3.133 \(\int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx\)

Optimal. Leaf size=572 \[ -\frac {(d g-c h) (c+d x)^{-m-2} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+a b (c f (m+1)-d e (m+3))+b^2 c e\right )\right )}{d^3 f (m+2) (m+3) (d e-c f)^2}+\frac {(d g-c h) (c+d x)^{-m-1} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+a b (c f (m+1)-d e (m+3))+b^2 c e\right )\right )}{d^3 (m+1) (m+2) (m+3) (d e-c f)^3}+\frac {(b c-a d) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b (c f (m+2)-d e (m+3)))}{d^3 f (m+3) (d e-c f)}-\frac {h (b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^3 (m+2) (d e-c f)}-\frac {h (b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f-b (2 d e (m+2)-c f (2 m+3)))}{d^3 (m+1) (m+2) (d e-c f)^2}-\frac {b (a+b x) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d^2 f}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m} \]

[Out]

(-a*d+b*c)*(-c*h+d*g)*(a*d*f+b*(c*f*(2+m)-d*e*(3+m)))*(d*x+c)^(-3-m)*(f*x+e)^(1+m)/d^3/f/(-c*f+d*e)/(3+m)-b*(-
c*h+d*g)*(b*x+a)*(d*x+c)^(-3-m)*(f*x+e)^(1+m)/d^2/f-(-a*d+b*c)^2*h*(d*x+c)^(-2-m)*(f*x+e)^(1+m)/d^3/(-c*f+d*e)
/(2+m)-(-c*h+d*g)*(b^2*(-c*f+d*e)*(2+m)*(c*f*(1+m)-d*e*(3+m))-2*d*f*(b^2*c*e+a^2*d*f+a*b*(c*f*(1+m)-d*e*(3+m))
))*(d*x+c)^(-2-m)*(f*x+e)^(1+m)/d^3/f/(-c*f+d*e)^2/(2+m)/(3+m)-(-a*d+b*c)*h*(a*d*f-b*(2*d*e*(2+m)-c*f*(3+2*m))
)*(d*x+c)^(-1-m)*(f*x+e)^(1+m)/d^3/(-c*f+d*e)^2/(1+m)/(2+m)+(-c*h+d*g)*(b^2*(-c*f+d*e)*(2+m)*(c*f*(1+m)-d*e*(3
+m))-2*d*f*(b^2*c*e+a^2*d*f+a*b*(c*f*(1+m)-d*e*(3+m))))*(d*x+c)^(-1-m)*(f*x+e)^(1+m)/d^3/(-c*f+d*e)^3/(1+m)/(2
+m)/(3+m)-b^2*h*(f*x+e)^m*hypergeom([-m, -m],[1-m],-f*(d*x+c)/(-c*f+d*e))/d^4/m/((d*x+c)^m)/((d*(f*x+e)/(-c*f+
d*e))^m)

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Rubi [A]  time = 0.66, antiderivative size = 566, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {159, 89, 79, 70, 69, 90, 45, 37} \[ -\frac {(d g-c h) (c+d x)^{-m-2} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )\right )}{d^3 f (m+2) (m+3) (d e-c f)^2}+\frac {(d g-c h) (c+d x)^{-m-1} (e+f x)^{m+1} \left (b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))-2 d f \left (a^2 d f+b (a c f (m+1)-a d e (m+3)+b c e)\right )\right )}{d^3 (m+1) (m+2) (m+3) (d e-c f)^3}+\frac {(b c-a d) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1} (a d f+b c f (m+2)-b d e (m+3))}{d^3 f (m+3) (d e-c f)}-\frac {b (a+b x) (d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d^2 f}-\frac {h (b c-a d)^2 (c+d x)^{-m-2} (e+f x)^{m+1}}{d^3 (m+2) (d e-c f)}-\frac {h (b c-a d) (c+d x)^{-m-1} (e+f x)^{m+1} (a d f+b c f (2 m+3)-2 b d e (m+2))}{d^3 (m+1) (m+2) (d e-c f)^2}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

