3.1877 \(\int (a+b x)^n (c+d x)^{-5-n} \, dx\)

Optimal. Leaf size=185 \[ \frac {6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac {6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac {3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac {6 b^2 (a+b x)^{n+1} (c+d x)^{-n-2}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac {6 b^3 (a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}+\frac {(a+b x)^{n+1} (c+d x)^{-n-4}}{(n+4) (b c-a d)}+\frac {3 b (a+b x)^{n+1} (c+d x)^{-n-3}}{(n+3) (n+4) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n*(c + d*x)^(-5 - n),x]

[Out]

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

Rubi steps

\begin {align*} \int (a+b x)^n (c+d x)^{-5-n} \, dx &=\frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {(3 b) \int (a+b x)^n (c+d x)^{-4-n} \, dx}{(b c-a d) (4+n)}\\ &=\frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {\left (6 b^2\right ) \int (a+b x)^n (c+d x)^{-3-n} \, dx}{(b c-a d)^2 (3+n) (4+n)}\\ &=\frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {\left (6 b^3\right ) \int (a+b x)^n (c+d x)^{-2-n} \, dx}{(b c-a d)^3 (2+n) (3+n) (4+n)}\\ &=\frac {(a+b x)^{1+n} (c+d x)^{-4-n}}{(b c-a d) (4+n)}+\frac {3 b (a+b x)^{1+n} (c+d x)^{-3-n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {6 b^2 (a+b x)^{1+n} (c+d x)^{-2-n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {6 b^3 (a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d)^4 (1+n) (2+n) (3+n) (4+n)}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 195, normalized size = 1.05 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-4} \left (-a^3 d^3 \left (n^3+6 n^2+11 n+6\right )+3 a^2 b d^2 \left (n^2+3 n+2\right ) (c (n+4)+d x)-3 a b^2 d (n+1) \left (c^2 \left (n^2+7 n+12\right )+2 c d (n+4) x+2 d^2 x^2\right )+b^3 \left (c^3 \left (n^3+9 n^2+26 n+24\right )+3 c^2 d \left (n^2+7 n+12\right ) x+6 c d^2 (n+4) x^2+6 d^3 x^3\right )\right )}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(-4 - n)*(-(a^3*d^3*(6 + 11*n + 6*n^2 + n^3)) + 3*a^2*b*d^2*(2 + 3*n + n^2)*(c*(4
+ n) + d*x) - 3*a*b^2*d*(1 + n)*(c^2*(12 + 7*n + n^2) + 2*c*d*(4 + n)*x + 2*d^2*x^2) + b^3*(c^3*(24 + 26*n +
9*n^2 + n^3) + 3*c^2*d*(12 + 7*n + n^2)*x + 6*c*d^2*(4 + n)*x^2 + 6*d^3*x^3)))/((b*c - a*d)^4*(1 + n)*(2 + n)*
(3 + n)*(4 + n))

