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

Optimal. Leaf size=186 \[ \frac {6 d^3 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}-\frac {6 d^2 (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (n+4) (b c-a d)^3}-\frac {(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}+\frac {3 d (a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (n+4) (b c-a d)^2} \]

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Rubi [A]  time = 0.09, antiderivative size = 186, 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} \begin {gather*} -\frac {6 d^2 (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac {6 d^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-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}+\frac {3 d (a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (n+4) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-5 - n)*(c + d*x)^n,x]

[Out]

-(((a + b*x)^(-4 - n)*(c + d*x)^(1 + n))/((b*c - a*d)*(4 + n))) + (3*d*(a + b*x)^(-3 - n)*(c + d*x)^(1 + n))/(
(b*c - a*d)^2*(3 + n)*(4 + n)) - (6*d^2*(a + b*x)^(-2 - n)*(c + d*x)^(1 + n))/((b*c - a*d)^3*(2 + n)*(3 + n)*(
4 + n)) + (6*d^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)^{-5-n} (c+d x)^n \, dx &=-\frac {(a+b x)^{-4-n} (c+d x)^{1+n}}{(b c-a d) (4+n)}-\frac {(3 d) \int (a+b x)^{-4-n} (c+d x)^n \, dx}{(b c-a d) (4+n)}\\ &=-\frac {(a+b x)^{-4-n} (c+d x)^{1+n}}{(b c-a d) (4+n)}+\frac {3 d (a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d)^2 (3+n) (4+n)}+\frac {\left (6 d^2\right ) \int (a+b x)^{-3-n} (c+d x)^n \, dx}{(b c-a d)^2 (3+n) (4+n)}\\ &=-\frac {(a+b x)^{-4-n} (c+d x)^{1+n}}{(b c-a d) (4+n)}+\frac {3 d (a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d)^2 (3+n) (4+n)}-\frac {6 d^2 (a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}-\frac {\left (6 d^3\right ) \int (a+b x)^{-2-n} (c+d x)^n \, dx}{(b c-a d)^3 (2+n) (3+n) (4+n)}\\ &=-\frac {(a+b x)^{-4-n} (c+d x)^{1+n}}{(b c-a d) (4+n)}+\frac {3 d (a+b x)^{-3-n} (c+d x)^{1+n}}{(b c-a d)^2 (3+n) (4+n)}-\frac {6 d^2 (a+b x)^{-2-n} (c+d x)^{1+n}}{(b c-a d)^3 (2+n) (3+n) (4+n)}+\frac {6 d^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 = 0.12, size = 195, normalized size = 1.05 \begin {gather*} \frac {(a+b x)^{-n-4} (c+d x)^{n+1} \left (a^3 d^3 \left (n^3+9 n^2+26 n+24\right )-3 a^2 b d^2 \left (n^2+7 n+12\right ) (c n+c-d x)+3 a b^2 d (n+4) \left (c^2 \left (n^2+3 n+2\right )-2 c d (n+1) x+2 d^2 x^2\right )-\left (b^3 \left (c^3 \left (n^3+6 n^2+11 n+6\right )-3 c^2 d \left (n^2+3 n+2\right ) x+6 c d^2 (n+1) x^2-6 d^3 x^3\right )\right )\right )}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(-4 - n)*(c + d*x)^(1 + n)*(a^3*d^3*(24 + 26*n + 9*n^2 + n^3) - 3*a^2*b*d^2*(12 + 7*n + n^2)*(c + c
*n - d*x) + 3*a*b^2*d*(4 + n)*(c^2*(2 + 3*n + n^2) - 2*c*d*(1 + n)*x + 2*d^2*x^2) - b^3*(c^3*(6 + 11*n + 6*n^2
+ n^3) - 3*c^2*d*(2 + 3*n + n^2)*x + 6*c*d^2*(1 + 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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IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^{-5-n} (c+d x)^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.40, size = 959, normalized size = 5.16 \begin {gather*} \frac {{\left (6 \, b^{4} d^{4} x^{5} - 6 \, a b^{3} c^{4} + 24 \, a^{2} b^{2} c^{3} d - 36 \, a^{3} b c^{2} d^{2} + 24 \, a^{4} c d^{3} + 6 \, {\left (5 \, a b^{3} d^{4} - {\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 \, a^{2} b^{2} d^{4} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} n^{2} + {\left (b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + 9 \, a^{2} b^{2} d^{4}\right )} n\right )} x^{3} - 3 \, {\left (2 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d + 8 \, a^{3} b c^{2} d^{2} - 3 \, a^{4} c d^{3}\right )} n^{2} + {\left (60 \, a^{3} b d^{4} - {\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 (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 9 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} n^{2} - {\left (2 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 60 \, a^{2} b^{2} c d^{3} - 47 \, a^{3} b d^{4}\right )} n\right )} x^{2} - {\left (11 \, a b^{3} c^{4} - 42 \, a^{2} b^{2} c^{3} d + 57 \, a^{3} b c^{2} d^{2} - 26 \, a^{4} c d^{3}\right )} n - {\left (6 \, 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} - 24 \, 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 (2 \, b^{4} c^{4} - 6 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}\right )} n^{2} + {\left (11 \, b^{4} c^{4} - 40 \, a b^{3} c^{3} d + 45 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} - 26 \, a^{4} d^{4}\right )} n\right )} x\right )} {\left (b x + a\right )}^{-n - 5} {\left (d x + c\right )}^{n}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6*b^4*d^4*x^5 - 6*a*b^3*c^4 + 24*a^2*b^2*c^3*d - 36*a^3*b*c^2*d^2 + 24*a^4*c*d^3 + 6*(5*a*b^3*d^4 - (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*a^2*b^2*d^4 +