((b*c - a*d)*(d*g - c*h)*(a*d*f + b*c*f*(2 + m) - b*d*e*(3 + m))*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d^3*f*
(d*e - c*f)*(3 + m)) - (b*(d*g - c*h)*(a + b*x)*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d^2*f) - ((b*c - a*d)^2
*h*(c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)*(2 + m)) - ((d*g - c*h)*(b^2*(d*e - c*f)*(2 + m)*(c*
f*(1 + m) - d*e*(3 + m)) - 2*d*f*(a^2*d*f + b*(b*c*e + a*c*f*(1 + m) - a*d*e*(3 + m))))*(c + d*x)^(-2 - m)*(e
+ f*x)^(1 + m))/(d^3*f*(d*e - c*f)^2*(2 + m)*(3 + m)) - ((b*c - a*d)*h*(a*d*f - 2*b*d*e*(2 + m) + b*c*f*(3 + 2
*m))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)^2*(1 + m)*(2 + m)) + ((d*g - c*h)*(b^2*(d*e - c*f)
*(2 + m)*(c*f*(1 + m) - d*e*(3 + m)) - 2*d*f*(a^2*d*f + b*(b*c*e + a*c*f*(1 + m) - a*d*e*(3 + m))))*(c + d*x)^
(-1 - m)*(e + f*x)^(1 + m))/(d^3*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)) - (b^2*h*(e + f*x)^m*Hypergeometric2F1
[-m, -m, 1 - m, -((f*(c + d*x))/(d*e - c*f))])/(d^4*m*(c + d*x)^m*((d*(e + f*x))/(d*e - c*f))^m)

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 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 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 89

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

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Dist[h/b, Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(a + b*x)^m*(
c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (SumSimplerQ[m, 1] || ( !SumS
implerQ[n, 1] &&  !SumSimplerQ[p, 1]))

Rubi steps

\begin {align*} \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx &=\frac {h \int (a+b x)^2 (c+d x)^{-3-m} (e+f x)^m \, dx}{d}+\frac {(d g-c h) \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m \, dx}{d}\\ &=-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \int (c+d x)^{-4-m} (e+f x)^m \left (-a^2 d f-b (b c e+a c f (1+m)-a d e (3+m))+b^2 (d e-c f) (2+m) x\right ) \, dx}{d^2 f}+\frac {h \int (c+d x)^{-2-m} (e+f x)^m \left (-a^2 d^2 f+b^2 c (c f (1+m)-d e (2+m))-2 a b d (c f (1+m)-d e (2+m))+b^2 d (d e-c f) (2+m) x\right ) \, dx}{d^3 (d e-c f) (2+m)}\\ &=\frac {(b c-a d) (d g-c h) (a d f+b c f (2+m)-b d e (3+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(b c-a d) h (a d f-2 b d e (2+m)+b c f (3+2 m)) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}+\frac {\left (b^2 h\right ) \int (c+d x)^{-1-m} (e+f x)^m \, dx}{d^3}+\frac {\left ((d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right )\right ) \int (c+d x)^{-3-m} (e+f x)^m \, dx}{d^2 f (3+m)}\\ &=\frac {(b c-a d) (d g-c h) (a d f+b c f (2+m)-b d e (3+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right ) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^2 f (d e-c f) (2+m) (3+m)}-\frac {(b c-a d) h (a d f-2 b d e (2+m)+b c f (3+2 m)) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}-\frac {\left ((d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right )\right ) \int (c+d x)^{-2-m} (e+f x)^m \, dx}{d^2 (d e-c f) (2+m) (3+m)}+\frac {\left (b^2 h (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m}\right ) \int (c+d x)^{-1-m} \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^m \, dx}{d^3}\\ &=\frac {(b c-a d) (d g-c h) (a d f+b c f (2+m)-b d e (3+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f (d e-c f) (3+m)}-\frac {b (d g-c h) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^2 f}-\frac {(b c-a d)^2 h (c+d x)^{-2-m} (e+f x)^{1+m}}{d^3 (d e-c f) (2+m)}-\frac {(d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right ) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^2 f (d e-c f) (2+m) (3+m)}-\frac {(b c-a d) h (a d f-2 b d e (2+m)+b c f (3+2 m)) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^3 (d e-c f)^2 (1+m) (2+m)}+\frac {(d g-c h) \left (\frac {b^2 (2+m) (c f (1+m)-d e (3+m))}{d}-\frac {2 f \left (b^2 c e+a^2 d f+a b (c f (1+m)-d e (3+m))\right )}{d e-c f}\right ) (c+d x)^{-1-m} (e+f x)^{1+m}}{d^2 (d e-c f)^2 (1+m) (2+m) (3+m)}-\frac {b^2 h (c+d x)^{-m} (e+f x)^m \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {f (c+d x)}{d e-c f}\right )}{d^4 m}\\ \end {align*}

