3.131 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x) (g+h x) \, dx\)

Optimal. Leaf size=507 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{2 b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left (-d f h (m+4) x (b c-a d)+a c d f h (m+4)+b \left (c^2 (-f) h (m+2)-2 c d (e h+f g)+2 d^2 e g\right )\right )}{2 b d^2 (m+4) (b c-a d)} \]

[Out]

((a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 + m)) + b^2*(6*d^2*e*g + 2*c*d*(f*g
+ e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(2*b*d^2*(b*c - a*d)^2*(3 +
m)*(4 + m)) + ((a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 + m)) + b^2*(6*d^2*e*
g + 2*c*d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^2*(b*c - a*
d)^3*(2 + m)*(3 + m)*(4 + m)) + (b*(a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 +
m)) + b^2*(6*d^2*e*g + 2*c*d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1
- m))/(d^2*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) + ((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(a*c*d*f*h*(
4 + m) + b*(2*d^2*e*g - 2*c*d*(f*g + e*h) - c^2*f*h*(2 + m)) - d*(b*c - a*d)*f*h*(4 + m)*x))/(2*b*d^2*(b*c - a
*d)*(4 + m))

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Rubi [A]  time = 0.585929, antiderivative size = 507, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {146, 45, 37} \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{2 b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left (-d f h (m+4) x (b c-a d)+a c d f h (m+4)+b \left (c^2 (-f) h (m+2)-2 c d (e h+f g)+2 d^2 e g\right )\right )}{2 b d^2 (m+4) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)*(g + h*x),x]

[Out]

((a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 + m)) + b^2*(6*d^2*e*g + 2*c*d*(f*g
+ e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/(2*b*d^2*(b*c - a*d)^2*(3 +
m)*(4 + m)) + ((a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 + m)) + b^2*(6*d^2*e*
g + 2*c*d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 - m))/(d^2*(b*c - a*
d)^3*(2 + m)*(3 + m)*(4 + m)) + (b*(a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 +
m)) + b^2*(6*d^2*e*g + 2*c*d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1
- m))/(d^2*(b*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) + ((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(a*c*d*f*h*(
4 + m) + b*(2*d^2*e*g - 2*c*d*(f*g + e*h) - c^2*f*h*(2 + m)) - d*(b*c - a*d)*f*h*(4 + m)*x))/(2*b*d^2*(b*c - a
*d)*(4 + m))

Rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(
b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)), x] - Dist[
(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m +
1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d*(b*c - a*d)*(m +
1)*(m + n + 3)), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