(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*n^2 + (b^4*c^2*d^2 - 10*a*b^3*c*d^3 + 9*a^2*b^2*d^4)*n)*x^3 - 3*(2
*a*b^3*c^4 - 7*a^2*b^2*c^3*d + 8*a^3*b*c^2*d^2 - 3*a^4*c*d^3)*n^2 + (60*a^3*b*d^4 - (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*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 9*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*n^2 -
(2*b^4*c^3*d - 15*a*b^3*c^2*d^2 + 60*a^2*b^2*c*d^3 - 47*a^3*b*d^4)*n)*x^2 - (11*a*b^3*c^4 - 42*a^2*b^2*c^3*d +
57*a^3*b*c^2*d^2 - 26*a^4*c*d^3)*n - (6*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 - 24*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*(2*b^4*c^4 - 6*a*b^3*c^3*d + 3*a^2*b^2*c^
2*d^2 + 4*a^3*b*c*d^3 - 3*a^4*d^4)*n^2 + (11*b^4*c^4 - 40*a*b^3*c^3*d + 45*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 -
26*a^4*d^4)*n)*x)*(b*x + a)^(-n - 5)*(d*x + c)^n/(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 \begin {gather*} \int {\left (b x + a\right )}^{-n - 5} {\left (d x + c\right )}^{n}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 661, normalized size = 3.55 \begin {gather*} \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}+9 a^{3} d^{3} n^{2}-24 a^{2} b c \,d^{2} n^{2}+21 a^{2} b \,d^{3} n x +21 a \,b^{2} c^{2} d \,n^{2}-30 a \,b^{2} c \,d^{2} n x +24 a \,b^{2} d^{3} x^{2}-6 b^{3} c^{3} n^{2}+9 b^{3} c^{2} d n x -6 b^{3} c \,d^{2} x^{2}+26 a^{3} d^{3} n -57 a^{2} b c \,d^{2} n +36 a^{2} b \,d^{3} x +42 a \,b^{2} c^{2} d n -24 a \,b^{2} c \,d^{2} x -11 b^{3} c^{3} n +6 b^{3} c^{2} d x +24 a^{3} d^{3}-36 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \left (b x +a \right )^{-n -4} \left (d x +c \right )^{n +1}}{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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(-4-n)*(d*x+c)^(n+1)*(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+9*a^3*d^3*n^2-24*a^2*b*c
*d^2*n^2+21*a^2*b*d^3*n*x+21*a*b^2*c^2*d*n^2-30*a*b^2*c*d^2*n*x+24*a*b^2*d^3*x^2-6*b^3*c^3*n^2+9*b^3*c^2*d*n*x
-6*b^3*c*d^2*x^2+26*a^3*d^3*n-57*a^2*b*c*d^2*n+36*a^2*b*d^3*x+42*a*b^2*c^2*d*n-24*a*b^2*c*d^2*x-11*b^3*c^3*n+6
*b^3*c^2*d*x+24*a^3*d^3-36*a^2*b*c*d^2+24*a*b^2*c^2*d-6*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 \begin {gather*} \int {\left (b x + a\right )}^{-n - 5} {\left (d x + c\right )}^{n}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.64, size = 944, normalized size = 5.08 \begin {gather*} \frac {a\,c\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^3+9\,a^3\,d^3\,n^2+26\,a^3\,d^3\,n+24\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-24\,a^2\,b\,c\,d^2\,n^2-57\,a^2\,b\,c\,d^2\,n-36\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d\,n^3+21\,a\,b^2\,c^2\,d\,n^2+42\,a\,b^2\,c^2\,d\,n+24\,a\,b^2\,c^2\,d-b^3\,c^3\,n^3-6\,b^3\,c^3\,n^2-11\,b^3\,c^3\,n-6\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x\,{\left (c+d\,x\right )}^n\,\left (-a^4\,d^4\,n^3-9\,a^4\,d^4\,n^2-26\,a^4\,d^4\,n-24\,a^4\,d^4+2\,a^3\,b\,c\,d^3\,n^3+12\,a^3\,b\,c\,d^3\,n^2+10\,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^3-18\,a\,b^3\,c^3\,d\,n^2-40\,a\,b^3\,c^3\,d\,n-24\,a\,b^3\,c^3\,d+b^4\,c^4\,n^3+6\,b^4\,c^4\,n^2+11\,b^4\,c^4\,n+6\,b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,b^4\,d^4\,x^5\,{\left (c+d\,x\right )}^n}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,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 (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+9\,a^2\,d^2\,n+20\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-10\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+b^2\,c^2\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,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 (c+d\,x\right )}^n\,\left (5\,a\,d+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^3+12\,a^3\,d^3\,n^2+47\,a^3\,d^3\,n+60\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^3-27\,a^2\,b\,c\,d^2\,n^2-60\,a^2\,b\,c\,d^2\,n+3\,a\,b^2\,c^2\,d\,n^3+18\,a\,b^2\,c^2\,d\,n^2+15\,a\,b^2\,c^2\,d\,n-b^3\,c^3\,n^3-3\,b^3\,c^3\,n^2-2\,b^3\,c^3\,n\right )}{{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^{n+5}\,\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((c + d*x)^n/(a + b*x)^(n + 5),x)

