3.3089 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^3 \, dx\)

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))

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Rubi [A]  time = 1.34051, 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 \[ \frac{3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{b d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{3 (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )-2 a b d f (m+3) (c f (m+1)+d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{3 (b e-a f) (a+b x)^{m+1} (d e-c f) (c+d x)^{-m-3} (-a d f (m+3)+b c f (m+2)+b d e)}{b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(e+f x)^3 (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d)}-\frac{3 f (e+f x) (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-3}}{b d (m+4) (b c-a d)} \]

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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 2.9627, size = 610, normalized size = 1.33 \[ \frac{(a+b x)^m (c+d x)^{-m-4} \left (3 (m+1) (c+d x)^2 (b c-a d)^2 (c f-d e) \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+a b d f (m+4) (d e m-3 c f (m+2))+b^2 \left (2 c^2 f^2 \left (m^2+5 m+6\right )-c d e f m (m+2)-d^2 e^2 m\right )\right )-(c+d x)^3 (b c-a d) \left (-a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+3 a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (2 c f (m+1)-d e m)-3 a b^2 d f (m+4) \left (3 c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f m (m+1)-2 d^2 e^2 m\right )+b^3 \left (4 c^3 f^3 \left (m^3+6 m^2+11 m+6\right )-3 c^2 d e f^2 m \left (m^2+3 m+2\right )-6 c d^2 e^2 f m (m+1)-6 d^3 e^3 m\right )\right )+b (c+d x)^4 \left (-a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+3 a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (c f (m+1)+d e)-3 a b^2 d f (m+4) \left (c^2 f^2 \left (m^2+3 m+2\right )+2 c d e f (m+1)+2 d^2 e^2\right )+b^3 \left (c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+3 c^2 d e f^2 \left (m^2+3 m+2\right )+6 c d^2 e^2 f (m+1)+6 d^3 e^3\right )\right )-(m+1) (m+2) (c+d x) (b c-a d)^3 (d e-c f)^2 (-3 a d f (m+4)+4 b c f (m+3)-b d e m)+(m+1) (m+2) (m+3) (b c-a d)^4 (c f-d e)^3\right )}{d^4 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^m*(c + d*x)^(-4 - m)*((b*c - a*d)^4*(-(d*e) + c*f)^3*(1 + m)*(2 + m)*
(3 + m) - (b*c - a*d)^3*(d*e - c*f)^2*(1 + m)*(2 + m)*(-(b*d*e*m) + 4*b*c*f*(3 +
m) - 3*a*d*f*(4 + m))*(c + d*x) + 3*(b*c - a*d)^2*(-(d*e) + c*f)*(1 + m)*(a^2*d
^2*f^2*(12 + 7*m + m^2) + a*b*d*f*(4 + m)*(d*e*m - 3*c*f*(2 + m)) + b^2*(-(d^2*e
^2*m) - c*d*e*f*m*(2 + m) + 2*c^2*f^2*(6 + 5*m + m^2)))*(c + d*x)^2 - (b*c - a*d
)*(-(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3)) + 3*a^2*b*d^2*f^2*(12 + 7*m + m^2)*(
-(d*e*m) + 2*c*f*(1 + m)) - 3*a*b^2*d*f*(4 + m)*(-2*d^2*e^2*m - 2*c*d*e*f*m*(1 +
m) + 3*c^2*f^2*(2 + 3*m + m^2)) + b^3*(-6*d^3*e^3*m - 6*c*d^2*e^2*f*m*(1 + m) -
3*c^2*d*e*f^2*m*(2 + 3*m + m^2) + 4*c^3*f^3*(6 + 11*m + 6*m^2 + m^3)))*(c + d*x
)^3 + b*(-(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3)) + 3*a^2*b*d^2*f^2*(12 + 7*m +
m^2)*(d*e + c*f*(1 + m)) - 3*a*b^2*d*f*(4 + m)*(2*d^2*e^2 + 2*c*d*e*f*(1 + m) +
c^2*f^2*(2 + 3*m + m^2)) + b^3*(6*d^3*e^3 + 6*c*d^2*e^2*f*(1 + m) + 3*c^2*d*e*f^
2*(2 + 3*m + m^2) + c^3*f^3*(6 + 11*m + 6*m^2 + m^3)))*(c + d*x)^4))/(d^4*(b*c -
a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m))

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Maple [B]  time = 0.017, size = 2481, normalized size = 5.4 \[ \text{result too large to display} \]

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-69*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-174*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. \[ \int{\left (f x + e\right )}^{3}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.268794, size = 4618, normalized size = 10.04 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5),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*(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)*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 - (6
0*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 - 22*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. \[ \text{Timed out} \]

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/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^3*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError