3.3109 \(\int (a+b x)^m (c+d x)^{-5-m} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0586535, antiderivative size = 185, 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} \[ \frac{6 b^2 (a+b x)^{m+1} (c+d x)^{-m-2}}{(m+2) (m+3) (m+4) (b c-a d)^3}+\frac{6 b^3 (a+b x)^{m+1} (c+d x)^{-m-1}}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}+\frac{3 b (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (m+4) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-5 - m),x]

[Out]

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

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

Mathematica [A]  time = 0.0888319, size = 195, normalized size = 1.05 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left (3 a^2 b d^2 \left (m^2+3 m+2\right ) (c (m+4)+d x)-a^3 d^3 \left (m^3+6 m^2+11 m+6\right )-3 a b^2 d (m+1) \left (c^2 \left (m^2+7 m+12\right )+2 c d (m+4) x+2 d^2 x^2\right )+b^3 \left (3 c^2 d \left (m^2+7 m+12\right ) x+c^3 \left (m^3+9 m^2+26 m+24\right )+6 c d^2 (m+4) x^2+6 d^3 x^3\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 662, normalized size = 3.6 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-4-m} \left ({a}^{3}{d}^{3}{m}^{3}-3\,{a}^{2}bc{d}^{2}{m}^{3}-3\,{a}^{2}b{d}^{3}{m}^{2}x+3\,a{b}^{2}{c}^{2}d{m}^{3}+6\,a{b}^{2}c{d}^{2}{m}^{2}x+6\,a{b}^{2}{d}^{3}m{x}^{2}-{b}^{3}{c}^{3}{m}^{3}-3\,{b}^{3}{c}^{2}d{m}^{2}x-6\,{b}^{3}c{d}^{2}m{x}^{2}-6\,{b}^{3}{d}^{3}{x}^{3}+6\,{a}^{3}{d}^{3}{m}^{2}-21\,{a}^{2}bc{d}^{2}{m}^{2}-9\,{a}^{2}b{d}^{3}mx+24\,a{b}^{2}{c}^{2}d{m}^{2}+30\,a{b}^{2}c{d}^{2}mx+6\,a{b}^{2}{d}^{3}{x}^{2}-9\,{b}^{3}{c}^{3}{m}^{2}-21\,{b}^{3}{c}^{2}dmx-24\,{b}^{3}c{d}^{2}{x}^{2}+11\,{a}^{3}{d}^{3}m-42\,{a}^{2}bc{d}^{2}m-6\,{a}^{2}b{d}^{3}x+57\,a{b}^{2}{c}^{2}dm+24\,a{b}^{2}c{d}^{2}x-26\,{b}^{3}{c}^{3}m-36\,{b}^{3}{c}^{2}dx+6\,{a}^{3}{d}^{3}-24\,{a}^{2}cb{d}^{2}+36\,a{b}^{2}{c}^{2}d-24\,{b}^{3}{c}^{3} \right ) }{{a}^{4}{d}^{4}{m}^{4}-4\,{a}^{3}bc{d}^{3}{m}^{4}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{m}^{4}-4\,a{b}^{3}{c}^{3}d{m}^{4}+{b}^{4}{c}^{4}{m}^{4}+10\,{a}^{4}{d}^{4}{m}^{3}-40\,{a}^{3}bc{d}^{3}{m}^{3}+60\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{m}^{3}-40\,a{b}^{3}{c}^{3}d{m}^{3}+10\,{b}^{4}{c}^{4}{m}^{3}+35\,{a}^{4}{d}^{4}{m}^{2}-140\,{a}^{3}bc{d}^{3}{m}^{2}+210\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}{m}^{2}-140\,a{b}^{3}{c}^{3}d{m}^{2}+35\,{b}^{4}{c}^{4}{m}^{2}+50\,{a}^{4}{d}^{4}m-200\,{a}^{3}bc{d}^{3}m+300\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}m-200\,a{b}^{3}{c}^{3}dm+50\,{b}^{4}{c}^{4}m+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)^m*(d*x+c)^(-5-m),x)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-4-m)*(a^3*d^3*m^3-3*a^2*b*c*d^2*m^3-3*a^2*b*d^3*m^2*x+3*a*b^2*c^2*d*m^3+6*a*b^2*c*d^2
*m^2*x+6*a*b^2*d^3*m*x^2-b^3*c^3*m^3-3*b^3*c^2*d*m^2*x-6*b^3*c*d^2*m*x^2-6*b^3*d^3*x^3+6*a^3*d^3*m^2-21*a^2*b*
c*d^2*m^2-9*a^2*b*d^3*m*x+24*a*b^2*c^2*d*m^2+30*a*b^2*c*d^2*m*x+6*a*b^2*d^3*x^2-9*b^3*c^3*m^2-21*b^3*c^2*d*m*x
-24*b^3*c*d^2*x^2+11*a^3*d^3*m-42*a^2*b*c*d^2*m-6*a^2*b*d^3*x+57*a*b^2*c^2*d*m+24*a*b^2*c*d^2*x-26*b^3*c^3*m-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*m^4-4*a^3*b*c*d^3*m^4+6*a^2*b^2*c^2
*d^2*m^4-4*a*b^3*c^3*d*m^4+b^4*c^4*m^4+10*a^4*d^4*m^3-40*a^3*b*c*d^3*m^3+60*a^2*b^2*c^2*d^2*m^3-40*a*b^3*c^3*d
*m^3+10*b^4*c^4*m^3+35*a^4*d^4*m^2-140*a^3*b*c*d^3*m^2+210*a^2*b^2*c^2*d^2*m^2-140*a*b^3*c^3*d*m^2+35*b^4*c^4*
m^2+50*a^4*d^4*m-200*a^3*b*c*d^3*m+300*a^2*b^2*c^2*d^2*m-200*a*b^3*c^3*d*m+50*b^4*c^4*m+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 )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.02365, size = 1945, normalized size = 10.51 \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)^m*(d*x+c)^(-5-m),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)*m)*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)*m^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)*m^2 + (9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*m)*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)*m^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)*m^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)*m^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)*m)*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)*m + (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)*m^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)*m^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)*m)*x)*(b*x + a)^m*(d*x + c)^(-m - 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)*m^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)*m^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)*m^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)*m)

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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)**m*(d*x+c)**(-5-m),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 )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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