3.132 \(\int (a+b x)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx\)
Optimal. Leaf size=815 \[ \frac{h (a+b x)^3 (e+f x)^{m+1} (c+d x)^{-m-3}}{d f}+\frac{(b c-a d)^2 (a d f+b (c f (m+2)-d e (m+3))) (c f h (m+4)-d (f g+e h (m+3))) (e+f x)^{m+1} (c+d x)^{-m-3}}{d^4 f^2 (d e-c f) (m+3)}-\frac{b (b c-a d) (c f h (m+4)-d (f g+e h (m+3))) (a+b x) (e+f x)^{m+1} (c+d x)^{-m-3}}{d^3 f^2}-\frac{(b c-a d)^2 (3 a d f h-b (c f h (m+4)-d (f g+e h m))) (e+f x)^{m+1} (c+d x)^{-m-2}}{d^4 f (d e-c f) (m+2)}+\frac{(b c-a d) (c f h (m+4)-d (f g+e h (m+3))) \left (\left (d^2 \left (m^2+5 m+6\right ) e^2-2 c d f \left (m^2+4 m+3\right ) e+c^2 f^2 \left (m^2+3 m+2\right )\right ) b^2+2 a d f (c f (m+1)-d e (m+3)) b+2 a^2 d^2 f^2\right ) (e+f x)^{m+1} (c+d x)^{-m-2}}{d^4 f^2 (d e-c f)^2 (m+2) (m+3)}-\frac{(b c-a d) (a d f-b (2 d e (m+2)-c f (2 m+3))) (3 a d f h-b (c f h (m+4)-d (f g+e h m))) (e+f x)^{m+1} (c+d x)^{-m-1}}{d^4 f (d e-c f)^2 (m+1) (m+2)}-\frac{(b c-a d) (c f h (m+4)-d (f g+e h (m+3))) \left (\left (d^2 \left (m^2+5 m+6\right ) e^2-2 c d f \left (m^2+4 m+3\right ) e+c^2 f^2 \left (m^2+3 m+2\right )\right ) b^2+2 a d f (c f (m+1)-d e (m+3)) b+2 a^2 d^2 f^2\right ) (e+f x)^{m+1} (c+d x)^{-m-1}}{d^4 f (d e-c f)^3 (m+1) (m+2) (m+3)}-\frac{b^2 (3 a d f h-b (c f h (m+4)-d (f g+e h m))) (e+f x)^m \left (\frac{d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{f (c+d x)}{d e-c f}\right ) (c+d x)^{-m}}{d^5 f m} \]
[Out]
((b*c - a*d)^2*(a*d*f + b*(c*f*(2 + m) - d*e*(3 + m)))*(c*f*h*(4 + m) - d*(f*g + e*h*(3 + m)))*(c + d*x)^(-3 -
m)*(e + f*x)^(1 + m))/(d^4*f^2*(d*e - c*f)*(3 + m)) - (b*(b*c - a*d)*(c*f*h*(4 + m) - d*(f*g + e*h*(3 + m)))*
(a + b*x)*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d^3*f^2) + (h*(a + b*x)^3*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m
))/(d*f) - ((b*c - a*d)^2*(3*a*d*f*h - b*(c*f*h*(4 + m) - d*(f*g + e*h*m)))*(c + d*x)^(-2 - m)*(e + f*x)^(1 +
m))/(d^4*f*(d*e - c*f)*(2 + m)) + ((b*c - a*d)*(c*f*h*(4 + m) - d*(f*g + e*h*(3 + m)))*(2*a^2*d^2*f^2 + 2*a*b*
d*f*(c*f*(1 + m) - d*e*(3 + m)) + b^2*(c^2*f^2*(2 + 3*m + m^2) - 2*c*d*e*f*(3 + 4*m + m^2) + d^2*e^2*(6 + 5*m
+ m^2)))*(c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^4*f^2*(d*e - c*f)^2*(2 + m)*(3 + m)) - ((b*c - a*d)*(a*d*f -
b*(2*d*e*(2 + m) - c*f*(3 + 2*m)))*(3*a*d*f*h - b*(c*f*h*(4 + m) - d*(f*g + e*h*m)))*(c + d*x)^(-1 - m)*(e +
f*x)^(1 + m))/(d^4*f*(d*e - c*f)^2*(1 + m)*(2 + m)) - ((b*c - a*d)*(c*f*h*(4 + m) - d*(f*g + e*h*(3 + m)))*(2*
a^2*d^2*f^2 + 2*a*b*d*f*(c*f*(1 + m) - d*e*(3 + m)) + b^2*(c^2*f^2*(2 + 3*m + m^2) - 2*c*d*e*f*(3 + 4*m + m^2)