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fricas [B]  time = 0.73, size = 954, normalized size = 5.16 \[ \frac {{\left (6 \, b^{4} d^{4} x^{5} + 24 \, a b^{3} c^{4} - 36 \, a^{2} b^{2} c^{3} d + 24 \, a^{3} b c^{2} d^{2} - 6 \, a^{4} c d^{3} + 6 \, {\left (5 \, b^{4} c d^{3} + {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} n\right )} x^{4} + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} n^{3} + 3 \, {\left (20 \, b^{4} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} n^{2} + {\left (9 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} n\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{4} - 8 \, a^{2} b^{2} c^{3} d + 7 \, a^{3} b c^{2} d^{2} - 2 \, a^{4} c d^{3}\right )} n^{2} + {\left (60 \, b^{4} c^{3} d + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} n^{3} + 3 \, {\left (4 \, b^{4} c^{3} d - 9 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} n^{2} + {\left (47 \, b^{4} c^{3} d - 60 \, a b^{3} c^{2} d^{2} + 15 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} n\right )} x^{2} + {\left (26 \, a b^{3} c^{4} - 57 \, a^{2} b^{2} c^{3} d + 42 \, a^{3} b c^{2} d^{2} - 11 \, a^{4} c d^{3}\right )} n + {\left (24 \, b^{4} c^{4} + 24 \, a b^{3} c^{3} d - 36 \, a^{2} b^{2} c^{2} d^{2} + 24 \, a^{3} b c d^{3} - 6 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} n^{3} + 3 \, {\left (3 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2} + 6 \, a^{3} b c d^{3} - 2 \, a^{4} d^{4}\right )} n^{2} + {\left (26 \, b^{4} c^{4} - 10 \, a b^{3} c^{3} d - 45 \, a^{2} b^{2} c^{2} d^{2} + 40 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5}}{24 \, b^{4} c^{4} - 96 \, a b^{3} c^{3} d + 144 \, a^{2} b^{2} c^{2} d^{2} - 96 \, a^{3} b c d^{3} + 24 \, a^{4} d^{4} + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n^{4} + 10 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n^{3} + 35 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n^{2} + 50 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*b^4*d^4*x^5 + 24*a*b^3*c^4 - 36*a^2*b^2*c^3*d + 24*a^3*b*c^2*d^2 - 6*a^4*c*d^3 + 6*(5*b^4*c*d^3 + (b^4*c*d^
3 - a*b^3*d^4)*n)*x^4 + (a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*n^3 + 3*(20*b^4*c^2*d^2 +
(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*n^2 + (9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*n)*x^3 + 3*(3
*a*b^3*c^4 - 8*a^2*b^2*c^3*d + 7*a^3*b*c^2*d^2 - 2*a^4*c*d^3)*n^2 + (60*b^4*c^3*d + (b^4*c^3*d - 3*a*b^3*c^2*d
^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*n^3 + 3*(4*b^4*c^3*d - 9*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 - a^3*b*d^4)*n^2 +
(47*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 15*a^2*b^2*c*d^3 - 2*a^3*b*d^4)*n)*x^2 + (26*a*b^3*c^4 - 57*a^2*b^2*c^3*d +
42*a^3*b*c^2*d^2 - 11*a^4*c*d^3)*n + (24*b^4*c^4 + 24*a*b^3*c^3*d - 36*a^2*b^2*c^2*d^2 + 24*a^3*b*c*d^3 - 6*a
^4*d^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*n^3 + 3*(3*b^4*c^4 - 4*a*b^3*c^3*d - 3*a^2*b^2*c^
2*d^2 + 6*a^3*b*c*d^3 - 2*a^4*d^4)*n^2 + (26*b^4*c^4 - 10*a*b^3*c^3*d - 45*a^2*b^2*c^2*d^2 + 40*a^3*b*c*d^3 -
11*a^4*d^4)*n)*x)*(b*x + a)^n*(d*x + c)^(-n - 5)/(24*b^4*c^4 - 96*a*b^3*c^3*d + 144*a^2*b^2*c^2*d^2 - 96*a^3*b
*c*d^3 + 24*a^4*d^4 + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*n^4 + 10*(b^4*c^
4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*n^3 + 35*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2
*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*n^2 + 50*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^