[Out]

(a*c*(c + d*x)^n*(24*a^3*d^3 - 6*b^3*c^3 + 26*a^3*d^3*n - 11*b^3*c^3*n + 9*a^3*d^3*n^2 - 6*b^3*c^3*n^2 + a^3*d
^3*n^3 - b^3*c^3*n^3 + 24*a*b^2*c^2*d - 36*a^2*b*c*d^2 + 42*a*b^2*c^2*d*n - 57*a^2*b*c*d^2*n + 21*a*b^2*c^2*d*
n^2 - 24*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*(a + b*x)^(n + 5)*(50*n + 35
*n^2 + 10*n^3 + n^4 + 24)) - (x*(c + d*x)^n*(6*b^4*c^4 - 24*a^4*d^4 - 26*a^4*d^4*n + 11*b^4*c^4*n - 9*a^4*d^4*
n^2 + 6*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 - 40*a*
b^3*c^3*d*n + 10*a^3*b*c*d^3*n + 9*a^2*b^2*c^2*d^2*n^2 - 18*a*b^3*c^3*d*n^2 + 12*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*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 +
n^4 + 24)) + (6*b^4*d^4*x^5*(c + d*x)^n)/((a*d - b*c)^4*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)
) + (3*b^2*d^2*x^3*(c + d*x)^n*(20*a^2*d^2 + 9*a^2*d^2*n + 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*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (6*b^3*d^3*x^4*(c
+ d*x)^n*(5*a*d + a*d*n - b*c*n))/((a*d - b*c)^4*(a + b*x)^(n + 5)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (b*d
*x^2*(c + d*x)^n*(60*a^3*d^3 + 47*a^3*d^3*n - 2*b^3*c^3*n + 12*a^3*d^3*n^2 - 3*b^3*c^3*n^2 + a^3*d^3*n^3 - b^3
*c^3*n^3 + 15*a*b^2*c^2*d*n - 60*a^2*b*c*d^2*n + 18*a*b^2*c^2*d*n^2 - 27*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*(a + b*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 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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