3.3107 \(\int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx\)

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]

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

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Rubi [A]  time = 0.54, antiderivative size = 493, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {90, 79, 45, 37} \[ \frac {(a+b x)^{m+1} (c+d x)^{-m-3} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{2 b d^2 (m+3) (m+4) (b c-a d)^2}+\frac {(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d^2 (m+2) (m+3) (m+4) (b c-a d)^3}+\frac {b (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d^2 (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-4} (-a d f (m+4)+b c f (m+2)+2 b d e)}{2 b d^2 (m+4) (b c-a d)}-\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-4}}{2 b d} \]

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)

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]

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 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 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]

Rubi steps

\begin {align*} \int (a+b x)^m (c+d x)^{-5-m} (e+f x)^2 \, dx &=-\frac {f (a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)}{2 b d}-\frac {\int (a+b x)^m (c+d x)^{-5-m} (-b e (2 d e+c f (1+m))-a f (c f-d e (4+m))+f (a d f (3+m)-b (d e+c f (2+m))) x) \, dx}{2 b d}\\ &=\frac {(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)}-\frac {f (a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)}{2 b d}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 c d e f (1+m)+c^2 f^2 \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 {(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)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)}{2 b d}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 c d e f (1+m)+c^2 f^2 \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 {(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)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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}}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)}{2 b d}+\frac {\left (b \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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 {(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)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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)^{-3-m}}{2 b d^2 (b c-a d)^2 (3+m) (4+m)}+\frac {\left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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}}{d^2 (b c-a d)^3 (2+m) (3+m) (4+m)}+\frac {b \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )-2 a b d f (4+m) (2 d e+c f (1+m))+b^2 \left (6 d^2 e^2+4 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 {f (a+b x)^{1+m} (c+d x)^{-4-m} (e+f x)}{2 b d}\\ \end {align*}

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Mathematica [A]  time = 0.68, size = 269, normalized size = 0.54 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m-4} \left (\frac {(c+d x) \left (a^2 d^2 \left (m^2+3 m+2\right )-2 a b d (m+1) (c (m+3)+d x)+b^2 \left (c^2 \left (m^2+5 m+6\right )+2 c d (m+3) x+2 d^2 x^2\right )\right ) \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )-2 a b d f (m+4) (c f (m+1)+2 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )+4 c d e f (m+1)+6 d^2 e^2\right )\right )}{d (m+1) (m+2) (m+3) (m+4) (b c-a d)^4}-\frac {(d e-c f) (-a d f (m+4)+b c f (m+2)+2 b d e)}{d (m+4) (a d-b c)}-f (e+f x)\right )}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.88, size = 2662, normalized size = 5.39 \[ \text {result too large to display} \]

