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

Optimal. Leaf size=131 \[ -\frac {2 d^2 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (b c-a d)^3}-\frac {(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}+\frac {2 d (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (b c-a d)^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {2 d^2 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (b c-a d)^3}-\frac {(a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (b c-a d)}+\frac {2 d (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-4 - n)*(c + d*x)^n,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 113, normalized size = 0.86 \begin {gather*} -\frac {(a+b x)^{-n-3} (c+d x)^{n+1} \left (a^2 d^2 \left (n^2+5 n+6\right )-2 a b d (n+3) (c n+c-d x)+b^2 \left (c^2 \left (n^2+3 n+2\right )-2 c d (n+1) x+2 d^2 x^2\right )\right )}{(n+1) (n+2) (n+3) (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^{-4-n} (c+d x)^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.01, size = 509, normalized size = 3.89 \begin {gather*} -\frac {{\left (2 \, b^{3} d^{3} x^{4} + 2 \, a b^{2} c^{3} - 6 \, a^{2} b c^{2} d + 6 \, a^{3} c d^{2} + 2 \, {\left (4 \, a b^{2} d^{3} - {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} n\right )} x^{3} + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} n^{2} + {\left (12 \, a^{2} b d^{3} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} n^{2} + {\left (b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 7 \, a^{2} b d^{3}\right )} n\right )} x^{2} + {\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} n + {\left (2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} + 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} n^{2} + {\left (3 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d - a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{-n - 4} {\left (d x + c\right )}^{n}}{6 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 6 \, a^{3} d^{3} + {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{3} + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n^{2} + 11 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-4-n)*(d*x+c)^n,x, algorithm="fricas")

[Out]

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

[Out]

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

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maple [B]  time = 0.01, size = 318, normalized size = 2.43 \begin {gather*} \frac {\left (a^{2} d^{2} n^{2}-2 a b c d \,n^{2}+2 a b \,d^{2} n x +b^{2} c^{2} n^{2}-2 b^{2} c d n x +2 b^{2} x^{2} d^{2}+5 a^{2} d^{2} n -8 a b c d n +6 a b \,d^{2} x +3 b^{2} c^{2} n -2 b^{2} c d x +6 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) \left (b x +a \right )^{-n -3} \left (d x +c \right )^{n +1}}{a^{3} d^{3} n^{3}-3 a^{2} b c \,d^{2} n^{3}+3 a \,b^{2} c^{2} d \,n^{3}-b^{3} c^{3} n^{3}+6 a^{3} d^{3} n^{2}-18 a^{2} b c \,d^{2} n^{2}+18 a \,b^{2} c^{2} d \,n^{2}-6 b^{3} c^{3} n^{2}+11 a^{3} d^{3} n -33 a^{2} b c \,d^{2} n +33 a \,b^{2} c^{2} d n -11 b^{3} c^{3} n +6 a^{3} d^{3}-18 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -6 b^{3} c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(-3-n)*(d*x+c)^(n+1)*(a^2*d^2*n^2-2*a*b*c*d*n^2+2*a*b*d^2*n*x+b^2*c^2*n^2-2*b^2*c*d*n*x+2*b^2*d^2*x^2+
5*a^2*d^2*n-8*a*b*c*d*n+6*a*b*d^2*x+3*b^2*c^2*n-2*b^2*c*d*x+6*a^2*d^2-6*a*b*c*d+2*b^2*c^2)/(a^3*d^3*n^3-3*a^2*
b*c*d^2*n^3+3*a*b^2*c^2*d*n^3-b^3*c^3*n^3+6*a^3*d^3*n^2-18*a^2*b*c*d^2*n^2+18*a*b^2*c^2*d*n^2-6*b^3*c^3*n^2+11
*a^3*d^3*n-33*a^2*b*c*d^2*n+33*a*b^2*c^2*d*n-11*b^3*c^3*n+6*a^3*d^3-18*a^2*b*c*d^2+18*a*b^2*c^2*d-6*b^3*c^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x + a\right )}^{-n - 4} {\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)^(-4-n)*(d*x+c)^n,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.99, size = 525, normalized size = 4.01 \begin {gather*} \frac {x\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^2+5\,a^3\,d^3\,n+6\,a^3\,d^3-a^2\,b\,c\,d^2\,n^2-a^2\,b\,c\,d^2\,n+6\,a^2\,b\,c\,d^2-a\,b^2\,c^2\,d\,n^2-7\,a\,b^2\,c^2\,d\,n-6\,a\,b^2\,c^2\,d+b^3\,c^3\,n^2+3\,b^3\,c^3\,n+2\,b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {a\,c\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+5\,a^2\,d^2\,n+6\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-8\,a\,b\,c\,d\,n-6\,a\,b\,c\,d+b^2\,c^2\,n^2+3\,b^2\,c^2\,n+2\,b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {2\,b^3\,d^3\,x^4\,{\left (c+d\,x\right )}^n}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+7\,a^2\,d^2\,n+12\,a^2\,d^2-2\,a\,b\,c\,d\,n^2-8\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+b^2\,c^2\,n\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )}+\frac {2\,b^2\,d^2\,x^3\,{\left (c+d\,x\right )}^n\,\left (4\,a\,d+a\,d\,n-b\,c\,n\right )}{{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^{n+4}\,\left (n^3+6\,n^2+11\,n+6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^n/(a + b*x)^(n + 4),x)

[Out]

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

[Out]

Timed out

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