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

Optimal. Leaf size=188 \[ -\frac {(d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d (m+3) (d e-c f)}+\frac {(c+d x)^{-m-2} (e+f x)^{m+1} (c f h (m+1)+d (2 f g-e h (m+3)))}{d (m+2) (m+3) (d e-c f)^2}-\frac {f (c+d x)^{-m-1} (e+f x)^{m+1} (c f h (m+1)+d (2 f g-e h (m+3)))}{d (m+1) (m+2) (m+3) (d e-c f)^3} \]

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Rubi [A]  time = 0.10, antiderivative size = 186, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 45, 37} \begin {gather*} -\frac {(d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d (m+3) (d e-c f)}+\frac {(c+d x)^{-m-2} (e+f x)^{m+1} (c f h (m+1)-d e h (m+3)+2 d f g)}{d (m+2) (m+3) (d e-c f)^2}-\frac {f (c+d x)^{-m-1} (e+f x)^{m+1} (c f h (m+1)-d e h (m+3)+2 d f g)}{d (m+1) (m+2) (m+3) (d e-c f)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*g - c*h)*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d*(d*e - c*f)*(3 + m))) + ((2*d*f*g + c*f*h*(1 + m) - d*
e*h*(3 + m))*(c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/(d*(d*e - c*f)^2*(2 + m)*(3 + m)) - (f*(2*d*f*g + c*f*h*(1
+ m) - d*e*h*(3 + m))*(c + d*x)^(-1 - m)*(e + f*x)^(1 + m))/(d*(d*e - c*f)^3*(1 + m)*(2 + m)*(3 + 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 181, normalized size = 0.96 \begin {gather*} \frac {(d g-c h) (c+d x)^{-m-3} (e+f x)^{m+1}}{d (-m-3) (d e-c f)}-\frac {\left (\frac {(c+d x)^{-m-2} (e+f x)^{m+1}}{(-m-2) (d e-c f)}+\frac {f (c+d x)^{-m-1} (e+f x)^{m+1}}{(-m-2) (-m-1) (d e-c f)^2}\right ) (-h (c f (m+1)+d e (-m-3))-2 d f g)}{d (-m-3) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*g - c*h)*(c + d*x)^(-3 - m)*(e + f*x)^(1 + m))/(d*(d*e - c*f)*(-3 - m)) - ((-2*d*f*g - h*(d*e*(-3 - m) + c
*f*(1 + m)))*(((c + d*x)^(-2 - m)*(e + f*x)^(1 + m))/((d*e - c*f)*(-2 - m)) + (f*(c + d*x)^(-1 - m)*(e + f*x)^
(1 + m))/((d*e - c*f)^2*(-2 - m)*(-1 - m))))/(d*(d*e - c*f)*(-3 - m))

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IntegrateAlgebraic [F]  time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (c+d x)^{-4-m} (e+f x)^m (g+h x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x),x]

[Out]

Defer[IntegrateAlgebraic][(c + d*x)^(-4 - m)*(e + f*x)^m*(g + h*x), x]

