3.3106 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^3 \, dx\)

Optimal. Leaf size=460 \[ \frac{3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{b d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}-\frac{3 (b e-a f) (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (a d f (m+3)-b (c f (m+2)+d e))}{b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(e+f x)^3 (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}-\frac{3 f (e+f x) (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d (m+4) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.462102, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {94, 90, 79, 45, 37} \[ \frac{3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{b d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{3 (b e-a f) (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(e+f x)^3 (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}-\frac{3 f (e+f x) (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d (m+4) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^3,x]

[Out]

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

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 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])

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

Mathematica [A]  time = 0.915036, size = 263, normalized size = 0.57 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-4} \left ((e+f x)^3-\frac{3 (c+d x) (b e-a f) \left (-(c+d x) (-a d (m+1)+b c (m+2)+b d x) \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )+d f (m+1) (m+2) (m+3) (e+f x) (b c-a d)^3+(m+1) (m+2) (b c-a d)^2 (c f-d e) (-a d f (m+3)+b c f (m+2)+b d e)\right )}{b d^2 (m+1) (m+2) (m+3) (b c-a d)^3}\right )}{(m+4) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^3,x]

[Out]

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

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Maple [B]  time = 0.011, size = 2481, normalized size = 5.4 \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)^3,x)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-4-m)*(a^3*d^3*f^3*m^3*x^3-3*a^2*b*c*d^2*f^3*m^3*x^3+3*a*b^2*c^2*d*f^3*m^3*x^3-b^3*c^3
*f^3*m^3*x^3+3*a^3*d^3*e*f^2*m^3*x^2+9*a^3*d^3*f^3*m^2*x^3-9*a^2*b*c*d^2*e*f^2*m^3*x^2-24*a^2*b*c*d^2*f^3*m^2*
x^3-3*a^2*b*d^3*e*f^2*m^2*x^3+9*a*b^2*c^2*d*e*f^2*m^3*x^2+21*a*b^2*c^2*d*f^3*m^2*x^3+6*a*b^2*c*d^2*e*f^2*m^2*x
^3-3*b^3*c^3*e*f^2*m^3*x^2-6*b^3*c^3*f^3*m^2*x^3-3*b^3*c^2*d*e*f^2*m^2*x^3+3*a^3*c*d^2*f^3*m^2*x^2+3*a^3*d^3*e
^2*f*m^3*x+24*a^3*d^3*e*f^2*m^2*x^2+26*a^3*d^3*f^3*m*x^3-6*a^2*b*c^2*d*f^3*m^2*x^2-9*a^2*b*c*d^2*e^2*f*m^3*x-6
9*a^2*b*c*d^2*e*f^2*m^2*x^2-57*a^2*b*c*d^2*f^3*m*x^3-6*a^2*b*d^3*e^2*f*m^2*x^2-21*a^2*b*d^3*e*f^2*m*x^3+3*a*b^