+ d^2*e^2*(6 + 5*m + m^2)))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d^4*f*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m
)) - (b^2*(3*a*d*f*h - b*(c*f*h*(4 + m) - d*(f*g + e*h*m)))*(e + f*x)^m*Hypergeometric2F1[-m, -m, 1 - m, -((f*
(c + d*x))/(d*e - c*f))])/(d^5*f*m*(c + d*x)^m*((d*(e + f*x))/(d*e - c*f))^m)
________________________________________________________________________________________
Rubi [A] time = 1.43461, antiderivative size = 803, normalized size of antiderivative = 0.99,
number of steps used = 10, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used =
{153, 159, 89, 79, 70, 69, 90, 45, 37} \[ \frac{h (a+b x)^3 (e+f x)^{m+1} (c+d x)^{-m-3}}{d f}+\frac{(b c-a d)^2 (a d f+b c (m+2) f-b d e (m+3)) (c f h (m+4)-d (f g+e h (m+3))) (e+f x)^{m+1} (c+d x)^{-m-3}}{d^4 f^2 (d e-c f) (m+3)}-\frac{b (b c-a d) (c f h (m+4)-d (f g+e h (m+3))) (a+b x) (e+f x)^{m+1} (c+d x)^{-m-3}}{d^3 f^2}-\frac{(b c-a d)^2 (b d f g+3 a d f h+b d e h m-b c f h (m+4)) (e+f x)^{m+1} (c+d x)^{-m-2}}{d^4 f (d e-c f) (m+2)}-\frac{(b c-a d) (d f g+d e h (m+3)-c f h (m+4)) \left (\left (d^2 \left (m^2+5 m+6\right ) e^2-2 c d f \left (m^2+4 m+3\right ) e+c^2 f^2 \left (m^2+3 m+2\right )\right ) b^2+2 a d f (c f (m+1)-d e (m+3)) b+2 a^2 d^2 f^2\right ) (e+f x)^{m+1} (c+d x)^{-m-2}}{d^4 f^2 (d e-c f)^2 (m+2) (m+3)}-\frac{(b c-a d) (b d f g+3 a d f h+b d e h m-b c f h (m+4)) (a d f+b c (2 m+3) f-2 b d e (m+2)) (e+f x)^{m+1} (c+d x)^{-m-1}}{d^4 f (d e-c f)^2 (m+1) (m+2)}+\frac{(b c-a d) (d f g+d e h (m+3)-c f h (m+4)) \left (\left (d^2 \left (m^2+5 m+6\right ) e^2-2 c d f \left (m^2+4 m+3\right ) e+c^2 f^2 \left (m^2+3 m+2\right )\right ) b^2+2 a d f (c f (m+1)-d e (m+3)) b+2 a^2 d^2 f^2\right ) (e+f x)^{m+1} (c+d x)^{-m-1}}{d^4 f (d e-c f)^3 (m+1) (m+2) (m+3)}-\frac{b^2 (b d f g+3 a d f h+b d e h m-b c f h (m+4)) (e+f x)^m \left (\frac{d (e+f x)}{d e-c f}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{f (c+d x)}{d e-c f}\right ) (c+d x)^{-m}}{d^5 f m} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]
[Out]
((b*c - a*d)^2*(a*d*f + b*c*f*(2 + m) - b*d*e*(3 + m))*(c*f*h*(4 + m) - d*(f*g + e*h*(3 + m)))*(c + d*x)^(-3 -
m)*(e + f*x)^(1 + m))/(d^4*f^2*(d*e - c*f)*(3 + m)) - (b*(b*c - a*d)*(c*f*h*(4 + m) - d*(f*g + e*h*(3 + m)))*
(a + b*x)*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d^3*f^2) + (h*(a + b*x)^3*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m
))/(d*f) - ((b*c - a*d)^2*(b*d*f*g + 3*a*d*f*h + b*d*e*h*m - b*c*f*h*(4 + m))*(c + d*x)^(-2 - m)*(e + f*x)^(1