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Mathematica [A]  time = 1.91, size = 422, normalized size = 0.74 \[ \frac {(c+d x)^{-m-3} (e+f x)^m \left (-h (m+3) (c+d x) (d e-c f) \left (d f (m+1) (e+f x) (b c-a d)^2 (d e-c f)-(c+d x) \left (d (e+f x) \left (a^2 d^2 f^2+2 a b d f (c f (m+1)-d e (m+2))+b^2 \left (d^2 e^2 (m+2)-c^2 f^2 (m+1)\right )\right )-b^2 (m+2) (d e-c f)^3 \left (\frac {d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m-1,-m-1;-m;\frac {f (c+d x)}{c f-d e}\right )\right )\right )-d (e+f x) (d g-c h) \left ((c+d x) (d (e m+e-f x)-c f (m+2)) \left (2 d f \left (a^2 (-d) f-b (a c f (m+1)-a d e (m+3)+b c e)\right )+b^2 (m+2) (d e-c f) (c f (m+1)-d e (m+3))\right )+b d (m+1) (m+2) (m+3) (a+b x) (d e-c f)^3-(m+1) (m+2) (b c-a d) (d e-c f)^2 (a d f+b c f (m+2)-b d e (m+3))\right )\right )}{d^4 f (m+1) (m+2) (m+3) (d e-c f)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

((c + d*x)^(-3 - m)*(e + f*x)^m*(-(d*(d*g - c*h)*(e + f*x)*(-((b*c - a*d)*(d*e - c*f)^2*(1 + m)*(2 + m)*(a*d*f
+ b*c*f*(2 + m) - b*d*e*(3 + m))) + b*d*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)*(a + b*x) + (b^2*(d*e - c*f)*(2
+ m)*(c*f*(1 + m) - d*e*(3 + m)) + 2*d*f*(-(a^2*d*f) - b*(b*c*e + a*c*f*(1 + m) - a*d*e*(3 + m))))*(c + d*x)*
(-(c*f*(2 + m)) + d*(e + e*m - f*x)))) - (d*e - c*f)*h*(3 + m)*(c + d*x)*(d*(b*c - a*d)^2*f*(d*e - c*f)*(1 + m
)*(e + f*x) - (c + d*x)*(d*(a^2*d^2*f^2 + 2*a*b*d*f*(c*f*(1 + m) - d*e*(2 + m)) + b^2*(-(c^2*f^2*(1 + m)) + d^
2*e^2*(2 + m)))*(e + f*x) - (b^2*(d*e - c*f)^3*(2 + m)*Hypergeometric2F1[-1 - m, -1 - m, -m, (f*(c + d*x))/(-(
d*e) + c*f)])/((d*(e + f*x))/(d*e - c*f))^m))))/(d^4*f*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m))

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fricas [F]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} h x^{3} + a^{2} g + {\left (b^{2} g + 2 \, a b h\right )} x^{2} + {\left (2 \, a b g + a^{2} h\right )} x\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="fricas")

[Out]

integral((b^2*h*x^3 + a^2*g + (b^2*g + 2*a*b*h)*x^2 + (2*a*b*g + a^2*h)*x)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="giac")

[Out]

integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{2} \left (h x +g \right ) \left (d x +c \right )^{-m -4} \left (f x +e \right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(d*x+c)^(-m-4)*(f*x+e)^m*(h*x+g),x)

[Out]

int((b*x+a)^2*(d*x+c)^(-m-4)*(f*x+e)^m*(h*x+g),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{2} {\left (h x + g\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="maxima")

[Out]

integrate((b*x + a)^2*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^m\,\left (g+h\,x\right )\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{m+4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)^m*(g + h*x)*(a + b*x)^2)/(c + d*x)^(m + 4),x)

[Out]

int(((e + f*x)^m*(g + h*x)*(a + b*x)^2)/(c + d*x)^(m + 4), x)

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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)**2*(d*x+c)**(-4-m)*(f*x+e)**m*(h*x+g),x)

[Out]

Timed out

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