4*d^4)*n)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 662, normalized size = 3.58 \[ -\frac {\left (a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}-3 a^{2} b \,d^{3} n^{2} x +3 a \,b^{2} c^{2} d \,n^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +6 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-3 b^{3} c^{2} d \,n^{2} x -6 b^{3} c \,d^{2} n \,x^{2}-6 b^{3} d^{3} x^{3}+6 a^{3} d^{3} n^{2}-21 a^{2} b c \,d^{2} n^{2}-9 a^{2} b \,d^{3} n x +24 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 a \,b^{2} d^{3} x^{2}-9 b^{3} c^{3} n^{2}-21 b^{3} c^{2} d n x -24 b^{3} c \,d^{2} x^{2}+11 a^{3} d^{3} n -42 a^{2} b c \,d^{2} n -6 a^{2} b \,d^{3} x +57 a \,b^{2} c^{2} d n +24 a \,b^{2} c \,d^{2} x -26 b^{3} c^{3} n -36 b^{3} c^{2} d x +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) \left (b x +a \right )^{n +1} \left (d x +c \right )^{-n -4}}{a^{4} d^{4} n^{4}-4 a^{3} b c \,d^{3} n^{4}+6 a^{2} b^{2} c^{2} d^{2} n^{4}-4 a \,b^{3} c^{3} d \,n^{4}+b^{4} c^{4} n^{4}+10 a^{4} d^{4} n^{3}-40 a^{3} b c \,d^{3} n^{3}+60 a^{2} b^{2} c^{2} d^{2} n^{3}-40 a \,b^{3} c^{3} d \,n^{3}+10 b^{4} c^{4} n^{3}+35 a^{4} d^{4} n^{2}-140 a^{3} b c \,d^{3} n^{2}+210 a^{2} b^{2} c^{2} d^{2} n^{2}-140 a \,b^{3} c^{3} d \,n^{2}+35 b^{4} c^{4} n^{2}+50 a^{4} d^{4} n -200 a^{3} b c \,d^{3} n +300 a^{2} b^{2} c^{2} d^{2} n -200 a \,b^{3} c^{3} d n +50 b^{4} c^{4} n +24 a^{4} d^{4}-96 a^{3} b c \,d^{3}+144 a^{2} b^{2} c^{2} d^{2}-96 a \,b^{3} c^{3} d +24 b^{4} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(n+1)*(d*x+c)^(-n-4)*(a^3*d^3*n^3-3*a^2*b*c*d^2*n^3-3*a^2*b*d^3*n^2*x+3*a*b^2*c^2*d*n^3+6*a*b^2*c*d^2
*n^2*x+6*a*b^2*d^3*n*x^2-b^3*c^3*n^3-3*b^3*c^2*d*n^2*x-6*b^3*c*d^2*n*x^2-6*b^3*d^3*x^3+6*a^3*d^3*n^2-21*a^2*b*
c*d^2*n^2-9*a^2*b*d^3*n*x+24*a*b^2*c^2*d*n^2+30*a*b^2*c*d^2*n*x+6*a*b^2*d^3*x^2-9*b^3*c^3*n^2-21*b^3*c^2*d*n*x
-24*b^3*c*d^2*x^2+11*a^3*d^3*n-42*a^2*b*c*d^2*n-6*a^2*b*d^3*x+57*a*b^2*c^2*d*n+24*a*b^2*c*d^2*x-26*b^3*c^3*n-3
6*b^3*c^2*d*x+6*a^3*d^3-24*a^2*b*c*d^2+36*a*b^2*c^2*d-24*b^3*c^3)/(a^4*d^4*n^4-4*a^3*b*c*d^3*n^4+6*a^2*b^2*c^2
*d^2*n^4-4*a*b^3*c^3*d*n^4+b^4*c^4*n^4+10*a^4*d^4*n^3-40*a^3*b*c*d^3*n^3+60*a^2*b^2*c^2*d^2*n^3-40*a*b^3*c^3*d
*n^3+10*b^4*c^4*n^3+35*a^4*d^4*n^2-140*a^3*b*c*d^3*n^2+210*a^2*b^2*c^2*d^2*n^2-140*a*b^3*c^3*d*n^2+35*b^4*c^4*
n^2+50*a^4*d^4*n-200*a^3*b*c*d^3*n+300*a^2*b^2*c^2*d^2*n-200*a*b^3*c^3*d*n+50*b^4*c^4*n+24*a^4*d^4-96*a^3*b*c*
d^3+144*a^2*b^2*c^2*d^2-96*a*b^3*c^3*d+24*b^4*c^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.61, size = 945, normalized size = 5.11 \[ \frac {6\,b^4\,d^4\,x^5\,{\left (a+b\,x\right )}^n}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^n\,\left (a^3\,d^3\,n^3+6\,a^3\,d^3\,n^2+11\,a^3\,d^3\,n+6\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-21\,a^2\,b\,c\,d^2\,n^2-42\,a^2\,b\,c\,d^2\,n-24\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,n^3+24\,a\,b^2\,c^2\,d\,n^2+57\,a\,b^2\,c^2\,d\,n+36\,a\,b^2\,c^2\,d-b^3\,c^3\,n^3-9\,b^3\,c^3\,n^2-26\,b^3\,c^3\,n-24\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x\,{\left (a+b\,x\right )}^n\,\left (a^4\,d^4\,n^3+6\,a^4\,d^4\,n^2+11\,a^4\,d^4\,n+6\,a^4\,d^4-2\,a^3\,b\,c\,d^3\,n^3-18\,a^3\,b\,c\,d^3\,n^2-40\,a^3\,b\,c\,d^3\,n-24\,a^3\,b\,c\,d^3+9\,a^2\,b^2\,c^2\,d^2\,n^2+45\,a^2\,b^2\,c^2\,d^2\,n+36\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d\,n^3+12\,a\,b^3\,c^3\,d\,n^2+10\,a\,b^3\,c^3\,d\,n-24\,a\,b^3\,c^3\,d-b^4\,c^4\,n^3-9\,b^4\,c^4\,n^2-26\,b^4\,c^4\,n-24\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,b^2\,d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (a^2\,d^2\,n^2+a^2\,d^2\,n-2\,a\,b\,c\,d\,n^2-10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+9\,b^2\,c^2\,n+20\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,b^3\,d^3\,x^4\,{\left (a+b\,x\right )}^n\,\left (5\,b\,c-a\,d\,n+b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n\,\left (-a^3\,d^3\,n^3-3\,a^3\,d^3\,n^2-2\,a^3\,d^3\,n+3\,a^2\,b\,c\,d^2\,n^3+18\,a^2\,b\,c\,d^2\,n^2+15\,a^2\,b\,c\,d^2\,n-3\,a\,b^2\,c^2\,d\,n^3-27\,a\,b^2\,c^2\,d\,n^2-60\,a\,b^2\,c^2\,d\,n+b^3\,c^3\,n^3+12\,b^3\,c^3\,n^2+47\,b^3\,c^3\,n+60\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/(c + d*x)^(n + 5),x)