+ m))/(d^4*f*(d*e - c*f)*(2 + m)) - ((b*c - a*d)*(d*f*g + d*e*h*(3 + m) - c*f*h*(4 + m))*(2*a^2*d^2*f^2 + 2*a*
b*d*f*(c*f*(1 + m) - d*e*(3 + m)) + b^2*(c^2*f^2*(2 + 3*m + m^2) - 2*c*d*e*f*(3 + 4*m + m^2) + d^2*e^2*(6 + 5*
m + m^2)))*(c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d^4*f^2*(d*e - c*f)^2*(2 + m)*(3 + m)) - ((b*c - a*d)*(b*d*f
*g + 3*a*d*f*h + b*d*e*h*m - b*c*f*h*(4 + m))*(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^4*f*(d*e - c*f)^2*(1 + m)*(2 + m)) + ((b*c - a*d)*(d*f*g + d*e*h*(3 + m) - c*f*h*(4 + m))
*(2*a^2*d^2*f^2 + 2*a*b*d*f*(c*f*(1 + m) - d*e*(3 + m)) + b^2*(c^2*f^2*(2 + 3*m + m^2) - 2*c*d*e*f*(3 + 4*m +
m^2) + d^2*e^2*(6 + 5*m + m^2)))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d^4*f*(d*e - c*f)^3*(1 + m)*(2 + m)*(3
+ m)) - (b^2*(b*d*f*g + 3*a*d*f*h + b*d*e*h*m - b*c*f*h*(4 + m))*(e + f*x)^m*Hypergeometric2F1[-m, -m, 1 - m,
-((f*(c + d*x))/(d*e - c*f))])/(d^5*f*m*(c + d*x)^m*((d*(e + f*x))/(d*e - c*f))^m)
Rule 153
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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]))
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 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 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 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 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 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)^3 (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx &=\frac{h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}+\frac{\int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m (-3 b c e h+a (d f g-c f h (1+m)+d e h (3+m))+(b d f g+3 a d f h+b d e h m-b c f h (4+m)) x) \, dx}{d f}\\ &=\frac{h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}-\frac{((b c-a d) (d f g+d e h (3+m)-c f h (4+m))) \int (a+b x)^2 (c+d x)^{-4-m} (e+f x)^m \, dx}{d^2 f}+\frac{(b d f g+3 a d f h+b d e h m-b c f h (4+m)) \int (a+b x)^2 (c+d x)^{-3-m} (e+f x)^m \, dx}{d^2 f}\\ &=\frac{b (b c-a d) (d f g+d e h (3+m)-c f h (4+m)) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f^2}+\frac{h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}-\frac{(b c-a d)^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^4 f (d e-c f) (2+m)}+\frac{((b c-a d) (d f g+d e h (3+m)-c f h (4+m))) \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^3 f^2}+\frac{(b d f g+3 a d f h+b d e h m-b c f h (4+m)) \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^4 f (d e-c f) (2+m)}\\ &=-\frac{(b c-a d)^2 (a d f+b c f (2+m)-b d e (3+m)) (d f g+d e h (3+m)-c