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} (e+f x) (g+h x) \, dx &=\frac{(a+b x)^{1+m} (c+d x)^{-4-m} \left (a c d f h (4+m)+b \left (2 d^2 e g-2 c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (4+m) x\right )}{2 b d^2 (b c-a d) (4+m)}+\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{-4-m} \, dx}{2 b d^2 (b c-a d) (4+m)}\\ &=\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac{(a+b x)^{1+m} (c+d x)^{-4-m} \left (a c d f h (4+m)+b \left (2 d^2 e g-2 c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (4+m) x\right )}{2 b d^2 (b c-a d) (4+m)}+\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{-3-m} \, dx}{d^2 (b c-a d)^2 (3+m) (4+m)}\\ &=\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac{(a+b x)^{1+m} (c+d x)^{-4-m} \left (a c d f h (4+m)+b \left (2 d^2 e g-2 c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (4+m) x\right )}{2 b d^2 (b c-a d) (4+m)}+\frac{\left (b \left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right )\right ) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}\\ &=\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac{\left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac{b \left (a^2 d^2 f h \left (12+7 m+m^2\right )-2 a b d (4+m) (d (f g+e h)+c f h (1+m))+b^2 \left (6 d^2 e g+2 c d (f g+e h) (1+m)+c^2 f h \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^4 (1+m) (2+m) (3+m) (4+m)}+\frac{(a+b x)^{1+m} (c+d x)^{-4-m} \left (a c d f h (4+m)+b \left (2 d^2 e g-2 c d (f g+e h)-c^2 f h (2+m)\right )-d (b c-a d) f h (4+m) x\right )}{2 b d^2 (b c-a d) (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.76909, size = 279, normalized size = 0.55 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left (\frac{(c+d x) \left (a^2 d^2 \left (m^2+3 m+2\right )-2 a b d (m+1) (c (m+3)+d x)+b^2 \left (c^2 \left (m^2+5 m+6\right )+2 c d (m+3) x+2 d^2 x^2\right )\right ) \left (a^2 d^2 f h \left (m^2+7 m+12\right )-2 a b d (m+4) (c f h (m+1)+d (e h+f g))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )+2 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{(m+1) (m+2) (m+3) (b c-a d)^3}+a d f h (m+4) (c+d x)+b \left (c^2 (-f) h (m+2)-c d (2 e h+2 f g+f h (m+4) x)+2 d^2 e g\right )\right )}{2 b d^2 (m+4) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(a*d*f*h*(4 + m)*(c + d*x) + b*(2*d^2*e*g - c^2*f*h*(2 + m) - c*d*(2*f*g
+ 2*e*h + f*h*(4 + m)*x)) + ((a^2*d^2*f*h*(12 + 7*m + m^2) - 2*a*b*d*(4 + m)*(d*(f*g + e*h) + c*f*h*(1 + m))
+ b^2*(6*d^2*e*g + 2*c*d*(f*g + e*h)*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(c + d*x)*(a^2*d^2*(2 + 3*m + m^2) -
2*a*b*d*(1 + m)*(c*(3 + m) + d*x) + b^2*(c^2*(6 + 5*m + m^2) + 2*c*d*(3 + m)*x + 2*d^2*x^2)))/((b*c - a*d)^3*(
1 + m)*(2 + m)*(3 + m))))/(2*b*d^2*(b*c - a*d)*(4 + m))