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, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.02, size = 1884, normalized size = 3.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x+a)^(m+1)*(d*x+c)^(-m-4)*(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+5
0*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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.79, size = 2932, normalized size = 5.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3*(a + b*x)^m*(8*b^4*c^4*f^2 - 12*a^4*d^4*f^2 - 19*a^4*d^4*f^2*m + 14*b^4*c^4*f^2*m + 60*b^4*c^2*d^2*e^2 -
8*a^4*d^4*f^2*m^2 + 7*b^4*c^4*f^2*m^2 - a^4*d^4*f^2*m^3 + b^4*c^4*f^2*m^3 + 40*b^4*c^3*d*e*f + 48*a^2*b^2*c^2*
d^2*f^2 + 3*a^2*b^2*d^4*e^2*m^2 + 3*b^4*c^2*d^2*e^2*m^2 - 32*a*b^3*c^3*d*f^2 + 48*a^3*b*c*d^3*f^2 + 3*a^2*b^2*
d^4*e^2*m + 27*b^4*c^2*d^2*e^2*m - 6*a*b^3*c*d^3*e^2*m^2 - 16*a*b^3*c^3*d*f^2*m^2 + 14*a^3*b*c*d^3*f^2*m^2 - 2
*a*b^3*c^3*d*f^2*m^3 + 2*a^3*b*c*d^3*f^2*m^3 - 8*a^3*b*d^4*e*f*m + 58*b^4*c^3*d*e*f*m + 15*a^2*b^2*c^2*d^2*f^2
*m - 160*a*b^3*c^2*d^2*e*f - 30*a*b^3*c*d^3*e^2*m - 46*a*b^3*c^3*d*f^2*m + 36*a^3*b*c*d^3*f^2*m - 10*a^3*b*d^4
*e*f*m^2 - 2*a^3*b*d^4*e*f*m^3 + 20*b^4*c^3*d*e*f*m^2 + 2*b^4*c^3*d*e*f*m^3 + 3*a^2*b^2*c^2*d^2*f^2*m^2 - 132*
a*b^3*c^2*d^2*e*f*m + 82*a^2*b^2*c*d^3*e*f*m - 50*a*b^3*c^2*d^2*e*f*m^2 + 40*a^2*b^2*c*d^3*e*f*m^2 - 6*a*b^3*c
^2*d^2*e*f*m^3 + 6*a^2*b^2*c*d^3*e*f*m^3))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24
)) - ((a + b*x)^m*(6*a^4*c*d^3*e^2 - 8*a^3*b*c^4*f^2 - 24*a*b^3*c^4*e^2 + 2*a^4*c^3*d*f^2 + 24*a^2*b^2*c^4*e*f
+ 4*a^4*c^2*d^2*e*f - 26*a*b^3*c^4*e^2*m - 2*a^3*b*c^4*f^2*m + 11*a^4*c*d^3*e^2*m + 2*a^4*c^3*d*f^2*m + 36*a^
2*b^2*c^3*d*e^2 - 24*a^3*b*c^2*d^2*e^2 - 9*a*b^3*c^4*e^2*m^2 - a*b^3*c^4*e^2*m^3 + 6*a^4*c*d^3*e^2*m^2 + a^4*c
*d^3*e^2*m^3 + 57*a^2*b^2*c^3*d*e^2*m - 42*a^3*b*c^2*d^2*e^2*m + 2*a^2*b^2*c^4*e*f*m^2 + 2*a^4*c^2*d^2*e*f*m^2
- 16*a^3*b*c^3*d*e*f + 24*a^2*b^2*c^3*d*e^2*m^2 - 21*a^3*b*c^2*d^2*e^2*m^2 + 3*a^2*b^2*c^3*d*e^2*m^3 - 3*a^3*
b*c^2*d^2*e^2*m^3 + 14*a^2*b^2*c^4*e*f*m + 6*a^4*c^2*d^2*e*f*m - 4*a^3*b*c^3*d*e*f*m^2 - 20*a^3*b*c^3*d*e*f*m)
)/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - (x^2*(a + b*x)^m*(20*a^4*c*d^3*f^2 -
60*b^4*c^3*d*e^2 + 16*a^4*d^4*e*f - 24*b^4*c^4*e*f + 28*a^4*d^4*e*f*m - 38*b^4*c^4*e*f*m - 4*a*b^3*c^4*f^2*m
+ 2*a^3*b*d^4*e^2*m + 29*a^4*c*d^3*f^2*m - 47*b^4*c^3*d*e^2*m + 14*a^4*d^4*e*f*m^2 - 16*b^4*c^4*e*f*m^2 + 2*a^
4*d^4*e*f*m^3 - 2*b^4*c^4*e*f*m^3 - 80*a^3*b*c^2*d^2*f^2 - 5*a*b^3*c^4*f^2*m^2 + 3*a^3*b*d^4*e^2*m^2 - a*b^3*c