f h (4+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^4 f^2 (d e-c f) (3+m)}+\frac{b (b c-a d) (d f g+d e h (3+m)-c f h (4+m)) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f^2}+\frac{h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}-\frac{(b c-a d)^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^4 f (d e-c f) (2+m)}-\frac{(b c-a d) (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (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^4 f (d e-c f)^2 (1+m) (2+m)}+\frac{\left (b^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m))\right ) \int (c+d x)^{-1-m} (e+f x)^m \, dx}{d^4 f}-\frac{\left ((b c-a d) (d f g+d e h (3+m)-c f h (4+m)) \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^3 f^2 (3+m)}\\ &=-\frac{(b c-a d)^2 (a d f+b c f (2+m)-b d e (3+m)) (d f g+d e h (3+m)-c f h (4+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^4 f^2 (d e-c f) (3+m)}+\frac{b (b c-a d) (d f g+d e h (3+m)-c f h (4+m)) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f^2}+\frac{h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}-\frac{(b c-a d)^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^4 f (d e-c f) (2+m)}+\frac{(b c-a d) (d f g+d e h (3+m)-c f h (4+m)) \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^3 f^2 (d e-c f) (2+m) (3+m)}-\frac{(b c-a d) (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (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^4 f (d e-c f)^2 (1+m) (2+m)}+\frac{\left ((b c-a d) (d f g+d e h (3+m)-c f h (4+m)) \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^3 f (d e-c f) (2+m) (3+m)}+\frac{\left (b^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (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^4 f}\\ &=-\frac{(b c-a d)^2 (a d f+b c f (2+m)-b d e (3+m)) (d f g+d e h (3+m)-c f h (4+m)) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^4 f^2 (d e-c f) (3+m)}+\frac{b (b c-a d) (d f g+d e h (3+m)-c f h (4+m)) (a+b x) (c+d x)^{-3-m} (e+f x)^{1+m}}{d^3 f^2}+\frac{h (a+b x)^3 (c+d x)^{-3-m} (e+f x)^{1+m}}{d f}-\frac{(b c-a d)^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (c+d x)^{-2-m} (e+f x)^{1+m}}{d^4 f (d e-c f) (2+m)}+\frac{(b c-a d) (d f g+d e h (3+m)-c f h (4+m)) \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^3 f^2 (d e-c f) (2+m) (3+m)}-\frac{(b c-a d) (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (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^4 f (d e-c f)^2 (1+m) (2+m)}-\frac{(b c-a d) (d f g+d e h (3+m)-c f h (4+m)) \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^3 f (d e-c f)^2 (1+m) (2+m) (3+m)}-\frac{b^2 (b d f g+3 a d f h+b d e h m-b c f h (4+m)) (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^5 f m}\\ \end{align*}