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Maple [B]  time = 0.01, size = 2343, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)*(h*x+g),x)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-4-m)*(a^3*d^3*f*h*m^3*x^2-3*a^2*b*c*d^2*f*h*m^3*x^2-a^2*b*d^3*f*h*m^2*x^3+3*a*b^2*c^2
*d*f*h*m^3*x^2+2*a*b^2*c*d^2*f*h*m^2*x^3-b^3*c^3*f*h*m^3*x^2-b^3*c^2*d*f*h*m^2*x^3+a^3*d^3*e*h*m^3*x+a^3*d^3*f
*g*m^3*x+8*a^3*d^3*f*h*m^2*x^2-3*a^2*b*c*d^2*e*h*m^3*x-3*a^2*b*c*d^2*f*g*m^3*x-23*a^2*b*c*d^2*f*h*m^2*x^2-2*a^
2*b*d^3*e*h*m^2*x^2-2*a^2*b*d^3*f*g*m^2*x^2-7*a^2*b*d^3*f*h*m*x^3+3*a*b^2*c^2*d*e*h*m^3*x+3*a*b^2*c^2*d*f*g*m^
3*x+22*a*b^2*c^2*d*f*h*m^2*x^2+4*a*b^2*c*d^2*e*h*m^2*x^2+4*a*b^2*c*d^2*f*g*m^2*x^2+10*a*b^2*c*d^2*f*h*m*x^3+2*
a*b^2*d^3*e*h*m*x^3+2*a*b^2*d^3*f*g*m*x^3-b^3*c^3*e*h*m^3*x-b^3*c^3*f*g*m^3*x-7*b^3*c^3*f*h*m^2*x^2-2*b^3*c^2*
d*e*h*m^2*x^2-2*b^3*c^2*d*f*g*m^2*x^2-3*b^3*c^2*d*f*h*m*x^3-2*b^3*c*d^2*e*h*m*x^3-2*b^3*c*d^2*f*g*m*x^3+2*a^3*
c*d^2*f*h*m^2*x+a^3*d^3*e*g*m^3+7*a^3*d^3*e*h*m^2*x+7*a^3*d^3*f*g*m^2*x+19*a^3*d^3*f*h*m*x^2-4*a^2*b*c^2*d*f*h
*m^2*x-3*a^2*b*c*d^2*e*g*m^3-22*a^2*b*c*d^2*e*h*m^2*x-22*a^2*b*c*d^2*f*g*m^2*x-58*a^2*b*c*d^2*f*h*m*x^2-3*a^2*
b*d^3*e*g*m^2*x-10*a^2*b*d^3*e*h*m*x^2-10*a^2*b*d^3*f*g*m*x^2-12*a^2*b*d^3*f*h*x^3+2*a*b^2*c^3*f*h*m^2*x+3*a*b
^2*c^2*d*e*g*m^3+23*a*b^2*c^2*d*e*h*m^2*x+23*a*b^2*c^2*d*f*g*m^2*x+53*a*b^2*c^2*d*f*h*m*x^2+6*a*b^2*c*d^2*e*g*
m^2*x+20*a*b^2*c*d^2*e*h*m*x^2+20*a*b^2*c*d^2*f*g*m*x^2+8*a*b^2*c*d^2*f*h*x^3+6*a*b^2*d^3*e*g*m*x^2+8*a*b^2*d^
3*e*h*x^3+8*a*b^2*d^3*f*g*x^3-b^3*c^3*e*g*m^3-8*b^3*c^3*e*h*m^2*x-8*b^3*c^3*f*g*m^2*x-14*b^3*c^3*f*h*m*x^2-3*b
^3*c^2*d*e*g*m^2*x-10*b^3*c^2*d*e*h*m*x^2-10*b^3*c^2*d*f*g*m*x^2-2*b^3*c^2*d*f*h*x^3-6*b^3*c*d^2*e*g*m*x^2-2*b
