3.3090 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx\)

Optimal. Leaf size=494 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{2 b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}-\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (a d f (m+4)-b (c f (m+2)+2 d e))}{2 b d^2 (m+4) (b c-a d)}-\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d} \]

[Out]

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

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Rubi [A]  time = 1.45797, antiderivative size = 493, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{2 b d^2 (m+3) (m+4) (b c-a d)^2}+\frac{(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac{b (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (-a d f (m+4)+b c f (m+2)+2 b d e)}{2 b d^2 (m+4) (b c-a d)}-\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^2,x]

[Out]

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

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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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.70017, size = 444, normalized size = 0.9 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{-a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (c f (2 m+3)-d e m)+b^2 \left (-3 c^2 f^2 (m+2)^2+2 c d e f m (m+1)+3 d^2 e^2 m\right )}{(m+2) \left (m^2+7 m+12\right ) (c+d x)^2 (b c-a d)^2}+\frac{b^2 \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{(m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac{b m \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{(m+1) \left (m^3+9 m^2+26 m+24\right ) (c+d x) (b c-a d)^3}+\frac{(d e-c f) (2 a d f (m+4)-b c f (3 m+8)+b d e m)}{(m+3) (m+4) (c+d x)^3 (b c-a d)}-\frac{(d e-c f)^2}{(m+4) (c+d x)^4}\right )}{d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^m*(c + d*x)^(-5 - m)*(e + f*x)^2,x]

[Out]

((a + b*x)^m*((b^2*(a^2*d^2*f^2*(12 + 7*m + m^2) - 2*a*b*d*f*(4 + m)*(2*d*e + c*
f*(1 + m)) + b^2*(6*d^2*e^2 + 4*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2))))/((b
*c - a*d)^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)) - (d*e - c*f)^2/((4 + m)*(c + d*x)^
4) + ((d*e - c*f)*(b*d*e*m + 2*a*d*f*(4 + m) - b*c*f*(8 + 3*m)))/((b*c - a*d)*(3
+ m)*(4 + m)*(c + d*x)^3) + (-(a^2*d^2*f^2*(12 + 7*m + m^2)) + b^2*(3*d^2*e^2*m
+ 2*c*d*e*f*m*(1 + m) - 3*c^2*f^2*(2 + m)^2) + 2*a*b*d*f*(4 + m)*(-(d*e*m) + c*
f*(3 + 2*m)))/((b*c - a*d)^2*(2 + m)*(12 + 7*m + m^2)*(c + d*x)^2) + (b*m*(a^2*d
^2*f^2*(12 + 7*m + m^2) - 2*a*b*d*f*(4 + m)*(2*d*e + c*f*(1 + m)) + b^2*(6*d^2*e
^2 + 4*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2))))/((b*c - a*d)^3*(1 + m)*(24 +
26*m + 9*m^2 + m^3)*(c + d*x))))/(d^3*(c + d*x)^m)