Mathematica [C] time = 48.597, size = 3579, normalized size = 4.39 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*x)^3*(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]
[Out]
(3*a*b^2*c^3*g*(c + d*x)^(-7 - m)*((c + d*x)/c)^(4 + m)*(e + f*x)^(3 + m)*(-2*c^3*e^3 - 6*c^2*d*e^3*x - 2*c^2*
d*e^3*m*x + 2*c^3*e^2*f*m*x - 6*c*d^2*e^3*x^2 - 5*c*d^2*e^3*m*x^2 + 6*c^2*d*e^2*f*m*x^2 - c^3*e*f^2*m*x^2 - c*
d^2*e^3*m^2*x^2 + 2*c^2*d*e^2*f*m^2*x^2 - c^3*e*f^2*m^2*x^2 - 6*c*d^2*e^2*f*x^3 + 6*c^2*d*e*f^2*x^3 - 2*c^3*f^
3*x^3 - 5*c*d^2*e^2*f*m*x^3 + 8*c^2*d*e*f^2*m*x^3 - 3*c^3*f^3*m*x^3 - c*d^2*e^2*f*m^2*x^3 + 2*c^2*d*e*f^2*m^2*
x^3 - c^3*f^3*m^2*x^3 + 2*c^3*e^3*((c*e + d*e*x)/(c*(e + f*x)))^m + 6*c^2*d*e^3*x*((c*e + d*e*x)/(c*(e + f*x))
)^m + 6*c*d^2*e^3*x^2*((c*e + d*e*x)/(c*(e + f*x)))^m + 2*d^3*e^3*x^3*((c*e + d*e*x)/(c*(e + f*x)))^m))/(e^3*(
d*e - c*f)^3*(1 + m)*(2 + m)*(3 + m)*((c*e + d*e*x)/(c*(e + f*x)))^m*((e + f*x)/e)^m*(1 + (f*x)/e)^3) + (3*a^2
*b*c^3*h*(c + d*x)^(-7 - m)*((c + d*x)/c)^(4 + m)*(e + f*x)^(3 + m)*(-2*c^3*e^3 - 6*c^2*d*e^3*x - 2*c^2*d*e^3*
m*x + 2*c^3*e^2*f*m*x - 6*c*d^2*e^3*x^2 - 5*c*d^2*e^3*m*x^2 + 6*c^2*d*e^2*f*m*x^2 - c^3*e*f^2*m*x^2 - c*d^2*e^
3*m^2*x^2 + 2*c^2*d*e^2*f*m^2*x^2 - c^3*e*f^2*m^2*x^2 - 6*c*d^2*e^2*f*x^3 + 6*c^2*d*e*f^2*x^3 - 2*c^3*f^3*x^3
- 5*c*d^2*e^2*f*m*x^3 + 8*c^2*d*e*f^2*m*x^3 - 3*c^3*f^3*m*x^3 - c*d^2*e^2*f*m^2*x^3 + 2*c^2*d*e*f^2*m^2*x^3 -
c^3*f^3*m^2*x^3 + 2*c^3*e^3*((c*e + d*e*x)/(c*(e + f*x)))^m + 6*c^2*d*e^3*x*((c*e + d*e*x)/(c*(e + f*x)))^m +
6*c*d^2*e^3*x^2*((c*e + d*e*x)/(c*(e + f*x)))^m + 2*d^3*e^3*x^3*((c*e + d*e*x)/(c*(e + f*x)))^m))/(e^3*(d*e -
c*f)^3*(1 + m)*(2 + m)*(3 + m)*((c*e + d*e*x)/(c*(e + f*x)))^m*((e + f*x)/e)^m*(1 + (f*x)/e)^3) + (b^3*g*x^4*(
c + d*x)^(-4 - m)*((c + d*x)/c)^(4 + m)*(e + f*x)^m*AppellF1[4, 4 + m, -m, 5, -((d*x)/c), -((f*x)/e)])/(4*((e
+ f*x)/e)^m) + (3*a*b^2*h*x^4*(c + d*x)^(-4 - m)*((c + d*x)/c)^(4 + m)*(e + f*x)^m*AppellF1[4, 4 + m, -m, 5, -
((d*x)/c), -((f*x)/e)])/(4*((e + f*x)/e)^m) + (b^3*h*x^5*(c + d*x)^(-4 - m)*((c + d*x)/c)^(4 + m)*(e + f*x)^m*
AppellF1[5, 4 + m, -m, 6, -((d*x)/c), -((f*x)/e)])/(5*((e + f*x)/e)^m) + (3*a^2*b*g*(c + d*x)^(-4 - m)*((c + d
*x)/c)^(4 + m)*(1 + (d*x)/c)^(-4 - m)*(e + f*x)^m*((c*(e + f*x))/(e*(c + d*x)))^(-1 - m)*(1 + (f*x)/e)^(1 + m)
*(c*(4 + m)*(3*e + f*x)*(-2*d^3*e^3*x^3 + c^3*(-2*e^2*f*m*x*((c*(e + f*x))/(e*(c + d*x)))^m + e*f^2*m*(1 + m)*