2*c^3*f^3*m^2*x^2+9*a*b^2*c^2*d*e^2*f*m^3*x+66*a*b^2*c^2*d*e*f^2*m^2*x^2+42*a*b^2*c^2*d*f^3*m*x^3+12*a*b^2*c*d
^2*e^2*f*m^2*x^2+30*a*b^2*c*d^2*e*f^2*m*x^3+6*a*b^2*d^3*e^2*f*m*x^3-3*b^3*c^3*e^2*f*m^3*x-21*b^3*c^3*e*f^2*m^2
*x^2-11*b^3*c^3*f^3*m*x^3-6*b^3*c^2*d*e^2*f*m^2*x^2-9*b^3*c^2*d*e*f^2*m*x^3-6*b^3*c*d^2*e^2*f*m*x^3+6*a^3*c*d^
2*e*f^2*m^2*x+21*a^3*c*d^2*f^3*m*x^2+a^3*d^3*e^3*m^3+21*a^3*d^3*e^2*f*m^2*x+57*a^3*d^3*e*f^2*m*x^2+24*a^3*d^3*
f^3*x^3-12*a^2*b*c^2*d*e*f^2*m^2*x-30*a^2*b*c^2*d*f^3*m*x^2-3*a^2*b*c*d^2*e^3*m^3-66*a^2*b*c*d^2*e^2*f*m^2*x-1
74*a^2*b*c*d^2*e*f^2*m*x^2-36*a^2*b*c*d^2*f^3*x^3-3*a^2*b*d^3*e^3*m^2*x-30*a^2*b*d^3*e^2*f*m*x^2-36*a^2*b*d^3*
e*f^2*x^3+6*a*b^2*c^3*e*f^2*m^2*x+9*a*b^2*c^3*f^3*m*x^2+3*a*b^2*c^2*d*e^3*m^3+69*a*b^2*c^2*d*e^2*f*m^2*x+159*a
*b^2*c^2*d*e*f^2*m*x^2+24*a*b^2*c^2*d*f^3*x^3+6*a*b^2*c*d^2*e^3*m^2*x+60*a*b^2*c*d^2*e^2*f*m*x^2+24*a*b^2*c*d^
2*e*f^2*x^3+6*a*b^2*d^3*e^3*m*x^2+24*a*b^2*d^3*e^2*f*x^3-b^3*c^3*e^3*m^3-24*b^3*c^3*e^2*f*m^2*x-42*b^3*c^3*e*f
^2*m*x^2-6*b^3*c^3*f^3*x^3-3*b^3*c^2*d*e^3*m^2*x-30*b^3*c^2*d*e^2*f*m*x^2-6*b^3*c^2*d*e*f^2*x^3-6*b^3*c*d^2*e^
3*m*x^2-6*b^3*c*d^2*e^2*f*x^3-6*b^3*d^3*e^3*x^3+6*a^3*c^2*d*f^3*m*x+3*a^3*c*d^2*e^2*f*m^2+30*a^3*c*d^2*e*f^2*m
*x+36*a^3*c*d^2*f^3*x^2+6*a^3*d^3*e^3*m^2+42*a^3*d^3*e^2*f*m*x+36*a^3*d^3*e*f^2*x^2-6*a^2*b*c^3*f^3*m*x-6*a^2*
b*c^2*d*e^2*f*m^2-60*a^2*b*c^2*d*e*f^2*m*x-24*a^2*b*c^2*d*f^3*x^2-21*a^2*b*c*d^2*e^3*m^2-159*a^2*b*c*d^2*e^2*f
*m*x-168*a^2*b*c*d^2*e*f^2*x^2-9*a^2*b*d^3*e^3*m*x-24*a^2*b*d^3*e^2*f*x^2+3*a*b^2*c^3*e^2*f*m^2+30*a*b^2*c^3*e
*f^2*m*x+6*a*b^2*c^3*f^3*x^2+24*a*b^2*c^2*d*e^3*m^2+174*a*b^2*c^2*d*e^2*f*m*x+102*a*b^2*c^2*d*e*f^2*x^2+30*a*b
^2*c*d^2*e^3*m*x+102*a*b^2*c*d^2*e^2*f*x^2+6*a*b^2*d^3*e^3*x^2-9*b^3*c^3*e^3*m^2-57*b^3*c^3*e^2*f*m*x-24*b^3*c
^3*e*f^2*x^2-21*b^3*c^2*d*e^3*m*x-24*b^3*c^2*d*e^2*f*x^2-24*b^3*c*d^2*e^3*x^2+6*a^3*c^2*d*e*f^2*m+24*a^3*c^2*d
*f^3*x+9*a^3*c*d^2*e^2*f*m+24*a^3*c*d^2*e*f^2*x+11*a^3*d^3*e^3*m+24*a^3*d^3*e^2*f*x-6*a^2*b*c^3*e*f^2*m-6*a^2*
b*c^3*f^3*x-30*a^2*b*c^2*d*e^2*f*m-102*a^2*b*c^2*d*e*f^2*x-42*a^2*b*c*d^2*e^3*m-102*a^2*b*c*d^2*e^2*f*x-6*a^2*
b*d^3*e^3*x+21*a*b^2*c^3*e^2*f*m+24*a*b^2*c^3*e*f^2*x+57*a*b^2*c^2*d*e^3*m+168*a*b^2*c^2*d*e^2*f*x+24*a*b^2*c*
d^2*e^3*x-26*b^3*c^3*e^3*m-36*b^3*c^3*e^2*f*x-36*b^3*c^2*d*e^3*x+6*a^3*c^3*f^3+6*a^3*c^2*d*e*f^2+6*a^3*c*d^2*e
^2*f+6*a^3*d^3*e^3-24*a^2*b*c^3*e*f^2-24*a^2*b*c^2*d*e^2*f-24*a^2*b*c*d^2*e^3+36*a*b^2*c^3*e^2*f+36*a*b^2*c^2*
d*e^3-24*b^3*c^3*e^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 (f x + e\right )}^{3}{\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)^3,x, algorithm="maxima")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Fricas [B]  time = 2.6983, size = 6888, normalized size = 14.97 \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)^3,x, algorithm="fricas")

