3.1871 \(\int (a+b x)^{-5-n} (c+d x)^n \, dx\)

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

[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))

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Rubi [A]  time = 0.0862265, antiderivative size = 186, normalized size of antiderivative = 1., 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} \[ -\frac{6 d^2 (a+b x)^{-n-2} (c+d x)^{n+1}}{(n+2) (n+3) (n+4) (b c-a d)^3}+\frac{6 d^3 (a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4}-\frac{(a+b x)^{-n-4} (c+d x)^{n+1}}{(n+4) (b c-a d)}+\frac{3 d (a+b x)^{-n-3} (c+d x)^{n+1}}{(n+3) (n+4) (b c-a d)^2} \]

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 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 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]

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*}

Mathematica [A]  time = 0.0870107, size = 195, normalized size = 1.05 \[ \frac{(a+b x)^{-n-4} (c+d x)^{n+1} \left (-3 a^2 b d^2 \left (n^2+7 n+12\right ) (c n+c-d x)+a^3 d^3 \left (n^3+9 n^2+26 n+24\right )+3 a b^2 d (n+4) \left (c^2 \left (n^2+3 n+2\right )-2 c d (n+1) x+2 d^2 x^2\right )+b^3 \left (-\left (-3 c^2 d \left (n^2+3 n+2\right ) x+c^3 \left (n^3+6 n^2+11 n+6\right )+6 c d^2 (n+1) x^2-6 d^3 x^3\right )\right )\right )}{(n+1) (n+2) (n+3) (n+4) (b c-a d)^4} \]

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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Maple [B]  time = 0.008, size = 661, normalized size = 3.6 \begin{align*}{\frac{ \left ( bx+a \right ) ^{-4-n} \left ( dx+c \right ) ^{1+n} \left ({a}^{3}{d}^{3}{n}^{3}-3\,{a}^{2}bc{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}bc{d}^{2}{n}^{2}+21\,{a}^{2}b{d}^{3}nx+21\,a{b}^{2}{c}^{2}d{n}^{2}-30\,a{b}^{2}c{d}^{2}nx+24\,a{b}^{2}{d}^{3}{x}^{2}-6\,{b}^{3}{c}^{3}{n}^{2}+9\,{b}^{3}{c}^{2}dnx-6\,{b}^{3}c{d}^{2}{x}^{2}+26\,{a}^{3}{d}^{3}n-57\,{a}^{2}bc{d}^{2}n+36\,{a}^{2}b{d}^{3}x+42\,a{b}^{2}{c}^{2}dn-24\,a{b}^{2}c{d}^{2}x-11\,{b}^{3}{c}^{3}n+6\,{b}^{3}{c}^{2}dx+24\,{a}^{3}{d}^{3}-36\,{a}^{2}cb{d}^{2}+24\,a{b}^{2}{c}^{2}d-6\,{b}^{3}{c}^{3} \right ) }{{a}^{4}{d}^{4}{n}^{4}-4\,{a}^{3}bc{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}bc{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}bc{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}bc{d}^{3}n+300\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}n-200\,a{b}^{3}{c}^{3}dn+50\,{b}^{4}{c}^{4}n+24\,{a}^{4}{d}^{4}-96\,{a}^{3}bc{d}^{3}+144\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-96\,a{b}^{3}{c}^{3}d+24\,{b}^{4}{c}^{4}}} \end{align*}

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)^(1+n)*(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., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{-n - 5}{\left (d x + c\right )}^{n}\,{d x} \end{align*}

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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Fricas [B]  time = 2.36492, size = 1945, normalized size = 10.46 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{-n - 5}{\left (d x + c\right )}^{n}\,{d x} \end{align*}

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)