x^2*((c*(e + f*x))/(e*(c + d*x)))^m + f^3*(2 + 3*m + m^2)*x^3*((c*(e + f*x))/(e*(c + d*x)))^m + 2*e^3*(-1 + ((
c*(e + f*x))/(e*(c + d*x)))^m)) - 2*c^2*d*e*x*(e*f*m*(3 + m)*x*((c*(e + f*x))/(e*(c + d*x)))^m + f^2*(3 + 4*m
+ m^2)*x^2*((c*(e + f*x))/(e*(c + d*x)))^m - e^2*(-3 + 3*((c*(e + f*x))/(e*(c + d*x)))^m + m*((c*(e + f*x))/(e
*(c + d*x)))^m)) + c*d^2*e^2*x^2*(f*(6 + 5*m + m^2)*x*((c*(e + f*x))/(e*(c + d*x)))^m + e*(-6 + 6*((c*(e + f*x
))/(e*(c + d*x)))^m + 5*m*((c*(e + f*x))/(e*(c + d*x)))^m + m^2*((c*(e + f*x))/(e*(c + d*x)))^m)))*Gamma[4 + m
] - (2*d^4*e^4*(1 + m)*x^4 - 2*c*d^3*e^3*x^3*(-3*e*m + f*(4 + m)*x) + c^4*(e^2*f^2*(-5 + m)*m*x^2*((c*(e + f*x
))/(e*(c + d*x)))^m + 2*e*f^3*m*(1 + m)*x^3*((c*(e + f*x))/(e*(c + d*x)))^m + f^4*(2 + 3*m + m^2)*x^4*((c*(e +
f*x))/(e*(c + d*x)))^m + 6*e^4*(-1 + ((c*(e + f*x))/(e*(c + d*x)))^m) - 2*e^3*f*x*(4 + m - 4*((c*(e + f*x))/(
e*(c + d*x)))^m + 2*m*((c*(e + f*x))/(e*(c + d*x)))^m)) - 2*c^3*d*e*x*(2*e*f^2*m*(4 + m)*x^2*((c*(e + f*x))/(e
*(c + d*x)))^m + f^3*(4 + 5*m + m^2)*x^3*((c*(e + f*x))/(e*(c + d*x)))^m + e^2*f*(4 + m)*x*(3 - 3*((c*(e + f*x
))/(e*(c + d*x)))^m + m*((c*(e + f*x))/(e*(c + d*x)))^m) - e^3*(-8 + m + 8*((c*(e + f*x))/(e*(c + d*x)))^m + 2
*m*((c*(e + f*x))/(e*(c + d*x)))^m)) + c^2*d^2*e^2*x^2*(f^2*(12 + 7*m + m^2)*x^2*((c*(e + f*x))/(e*(c + d*x)))
^m + 2*e*f*(4 + m)*x*(-3 + 3*((c*(e + f*x))/(e*(c + d*x)))^m + m*((c*(e + f*x))/(e*(c + d*x)))^m) + e^2*(m^2*(
(c*(e + f*x))/(e*(c + d*x)))^m + 12*(-1 + ((c*(e + f*x))/(e*(c + d*x)))^m) + m*(6 + 7*((c*(e + f*x))/(e*(c + d
*x)))^m))))*Gamma[5 + m]))/(2*c*e*(-(d*e) + c*f)^3*(1 + m)*(2 + m)*(3 + m)*(4 + m)*x*((e + f*x)/e)^m*Gamma[4 +
m]) + (a^3*h*(c + d*x)^(-4 - m)*((c + d*x)/c)^(4 + m)*(1 + (d*x)/c)^(-4 - m)*(e + f*x)^m*((c*(e + f*x))/(e*(c
+ d*x)))^(-1 - m)*(1 + (f*x)/e)^(1 + m)*(c*(4 + m)*(3*e + f*x)*(-2*d^3*e^3*x^3 + c^3*(-2*e^2*f*m*x*((c*(e + f
*x))/(e*(c + d*x)))^m + e*f^2*m*(1 + m)*x^2*((c*(e + f*x))/(e*(c + d*x)))^m + f^3*(2 + 3*m + m^2)*x^3*((c*(e +
f*x))/(e*(c + d*x)))^m + 2*e^3*(-1 + ((c*(e + f*x))/(e*(c + d*x)))^m)) - 2*c^2*d*e*x*(e*f*m*(3 + m)*x*((c*(e
+ f*x))/(e*(c + d*x)))^m + f^2*(3 + 4*m + m^2)*x^2*((c*(e + f*x))/(e*(c + d*x)))^m - e^2*(-3 + 3*((c*(e + f*x)
)/(e*(c + d*x)))^m + m*((c*(e + f*x))/(e*(c + d*x)))^m)) + c*d^2*e^2*x^2*(f*(6 + 5*m + m^2)*x*((c*(e + f*x))/(
e*(c + d*x)))^m + e*(-6 + 6*((c*(e + f*x))/(e*(c + d*x)))^m + 5*m*((c*(e + f*x))/(e*(c + d*x)))^m + m^2*((c*(e
+ f*x))/(e*(c + d*x)))^m)))*Gamma[4 + m] - (2*d^4*e^4*(1 + m)*x^4 - 2*c*d^3*e^3*x^3*(-3*e*m + f*(4 + m)*x) +