^3*c*d^2*e*h*x^3-2*b^3*c*d^2*f*g*x^3-6*b^3*d^3*e*g*x^3+a^3*c*d^2*e*h*m^2+a^3*c*d^2*f*g*m^2+10*a^3*c*d^2*f*h*m*
x+6*a^3*d^3*e*g*m^2+14*a^3*d^3*e*h*m*x+14*a^3*d^3*f*g*m*x+12*a^3*d^3*f*h*x^2-2*a^2*b*c^2*d*e*h*m^2-2*a^2*b*c^2
*d*f*g*m^2-20*a^2*b*c^2*d*f*h*m*x-21*a^2*b*c*d^2*e*g*m^2-53*a^2*b*c*d^2*e*h*m*x-53*a^2*b*c*d^2*f*g*m*x-56*a^2*
b*c*d^2*f*h*x^2-9*a^2*b*d^3*e*g*m*x-8*a^2*b*d^3*e*h*x^2-8*a^2*b*d^3*f*g*x^2+a*b^2*c^3*e*h*m^2+a*b^2*c^3*f*g*m^
2+10*a*b^2*c^3*f*h*m*x+24*a*b^2*c^2*d*e*g*m^2+58*a*b^2*c^2*d*e*h*m*x+58*a*b^2*c^2*d*f*g*m*x+34*a*b^2*c^2*d*f*h
*x^2+30*a*b^2*c*d^2*e*g*m*x+34*a*b^2*c*d^2*e*h*x^2+34*a*b^2*c*d^2*f*g*x^2+6*a*b^2*d^3*e*g*x^2-9*b^3*c^3*e*g*m^
2-19*b^3*c^3*e*h*m*x-19*b^3*c^3*f*g*m*x-8*b^3*c^3*f*h*x^2-21*b^3*c^2*d*e*g*m*x-8*b^3*c^2*d*e*h*x^2-8*b^3*c^2*d
*f*g*x^2-24*b^3*c*d^2*e*g*x^2+2*a^3*c^2*d*f*h*m+3*a^3*c*d^2*e*h*m+3*a^3*c*d^2*f*g*m+8*a^3*c*d^2*f*h*x+11*a^3*d
^3*e*g*m+8*a^3*d^3*e*h*x+8*a^3*d^3*f*g*x-2*a^2*b*c^3*f*h*m-10*a^2*b*c^2*d*e*h*m-10*a^2*b*c^2*d*f*g*m-34*a^2*b*
c^2*d*f*h*x-42*a^2*b*c*d^2*e*g*m-34*a^2*b*c*d^2*e*h*x-34*a^2*b*c*d^2*f*g*x-6*a^2*b*d^3*e*g*x+7*a*b^2*c^3*e*h*m
+7*a*b^2*c^3*f*g*m+8*a*b^2*c^3*f*h*x+57*a*b^2*c^2*d*e*g*m+56*a*b^2*c^2*d*e*h*x+56*a*b^2*c^2*d*f*g*x+24*a*b^2*c
*d^2*e*g*x-26*b^3*c^3*e*g*m-12*b^3*c^3*e*h*x-12*b^3*c^3*f*g*x-36*b^3*c^2*d*e*g*x+2*a^3*c^2*d*f*h+2*a^3*c*d^2*e
*h+2*a^3*c*d^2*f*g+6*a^3*d^3*e*g-8*a^2*b*c^3*f*h-8*a^2*b*c^2*d*e*h-8*a^2*b*c^2*d*f*g-24*a^2*b*c*d^2*e*g+12*a*b
^2*c^3*e*h+12*a*b^2*c^3*f*g+36*a*b^2*c^2*d*e*g-24*b^3*c^3*e*g)/(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 (f x + e\right )}{\left (h x + g\right )}{\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)*(f*x+e)*(h*x+g),x, algorithm="maxima")

