - 6))

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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)**(-4+n)/((f*x+e)**n),x)

[Out]

Timed out

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