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fricas [B]  time = 0.84, size = 905, normalized size = 4.81 \begin {gather*} -\frac {{\left ({\left (2 \, d^{3} f^{3} g - {\left (d^{3} e f^{2} - c d^{2} f^{3}\right )} h m - {\left (3 \, d^{3} e f^{2} - c d^{2} f^{3}\right )} h\right )} x^{4} + {\left (c d^{2} e^{3} - 2 \, c^{2} d e^{2} f + c^{3} e f^{2}\right )} g m^{2} + {\left (8 \, c d^{2} f^{3} g + {\left (d^{3} e^{2} f - 2 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} h m^{2} - 4 \, {\left (3 \, c d^{2} e f^{2} - c^{2} d f^{3}\right )} h - {\left (2 \, {\left (d^{3} e f^{2} - c d^{2} f^{3}\right )} g - {\left (3 \, d^{3} e^{2} f - 8 \, c d^{2} e f^{2} + 5 \, c^{2} d f^{3}\right )} h\right )} m\right )} x^{3} + {\left (12 \, c^{2} d f^{3} g + {\left ({\left (d^{3} e^{2} f - 2 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g + {\left (d^{3} e^{3} - c d^{2} e^{2} f - c^{2} d e f^{2} + c^{3} f^{3}\right )} h\right )} m^{2} + 3 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + c^{3} f^{3}\right )} h + {\left ({\left (d^{3} e^{2} f - 8 \, c d^{2} e f^{2} + 7 \, c^{2} d f^{3}\right )} g + 4 \, {\left (d^{3} e^{3} - c d^{2} e^{2} f - c^{2} d e f^{2} + c^{3} f^{3}\right )} h\right )} m\right )} x^{2} + 2 \, {\left (c d^{2} e^{3} - 3 \, c^{2} d e^{2} f + 3 \, c^{3} e f^{2}\right )} g + {\left (c^{2} d e^{3} - 3 \, c^{3} e^{2} f\right )} h + {\left ({\left (3 \, c d^{2} e^{3} - 8 \, c^{2} d e^{2} f + 5 \, c^{3} e f^{2}\right )} g + {\left (c^{2} d e^{3} - c^{3} e^{2} f\right )} h\right )} m + {\left ({\left ({\left (d^{3} e^{3} - c d^{2} e^{2} f - c^{2} d e f^{2} + c^{3} f^{3}\right )} g + {\left (c d^{2} e^{3} - 2 \, c^{2} d e^{2} f + c^{3} e f^{2}\right )} h\right )} m^{2} + 2 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} + 3 \, c^{3} f^{3}\right )} g + 4 \, {\left (c d^{2} e^{3} - 3 \, c^{2} d e^{2} f\right )} h + {\left ({\left (3 \, d^{3} e^{3} - 7 \, c d^{2} e^{2} f - c^{2} d e f^{2} + 5 \, c^{3} f^{3}\right )} g + {\left (5 \, c d^{2} e^{3} - 8 \, c^{2} d e^{2} f + 3 \, c^{3} e f^{2}\right )} h\right )} m\right )} x\right )} {\left (d x + c\right )}^{-m - 4} {\left (f x + e\right )}^{m}}{6 \, d^{3} e^{3} - 18 \, c d^{2} e^{2} f + 18 \, c^{2} d e f^{2} - 6 \, c^{3} f^{3} + {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} m^{3} + 6 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} m^{2} + 11 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((2*d^3*f^3*g - (d^3*e*f^2 - c*d^2*f^3)*h*m - (3*d^3*e*f^2 - c*d^2*f^3)*h)*x^4 + (c*d^2*e^3 - 2*c^2*d*e^2*f +
c^3*e*f^2)*g*m^2 + (8*c*d^2*f^3*g + (d^3*e^2*f - 2*c*d^2*e*f^2 + c^2*d*f^3)*h*m^2 - 4*(3*c*d^2*e*f^2 - c^2*d*
f^3)*h - (2*(d^3*e*f^2 - c*d^2*f^3)*g - (3*d^3*e^2*f - 8*c*d^2*e*f^2 + 5*c^2*d*f^3)*h)*m)*x^3 + (12*c^2*d*f^3*
g + ((d^3*e^2*f - 2*c*d^2*e*f^2 + c^2*d*f^3)*g + (d^3*e^3 - c*d^2*e^2*f - c^2*d*e*f^2 + c^3*f^3)*h)*m^2 + 3*(d
^3*e^3 - 3*c*d^2*e^2*f - 3*c^2*d*e*f^2 + c^3*f^3)*h + ((d^3*e^2*f - 8*c*d^2*e*f^2 + 7*c^2*d*f^3)*g + 4*(d^3*e^