^4*f^2*m^3 + a^3*b*d^4*e^2*m^3 + 10*a^4*c*d^3*f^2*m^2 - 12*b^4*c^3*d*e^2*m^2 + a^4*c*d^3*f^2*m^3 - b^4*c^3*d*e
^2*m^3 + 96*a^2*b^2*c^2*d^2*e*f + 60*a*b^3*c^2*d^2*e^2*m - 15*a^2*b^2*c*d^3*e^2*m + 41*a^2*b^2*c^3*d*f^2*m - 6
6*a^3*b*c^2*d^2*f^2*m + 96*a*b^3*c^3*d*e*f - 64*a^3*b*c*d^3*e*f + 27*a*b^3*c^2*d^2*e^2*m^2 - 18*a^2*b^2*c*d^3*
e^2*m^2 + 3*a*b^3*c^2*d^2*e^2*m^3 - 3*a^2*b^2*c*d^3*e^2*m^3 + 20*a^2*b^2*c^3*d*f^2*m^2 - 25*a^3*b*c^2*d^2*f^2*
m^2 + 3*a^2*b^2*c^3*d*f^2*m^3 - 3*a^3*b*c^2*d^2*f^2*m^3 + 28*a*b^3*c^3*d*e*f*m^2 - 32*a^3*b*c*d^3*e*f*m^2 + 4*
a*b^3*c^3*d*e*f*m^3 - 4*a^3*b*c*d^3*e*f*m^3 + 30*a^2*b^2*c^2*d^2*e*f*m + 72*a*b^3*c^3*d*e*f*m - 92*a^3*b*c*d^3
*e*f*m + 6*a^2*b^2*c^2*d^2*e*f*m^2))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) - (
x*(a + b*x)^m*(6*a^4*d^4*e^2 - 24*b^4*c^4*e^2 + 11*a^4*d^4*e^2*m - 26*b^4*c^4*e^2*m + 10*a^4*c^2*d^2*f^2 + 6*a
^4*d^4*e^2*m^2 - 9*b^4*c^4*e^2*m^2 + a^4*d^4*e^2*m^3 - b^4*c^4*e^2*m^3 + 20*a^4*c*d^3*e*f + 36*a^2*b^2*c^2*d^2
*e^2 + 2*a^2*b^2*c^4*f^2*m^2 + 2*a^4*c^2*d^2*f^2*m^2 - 24*a*b^3*c^3*d*e^2 - 24*a^3*b*c*d^3*e^2 - 40*a^3*b*c^3*
d*f^2 + 8*a^2*b^2*c^4*f^2*m + 12*a^4*c^2*d^2*f^2*m + 12*a*b^3*c^3*d*e^2*m^2 - 18*a^3*b*c*d^3*e^2*m^2 + 2*a*b^3
*c^3*d*e^2*m^3 - 2*a^3*b*c*d^3*e^2*m^3 - 4*a^3*b*c^3*d*f^2*m^2 - 24*a*b^3*c^4*e*f*m + 34*a^4*c*d^3*e*f*m + 45*
a^2*b^2*c^2*d^2*e^2*m + 120*a^2*b^2*c^3*d*e*f - 80*a^3*b*c^2*d^2*e*f + 10*a*b^3*c^3*d*e^2*m - 40*a^3*b*c*d^3*e
^2*m - 20*a^3*b*c^3*d*f^2*m - 14*a*b^3*c^4*e*f*m^2 - 2*a*b^3*c^4*e*f*m^3 + 16*a^4*c*d^3*e*f*m^2 + 2*a^4*c*d^3*
e*f*m^3 + 9*a^2*b^2*c^2*d^2*e^2*m^2 + 110*a^2*b^2*c^3*d*e*f*m - 120*a^3*b*c^2*d^2*e*f*m + 44*a^2*b^2*c^3*d*e*f
*m^2 - 46*a^3*b*c^2*d^2*e*f*m^2 + 6*a^2*b^2*c^3*d*e*f*m^3 - 6*a^3*b*c^2*d^2*e*f*m^3))/((a*d - b*c)^4*(c + d*x)
^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b^2*d^2*x^5*(a + b*x)^m*(12*a^2*d^2*f^2 + 2*b^2*c^2*f^2 + 6*b
^2*d^2*e^2 + 7*a^2*d^2*f^2*m + 3*b^2*c^2*f^2*m + a^2*d^2*f^2*m^2 + b^2*c^2*f^2*m^2 - 8*a*b*c*d*f^2 - 16*a*b*d^
2*e*f + 4*b^2*c*d*e*f - 10*a*b*c*d*f^2*m - 4*a*b*d^2*e*f*m + 4*b^2*c*d*e*f*m - 2*a*b*c*d*f^2*m^2))/((a*d - b*c
)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (b*d*x^4*(a + b*x)^m*(5*b*c - a*d*m + b*c*m)*(12*
a^2*d^2*f^2 + 2*b^2*c^2*f^2 + 6*b^2*d^2*e^2 + 7*a^2*d^2*f^2*m + 3*b^2*c^2*f^2*m + a^2*d^2*f^2*m^2 + b^2*c^2*f^
2*m^2 - 8*a*b*c*d*f^2 - 16*a*b*d^2*e*f + 4*b^2*c*d*e*f - 10*a*b*c*d*f^2*m - 4*a*b*d^2*e*f*m + 4*b^2*c*d*e*f*m
- 2*a*b*c*d*f^2*m^2))/((a*d - b*c)^4*(c + d*x)^(m + 5)*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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