[Out]

(6*b^4*d^4*x^5*(a + b*x)^n)/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) - (a*c*(a +
b*x)^n*(6*a^3*d^3 - 24*b^3*c^3 + 11*a^3*d^3*n - 26*b^3*c^3*n + 6*a^3*d^3*n^2 - 9*b^3*c^3*n^2 + a^3*d^3*n^3 - b
^3*c^3*n^3 + 36*a*b^2*c^2*d - 24*a^2*b*c*d^2 + 57*a*b^2*c^2*d*n - 42*a^2*b*c*d^2*n + 24*a*b^2*c^2*d*n^2 - 21*a
^2*b*c*d^2*n^2 + 3*a*b^2*c^2*d*n^3 - 3*a^2*b*c*d^2*n^3))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*
n^3 + n^4 + 24)) - (x*(a + b*x)^n*(6*a^4*d^4 - 24*b^4*c^4 + 11*a^4*d^4*n - 26*b^4*c^4*n + 6*a^4*d^4*n^2 - 9*b^
4*c^4*n^2 + a^4*d^4*n^3 - b^4*c^4*n^3 + 36*a^2*b^2*c^2*d^2 - 24*a*b^3*c^3*d - 24*a^3*b*c*d^3 + 10*a*b^3*c^3*d*
n - 40*a^3*b*c*d^3*n + 9*a^2*b^2*c^2*d^2*n^2 + 12*a*b^3*c^3*d*n^2 - 18*a^3*b*c*d^3*n^2 + 2*a*b^3*c^3*d*n^3 - 2
*a^3*b*c*d^3*n^3 + 45*a^2*b^2*c^2*d^2*n))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)
) + (3*b^2*d^2*x^3*(a + b*x)^n*(20*b^2*c^2 + a^2*d^2*n + 9*b^2*c^2*n + a^2*d^2*n^2 + b^2*c^2*n^2 - 10*a*b*c*d*
n - 2*a*b*c*d*n^2))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (6*b^3*d^3*x^4*(a
+ b*x)^n*(5*b*c - a*d*n + b*c*n))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (b*d
*x^2*(a + b*x)^n*(60*b^3*c^3 - 2*a^3*d^3*n + 47*b^3*c^3*n - 3*a^3*d^3*n^2 + 12*b^3*c^3*n^2 - a^3*d^3*n^3 + b^3
*c^3*n^3 - 60*a*b^2*c^2*d*n + 15*a^2*b*c*d^2*n - 27*a*b^2*c^2*d*n^2 + 18*a^2*b*c*d^2*n^2 - 3*a*b^2*c^2*d*n^3 +
3*a^2*b*c*d^2*n^3))/((a*d - b*c)^4*(c + d*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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