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Maple [B]  time = 0.02, size = 1884, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-5-m)*(f*x+e)^2,x)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-4-m)*(a^3*d^3*f^2*m^3*x^2-3*a^2*b*c*d^2*f^2*m^3*x^2-a^2
*b*d^3*f^2*m^2*x^3+3*a*b^2*c^2*d*f^2*m^3*x^2+2*a*b^2*c*d^2*f^2*m^2*x^3-b^3*c^3*f
^2*m^3*x^2-b^3*c^2*d*f^2*m^2*x^3+2*a^3*d^3*e*f*m^3*x+8*a^3*d^3*f^2*m^2*x^2-6*a^2
*b*c*d^2*e*f*m^3*x-23*a^2*b*c*d^2*f^2*m^2*x^2-4*a^2*b*d^3*e*f*m^2*x^2-7*a^2*b*d^
3*f^2*m*x^3+6*a*b^2*c^2*d*e*f*m^3*x+22*a*b^2*c^2*d*f^2*m^2*x^2+8*a*b^2*c*d^2*e*f
*m^2*x^2+10*a*b^2*c*d^2*f^2*m*x^3+4*a*b^2*d^3*e*f*m*x^3-2*b^3*c^3*e*f*m^3*x-7*b^
3*c^3*f^2*m^2*x^2-4*b^3*c^2*d*e*f*m^2*x^2-3*b^3*c^2*d*f^2*m*x^3-4*b^3*c*d^2*e*f*
m*x^3+2*a^3*c*d^2*f^2*m^2*x+a^3*d^3*e^2*m^3+14*a^3*d^3*e*f*m^2*x+19*a^3*d^3*f^2*
m*x^2-4*a^2*b*c^2*d*f^2*m^2*x-3*a^2*b*c*d^2*e^2*m^3-44*a^2*b*c*d^2*e*f*m^2*x-58*
a^2*b*c*d^2*f^2*m*x^2-3*a^2*b*d^3*e^2*m^2*x-20*a^2*b*d^3*e*f*m*x^2-12*a^2*b*d^3*
f^2*x^3+2*a*b^2*c^3*f^2*m^2*x+3*a*b^2*c^2*d*e^2*m^3+46*a*b^2*c^2*d*e*f*m^2*x+53*
a*b^2*c^2*d*f^2*m*x^2+6*a*b^2*c*d^2*e^2*m^2*x+40*a*b^2*c*d^2*e*f*m*x^2+8*a*b^2*c
*d^2*f^2*x^3+6*a*b^2*d^3*e^2*m*x^2+16*a*b^2*d^3*e*f*x^3-b^3*c^3*e^2*m^3-16*b^3*c
^3*e*f*m^2*x-14*b^3*c^3*f^2*m*x^2-3*b^3*c^2*d*e^2*m^2*x-20*b^3*c^2*d*e*f*m*x^2-2
*b^3*c^2*d*f^2*x^3-6*b^3*c*d^2*e^2*m*x^2-4*b^3*c*d^2*e*f*x^3-6*b^3*d^3*e^2*x^3+2
*a^3*c*d^2*e*f*m^2+10*a^3*c*d^2*f^2*m*x+6*a^3*d^3*e^2*m^2+28*a^3*d^3*e*f*m*x+12*
a^3*d^3*f^2*x^2-4*a^2*b*c^2*d*e*f*m^2-20*a^2*b*c^2*d*f^2*m*x-21*a^2*b*c*d^2*e^2*
m^2-106*a^2*b*c*d^2*e*f*m*x-56*a^2*b*c*d^2*f^2*x^2-9*a^2*b*d^3*e^2*m*x-16*a^2*b*
d^3*e*f*x^2+2*a*b^2*c^3*e*f*m^2+10*a*b^2*c^3*f^2*m*x+24*a*b^2*c^2*d*e^2*m^2+116*
a*b^2*c^2*d*e*f*m*x+34*a*b^2*c^2*d*f^2*x^2+30*a*b^2*c*d^2*e^2*m*x+68*a*b^2*c*d^2
*e*f*x^2+6*a*b^2*d^3*e^2*x^2-9*b^3*c^3*e^2*m^2-38*b^3*c^3*e*f*m*x-8*b^3*c^3*f^2*
x^2-21*b^3*c^2*d*e^2*m*x-16*b^3*c^2*d*e*f*x^2-24*b^3*c*d^2*e^2*x^2+2*a^3*c^2*d*f
^2*m+6*a^3*c*d^2*e*f*m+8*a^3*c*d^2*f^2*x+11*a^3*d^3*e^2*m+16*a^3*d^3*e*f*x-2*a^2
*b*c^3*f^2*m-20*a^2*b*c^2*d*e*f*m-34*a^2*b*c^2*d*f^2*x-42*a^2*b*c*d^2*e^2*m-68*a
^2*b*c*d^2*e*f*x-6*a^2*b*d^3*e^2*x+14*a*b^2*c^3*e*f*m+8*a*b^2*c^3*f^2*x+57*a*b^2
*c^2*d*e^2*m+112*a*b^2*c^2*d*e*f*x+24*a*b^2*c*d^2*e^2*x-26*b^3*c^3*e^2*m-24*b^3*
c^3*e*f*x-36*b^3*c^2*d*e^2*x+2*a^3*c^2*d*f^2+4*a^3*c*d^2*e*f+6*a^3*d^3*e^2-8*a^2
*b*c^3*f^2-16*a^2*b*c^2*d*e*f-24*a^2*b*c*d^2*e^2+24*a*b^2*c^3*e*f+36*a*b^2*c^2*d
*e^2-24*b^3*c^3*e^2)/(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+1
44*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. \[ \int{\left (f x + e\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="maxima")