[Out]

-(6*a^4*c^4*f^3 - (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^3*m^3 - (6*b^4*d^4*e^3 + (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^3*m^3 + 6*(b^4*c*d^3 - 4*a*b^3*d^4)*e^2*f + 6*(b^4*c^
2*d^2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*e*f^2 + 6*(b^4*c^3*d - 4*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 - 4*a^3*b*d^4)
*f^3 + 3*((b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e*f^2 + (2*b^4*c^3*d - 7*a*b^3*c^2*d^2 + 8*a^2*b^2*c*d^3
- 3*a^3*b*d^4)*f^3)*m^2 + (6*(b^4*c*d^3 - a*b^3*d^4)*e^2*f + 3*(3*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + 7*a^2*b^2*d^
4)*e*f^2 + (11*b^4*c^3*d - 42*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 - 26*a^3*b*d^4)*f^3)*m)*x^5 - (30*b^4*c*d^3*e^3
+ 30*(b^4*c^2*d^2 - 4*a*b^3*c*d^3)*e^2*f + 30*(b^4*c^3*d - 4*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3)*e*f^2 + 6*(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 - 4*a^4*d^4)*f^3 + (3*(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*f^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f^3)*m^3 + 3*(2*(b^4*c
^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^2*f + (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
)*e*f^2 + (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)*f^3)*m^2 + (6*(b^4*c*d^3
- a*b^3*d^4)*e^3 + 12*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*d^4)*e^2*f + 3*(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)*e*f^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)*f^3)*m)*x^4 - 6*(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^3 + 6*(6*a^2
*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*e^2*f - 6*(4*a^3*b*c^4 - a^4*c^3*d)*e*f^2 - (60*b^4*c^2*d^2*e^3 - 60*a
^4*c*d^3*f^3 + 60*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*e^2*f + 12*(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)*e*f^2 + (3*(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^2*f + 3*(b
^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*e*f^2 + (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^3)*m^3 + 3*((b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^3 + 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^2*f + (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
)*e*f^2 + (a*b^3*c^4 - 6*a^2*b^2*c^3*d + 9*a^3*b*c^2*d^2 - 4*a^4*c*d^3)*f^3)*m^2 + (3*(9*b^4*c^2*d^2 - 10*a*b^
3*c*d^3 + a^2*b^2*d^4)*e^3 + 3*(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^2*f + 3*(1
4*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)*e*f^2 + (2*a*b^3*c^4 - 15*a^2*b
^2*c^3*d + 60*a^3*b*c^2*d^2 - 47*a^4*c*d^3)*f^3)*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)*e^3 - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*e^2*f)*m^2 - (60*b^4*c^3*d*e^3 - 60*a^4*c^2*d
^2*f^3 + 12*(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^2*f + 60*(4*a^3*b*
c^2*d^2 - a^4*c*d^3)*e*f^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^3 + 3*(b^4*c^4 - 2
*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*e^2*f + 3*(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*f^2)*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^3 + (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^2*f + 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)*e*f^2 - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*f^3)*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)*e^3 + 3*(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^2*f + 3*(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)*e*f^2 - 3*(a^2*b^2*
c^4 - 10*a^3*b*c^3*d + 9*a^4*c^2*d^2)*f^3)*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)*e^3 - 3*(7*a^2*b^2*c^4 - 10*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^2*f + 6*(a^3*b*c^4 - a^4*c^3*d)*e*f^2)*m
+ (30*a^4*c^3*d*f^3 - 6*(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^3 + 30*(6
*a^2*b^2*c^3*d - 4*a^3*b*c^2*d^2 + a^4*c*d^3)*e^2*f - 30*(4*a^3*b*c^3*d - a^4*c^2*d^2)*e*f^2 - ((b^4*c^4 - 2*a
*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*e^3 + 3*(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^2*
f)*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^3 + (7*a*b^3*c^4 - 2
2*a^2*b^2*c^3*d + 23*a^3*b*c^2*d^2 - 8*a^4*c*d^3)*e^2*f - 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*e*f^2)
*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)*e^3 + 3*(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^2*f - 12*(2*a^2*b^2*c^4 - 5*a^3*b*c^3*d + 3*a^4*c^2*d
^2)*e*f^2 + 6*(a^3*b*c^4 - a^4*c^3*d)*f^3)*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)**3,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 )}^{3}{\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)^3,x, algorithm="giac")

[Out]

integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5), x)