3 - c*d^2*e^2*f - c^2*d*e*f^2 + c^3*f^3)*h)*m)*x^2 + 2*(c*d^2*e^3 - 3*c^2*d*e^2*f + 3*c^3*e*f^2)*g + (c^2*d*e^
3 - 3*c^3*e^2*f)*h + ((3*c*d^2*e^3 - 8*c^2*d*e^2*f + 5*c^3*e*f^2)*g + (c^2*d*e^3 - c^3*e^2*f)*h)*m + (((d^3*e^
3 - c*d^2*e^2*f - c^2*d*e*f^2 + c^3*f^3)*g + (c*d^2*e^3 - 2*c^2*d*e^2*f + c^3*e*f^2)*h)*m^2 + 2*(d^3*e^3 - 3*c
*d^2*e^2*f + 3*c^2*d*e*f^2 + 3*c^3*f^3)*g + 4*(c*d^2*e^3 - 3*c^2*d*e^2*f)*h + ((3*d^3*e^3 - 7*c*d^2*e^2*f - c^
2*d*e*f^2 + 5*c^3*f^3)*g + (5*c*d^2*e^3 - 8*c^2*d*e^2*f + 3*c^3*e*f^2)*h)*m)*x)*(d*x + c)^(-m - 4)*(f*x + e)^m
/(6*d^3*e^3 - 18*c*d^2*e^2*f + 18*c^2*d*e*f^2 - 6*c^3*f^3 + (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3
)*m^3 + 6*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*m^2 + 11*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^
2 - c^3*f^3)*m)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 509, normalized size = 2.71 \begin {gather*} -\frac {\left (-c^{2} f^{2} h \,m^{2} x +2 c d e f h \,m^{2} x -c d \,f^{2} h m \,x^{2}-d^{2} e^{2} h \,m^{2} x +d^{2} e f h m \,x^{2}-c^{2} f^{2} g \,m^{2}-4 c^{2} f^{2} h m x +2 c d e f g \,m^{2}+8 c d e f h m x -2 c d \,f^{2} g m x -c d \,f^{2} h \,x^{2}-d^{2} e^{2} g \,m^{2}-4 d^{2} e^{2} h m x +2 d^{2} e f g m x +3 d^{2} e f h \,x^{2}-2 d^{2} f^{2} g \,x^{2}+c^{2} e f h m -5 c^{2} f^{2} g m -3 c^{2} f^{2} h x -c d \,e^{2} h m +8 c d e f g m +10 c d e f h x -6 c d \,f^{2} g x -3 d^{2} e^{2} g m -3 d^{2} e^{2} h x +2 d^{2} e f g x +3 c^{2} e f h -6 c^{2} f^{2} g -c d \,e^{2} h +6 c d e f g -2 d^{2} e^{2} g \right ) \left (d x +c \right )^{-m -3} \left (f x +e \right )^{m +1}}{c^{3} f^{3} m^{3}-3 c^{2} d e \,f^{2} m^{3}+3 c \,d^{2} e^{2} f \,m^{3}-d^{3} e^{3} m^{3}+6 c^{3} f^{3} m^{2}-18 c^{2} d e \,f^{2} m^{2}+18 c \,d^{2} e^{2} f \,m^{2}-6 d^{3} e^{3} m^{2}+11 c^{3} f^{3} m -33 c^{2} d e \,f^{2} m +33 c \,d^{2} e^{2} f m -11 d^{3} e^{3} m +6 c^{3} f^{3}-18 c^{2} d e \,f^{2}+18 c \,d^{2} e^{2} f -6 d^{3} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x+c)^(-m-3)*(f*x+e)^(m+1)*(-c^2*f^2*h*m^2*x+2*c*d*e*f*h*m^2*x-c*d*f^2*h*m*x^2-d^2*e^2*h*m^2*x+d^2*e*f*h*m*
x^2-c^2*f^2*g*m^2-4*c^2*f^2*h*m*x+2*c*d*e*f*g*m^2+8*c*d*e*f*h*m*x-2*c*d*f^2*g*m*x-c*d*f^2*h*x^2-d^2*e^2*g*m^2-
4*d^2*e^2*h*m*x+2*d^2*e*f*g*m*x+3*d^2*e*f*h*x^2-2*d^2*f^2*g*x^2+c^2*e*f*h*m-5*c^2*f^2*g*m-3*c^2*f^2*h*x-c*d*e^
2*h*m+8*c*d*e*f*g*m+10*c*d*e*f*h*x-6*c*d*f^2*g*x-3*d^2*e^2*g*m-3*d^2*e^2*h*x+2*d^2*e*f*g*x+3*c^2*e*f*h-6*c^2*f
^2*g-c*d*e^2*h+6*c*d*e*f*g-2*d^2*e^2*g)/(c^3*f^3*m^3-3*c^2*d*e*f^2*m^3+3*c*d^2*e^2*f*m^3-d^3*e^3*m^3+6*c^3*f^3
*m^2-18*c^2*d*e*f^2*m^2+18*c*d^2*e^2*f*m^2-6*d^3*e^3*m^2+11*c^3*f^3*m-33*c^2*d*e*f^2*m+33*c*d^2*e^2*f*m-11*d^3