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 5), x)

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Fricas [A]  time = 0.266098, size = 3594, normalized size = 7.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 5),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^2*m^3 + (6*b^4*d^
4*e^2 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*f^2*m^2 + 4*(b^4*c*d^3 - 4*a
*b^3*d^4)*e*f + 2*(b^4*c^2*d^2 - 4*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*f^2 + (4*(b^4*c*
d^3 - a*b^3*d^4)*e*f + (3*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*f^2)*m)*
x^5 + (30*b^4*c*d^3*e^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)*f^2*m^3 + 20*(b^4*c^2*d^2 - 4*a*b^3*c*d^3)*e*f + 10*(b^4*c^3*d - 4*a*b^3*c
^2*d^2 + 6*a^2*b^2*c*d^3)*f^2 + (4*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e
*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)*f^2)*m^2
+ (6*(b^4*c*d^3 - a*b^3*d^4)*e^2 + 8*(3*b^4*c^2*d^2 - 5*a*b^3*c*d^3 + 2*a^2*b^2*
d^4)*e*f + (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
^2)*m)*x^4 + (60*b^4*c^2*d^2*e^2 + (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*f + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*f^2)
*m^3 + 40*(b^4*c^3*d - 4*a*b^3*c^2*d^2)*e*f + 4*(2*b^4*c^4 - 8*a*b^3*c^3*d + 12*
a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 - 3*a^4*d^4)*f^2 + (3*(b^4*c^2*d^2 - 2*a*b^3*c*
d^3 + a^2*b^2*d^4)*e^2 + 10*(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*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)*f^2)*m^2 + (3*(9*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + a^2*b^2*d^4)*e^2 +
2*(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*f + (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^2
)*m)*x^3 + 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^2 -
4*(6*a^2*b^2*c^4 - 4*a^3*b*c^3*d + a^4*c^2*d^2)*e*f + 2*(4*a^3*b*c^4 - a^4*c^3*
d)*f^2 + (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^2
- 2*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*e*f)*m^2 + (60*b^4*c^3*d*e^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^2 + 2*(b^4*c^4 - 2
*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*e*f + (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^2)*m^3 + 8*(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*f + 20*(4*a^3*b*c^2*d^2 - a^4*c*d^3)*f^2
+ (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^2 + 2*(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*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)*f^2)*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^2 + 2*(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*f + (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^2)*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^2 - 2*(7*a^
2*b^2*c^4 - 10*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e*f + 2*(a^3*b*c^4 - a^4*c^3*d)*f^2)
*m + (((b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*e^2 + 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)*e*f)*m^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^2 - 20*(6*a^2*b^2*c^3*d
- 4*a^3*b*c^2*d^2 + a^4*c*d^3)*e*f + 10*(4*a^3*b*c^3*d - a^4*c^2*d^2)*f^2 + (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^2 +
2*(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*f - 2*(a^
2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*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^2 + 2*(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*f - 4*(2*a^2*b^2*c^4 - 5*a^3
*b*c^3*d + 3*a^4*c^2*d^2)*f^2)*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 5),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 5), x)