[Out]

integrate((f*x + e)*(h*x + g)*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Fricas [B]  time = 1.97079, size = 6969, normalized size = 13.75 \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)*(f*x+e)*(h*x+g),x, algorithm="fricas")

[Out]

((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)*e*g*m^3 + ((b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2
*d^4)*f*h*m^2 + 2*(3*b^4*d^4*e + (b^4*c*d^3 - 4*a*b^3*d^4)*f)*g + 2*((b^4*c*d^3 - 4*a*b^3*d^4)*e + (b^4*c^2*d^
2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*f)*h + (2*(b^4*c*d^3 - a*b^3*d^4)*f*g + (2*(b^4*c*d^3 - a*b^3*d^4)*e + (3*b
^4*c^2*d^2 - 10*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*f)*h)*m)*x^5 + ((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)*f*h*m^3 + (2*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f*g + (2*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^
2*b^2*d^4)*e + (8*b^4*c^3*d - 23*a*b^3*c^2*d^2 + 22*a^2*b^2*c*d^3 - 7*a^3*b*d^4)*f)*h)*m^2 + 10*(3*b^4*c*d^3*e
+ (b^4*c^2*d^2 - 4*a*b^3*c*d^3)*f)*g + 10*((b^4*c^2*d^2 - 4*a*b^3*c*d^3)*e + (b^4*c^3*d - 4*a*b^3*c^2*d^2 + 6
*a^2*b^2*c*d^3)*f)*h + (2*(3*(b^4*c*d^3 - a*b^3*d^4)*e + 2*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*f)*
g + (4*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*e + (17*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 55*a^2*b^2*c*d^3
- 12*a^3*b*d^4)*f)*h)*m)*x^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)*f*g + ((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)*e + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f)*
h)*m^3 + ((3*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e + 5*(2*b^4*c^3*d - 5*a*b^3*c^2*d^2 + 4*a^2*b^2*c*d^
3 - a^3*b*d^4)*f)*g + (5*(2*b^4*c^3*d - 5*a*b^3*c^2*d^2 + 4*a^2*b^2*c*d^3 - a^3*b*d^4)*e + (7*b^4*c^4 - 16*a*b
^3*c^3*d + 3*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 8*a^4*d^4)*f)*h)*m^2 + 20*(3*b^4*c^2*d^2*e + (b^4*c^3*d - 4*a*
b^3*c^2*d^2)*f)*g + 4*(5*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*e + (2*b^4*c^4 - 8*a*b^3*c^3*d + 12*a^2*b^2*c^2*d^2 + 1
2*a^3*b*c*d^3 - 3*a^4*d^4)*f)*h + ((3*(9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*e + (29*b^4*c^3*d - 66*a*
b^3*c^2*d^2 + 41*a^2*b^2*c*d^3 - 4*a^3*b*d^4)*f)*g + ((29*b^4*c^3*d - 66*a*b^3*c^2*d^2 + 41*a^2*b^2*c*d^3 - 4*
a^3*b*d^4)*e + (14*b^4*c^4 - 46*a*b^3*c^3*d + 15*a^2*b^2*c^2*d^2 + 36*a^3*b*c*d^3 - 19*a^4*d^4)*f)*h)*m)*x^3 -
((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*e*h - (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)*e - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f)*g)*m^2 + ((((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)*e + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f)*g + ((b^4*c^4 - 2*a*b^3*c^3*d
+ 2*a^3*b*c*d^3 - a^4*d^4)*e + (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)*f)*h)*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)*e + (8*b^4*c^4 - 14*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^
2 + 16*a^3*b*c*d^3 - 7*a^4*d^4)*f)*g + ((8*b^4*c^4 - 14*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 - 7*a
^4*d^4)*e + 5*(a*b^3*c^4 - 4*a^2*b^2*c^3*d + 5*a^3*b*c^2*d^2 - 2*a^4*c*d^3)*f)*h)*m^2 + 4*(15*b^4*c^3*d*e + (3
*b^4*c^4 - 12*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 8*a^3*b*c*d^3 - 2*a^4*d^4)*f)*g + 4*((3*b^4*c^4 - 12*a*b^3*c^
3*d - 12*a^2*b^2*c^2*d^2 + 8*a^3*b*c*d^3 - 2*a^4*d^4)*e + 5*(4*a^3*b*c^2*d^2 - a^4*c*d^3)*f)*h + (((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)*e + (19*b^4*c^4 - 36*a*b^3*c^3*d - 15*a^2*b^2*c^2*d^2
+ 46*a^3*b*c*d^3 - 14*a^4*d^4)*f)*g + ((19*b^4*c^4 - 36*a*b^3*c^3*d - 15*a^2*b^2*c^2*d^2 + 46*a^3*b*c*d^3 - 14
*a^4*d^4)*e + (4*a*b^3*c^4 - 41*a^2*b^2*c^3*d + 66*a^3*b*c^2*d^2 - 29*a^4*c*d^3)*f)*h)*m)*x^2 + 2*(3*(4*a*b^3*
c^4 - 6*a^2*b^2*c^3*d + 4*a^3*b*c^2*d^2 - a^4*c*d^3)*e - (6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*f)*g -
2*((6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*e - (4*a^3*b*c^4 - a^4*c^3*d)*f)*h + (((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)*e - (7*a^2*b^2*c^4 - 10*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f)*g - ((7*
a^2*b^2*c^4 - 10*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e - 2*(a^3*b*c^4 - a^4*c^3*d)*f)*h)*m + (((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)*e*h + ((b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*e + (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)*f)*g)*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)*e + (7*a*b^3*c^4 - 22*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 8*a^4*c*d^3)*f)*g +
((7*a*b^3*c^4 - 22*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 8*a^4*c*d^3)*e - 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^
2*d^2)*f)*h)*m^2 + 2*(3*(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)*e - 5*(6*a^2
*b^2*c^3*d - 4*a^3*b*c^2*d^2 + a^4*c*d^3)*f)*g - 10*((6*a^2*b^2*c^3*d - 4*a^3*b*c^2*d^2 + a^4*c*d^3)*e - (4*a^
3*b*c^3*d - a^4*c^2*d^2)*f)*h + (((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)*e + (12*a*b^3*c^4 - 55*a^2*b^2*c^3*d + 60*a^3*b*c^2*d^2 - 17*a^4*c*d^3)*f)*g + ((12*a*b^3*c^4 - 55*a^2*b^
2*c^3*d + 60*a^3*b*c^2*d^2 - 17*a^4*c*d^3)*e - 4*(2*a^2*b^2*c^4 - 5*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f)*h)*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)*(f*x+e)*(h*x+g),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (h x + g\right )}{\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)*(f*x+e)*(h*x+g),x, algorithm="giac")

[Out]

integrate((f*x + e)*(h*x + g)*(b*x + a)^m*(d*x + c)^(-m - 5), x)