c^4*(e^2*f^2*(-5 + m)*m*x^2*((c*(e + f*x))/(e*(c + d*x)))^m + 2*e*f^3*m*(1 + m)*x^3*((c*(e + f*x))/(e*(c + d*x
)))^m + f^4*(2 + 3*m + m^2)*x^4*((c*(e + f*x))/(e*(c + d*x)))^m + 6*e^4*(-1 + ((c*(e + f*x))/(e*(c + d*x)))^m)
- 2*e^3*f*x*(4 + m - 4*((c*(e + f*x))/(e*(c + d*x)))^m + 2*m*((c*(e + f*x))/(e*(c + d*x)))^m)) - 2*c^3*d*e*x*
(2*e*f^2*m*(4 + m)*x^2*((c*(e + f*x))/(e*(c + d*x)))^m + f^3*(4 + 5*m + m^2)*x^3*((c*(e + f*x))/(e*(c + d*x)))
^m + e^2*f*(4 + m)*x*(3 - 3*((c*(e + f*x))/(e*(c + d*x)))^m + m*((c*(e + f*x))/(e*(c + d*x)))^m) - e^3*(-8 + m
+ 8*((c*(e + f*x))/(e*(c + d*x)))^m + 2*m*((c*(e + f*x))/(e*(c + d*x)))^m)) + c^2*d^2*e^2*x^2*(f^2*(12 + 7*m
+ m^2)*x^2*((c*(e + f*x))/(e*(c + d*x)))^m + 2*e*f*(4 + m)*x*(-3 + 3*((c*(e + f*x))/(e*(c + d*x)))^m + m*((c*(
e + f*x))/(e*(c + d*x)))^m) + e^2*(m^2*((c*(e + f*x))/(e*(c + d*x)))^m + 12*(-1 + ((c*(e + f*x))/(e*(c + d*x))
)^m) + m*(6 + 7*((c*(e + f*x))/(e*(c + d*x)))^m))))*Gamma[5 + m]))/(2*c*e*(-(d*e) + c*f)^3*(1 + m)*(2 + m)*(3
+ m)*(4 + m)*x*((e + f*x)/e)^m*Gamma[4 + m]) + (a^3*f^3*g*(e + f*x)^(1 + m)*(1 + (d*(e + f*x))/((c - (d*e)/f)*
f))^m*Hypergeometric2F1[1 + m, 4 + m, 2 + m, -((d*(e + f*x))/((c - (d*e)/f)*f))])/((-(d*e) + c*f)^4*(1 + m)*(c
- (d*e)/f + (d*(e + f*x))/f)^m)
________________________________________________________________________________________
Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{3} \left ( dx+c \right ) ^{-4-m} \left ( fx+e \right ) ^{m} \left ( hx+g \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x)
[Out]
int((b*x+a)^3*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3}{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="maxima")
[Out]
integrate((b*x + a)^3*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} h x^{4} + a^{3} g +{\left (b^{3} g + 3 \, a b^{2} h\right )} x^{3} + 3 \,{\left (a b^{2} g + a^{2} b h\right )} x^{2} +{\left (3 \, a^{2} b g + a^{3} h\right )} x\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="fricas")
[Out]
integral((b^3*h*x^4 + a^3*g + (b^3*g + 3*a*b^2*h)*x^3 + 3*(a*b^2*g + a^2*b*h)*x^2 + (3*a^2*b*g + a^3*h)*x)*(d*
x + c)^(-m - 4)*(f*x + e)^m, x)
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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)**3*(d*x+c)**(-4-m)*(f*x+e)**m*(h*x+g),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3}{\left (h x + g\right )}{\left (d x + c\right )}^{-m - 4}{\left (f x + e\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^(-4-m)*(f*x+e)^m*(h*x+g),x, algorithm="giac")
[Out]
integrate((b*x + a)^3*(h*x + g)*(d*x + c)^(-m - 4)*(f*x + e)^m, x)