*e^3*m+6*c^3*f^3-18*c^2*d*e*f^2+18*c*d^2*e^2*f-6*d^3*e^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.41, size = 869, normalized size = 4.62 \begin {gather*} \frac {x^2\,{\left (e+f\,x\right )}^m\,\left (h\,c^3\,f^3\,m^2+4\,h\,c^3\,f^3\,m+3\,h\,c^3\,f^3-h\,c^2\,d\,e\,f^2\,m^2-4\,h\,c^2\,d\,e\,f^2\,m-9\,h\,c^2\,d\,e\,f^2+g\,c^2\,d\,f^3\,m^2+7\,g\,c^2\,d\,f^3\,m+12\,g\,c^2\,d\,f^3-h\,c\,d^2\,e^2\,f\,m^2-4\,h\,c\,d^2\,e^2\,f\,m-9\,h\,c\,d^2\,e^2\,f-2\,g\,c\,d^2\,e\,f^2\,m^2-8\,g\,c\,d^2\,e\,f^2\,m+h\,d^3\,e^3\,m^2+4\,h\,d^3\,e^3\,m+3\,h\,d^3\,e^3+g\,d^3\,e^2\,f\,m^2+g\,d^3\,e^2\,f\,m\right )}{{\left (c\,f-d\,e\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {x\,{\left (e+f\,x\right )}^m\,\left (h\,c^3\,e\,f^2\,m^2+3\,h\,c^3\,e\,f^2\,m+g\,c^3\,f^3\,m^2+5\,g\,c^3\,f^3\,m+6\,g\,c^3\,f^3-2\,h\,c^2\,d\,e^2\,f\,m^2-8\,h\,c^2\,d\,e^2\,f\,m-12\,h\,c^2\,d\,e^2\,f-g\,c^2\,d\,e\,f^2\,m^2-g\,c^2\,d\,e\,f^2\,m+6\,g\,c^2\,d\,e\,f^2+h\,c\,d^2\,e^3\,m^2+5\,h\,c\,d^2\,e^3\,m+4\,h\,c\,d^2\,e^3-g\,c\,d^2\,e^2\,f\,m^2-7\,g\,c\,d^2\,e^2\,f\,m-6\,g\,c\,d^2\,e^2\,f+g\,d^3\,e^3\,m^2+3\,g\,d^3\,e^3\,m+2\,g\,d^3\,e^3\right )}{{\left (c\,f-d\,e\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {c\,e\,{\left (e+f\,x\right )}^m\,\left (-h\,c^2\,e\,f\,m-3\,h\,c^2\,e\,f+g\,c^2\,f^2\,m^2+5\,g\,c^2\,f^2\,m+6\,g\,c^2\,f^2+h\,c\,d\,e^2\,m+h\,c\,d\,e^2-2\,g\,c\,d\,e\,f\,m^2-8\,g\,c\,d\,e\,f\,m-6\,g\,c\,d\,e\,f+g\,d^2\,e^2\,m^2+3\,g\,d^2\,e^2\,m+2\,g\,d^2\,e^2\right )}{{\left (c\,f-d\,e\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d^2\,f^2\,x^4\,{\left (e+f\,x\right )}^m\,\left (c\,f\,h-3\,d\,e\,h+2\,d\,f\,g+c\,f\,h\,m-d\,e\,h\,m\right )}{{\left (c\,f-d\,e\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,f\,x^3\,{\left (e+f\,x\right )}^m\,\left (4\,c\,f+c\,f\,m-d\,e\,m\right )\,\left (c\,f\,h-3\,d\,e\,h+2\,d\,f\,g+c\,f\,h\,m-d\,e\,h\,m\right )}{{\left (c\,f-d\,e\right )}^3\,{\left (c+d\,x\right )}^{m+4}\,\left (m^3+6\,m^2+11\,m+6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*(e + f*x)^m*(3*c^3*f^3*h + 3*d^3*e^3*h + c^3*f^3*h*m^2 + d^3*e^3*h*m^2 + 12*c^2*d*f^3*g + 4*c^3*f^3*h*m +
4*d^3*e^3*h*m - 9*c*d^2*e^2*f*h - 9*c^2*d*e*f^2*h + 7*c^2*d*f^3*g*m + d^3*e^2*f*g*m + c^2*d*f^3*g*m^2 + d^3*e
^2*f*g*m^2 - 8*c*d^2*e*f^2*g*m - 4*c*d^2*e^2*f*h*m - 4*c^2*d*e*f^2*h*m - 2*c*d^2*e*f^2*g*m^2 - c*d^2*e^2*f*h*m
^2 - c^2*d*e*f^2*h*m^2))/((c*f - d*e)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) + (x*(e + f*x)^m*(6*c^3*f^
3*g + 2*d^3*e^3*g + c^3*f^3*g*m^2 + d^3*e^3*g*m^2 + 4*c*d^2*e^3*h + 5*c^3*f^3*g*m + 3*d^3*e^3*g*m - 6*c*d^2*e^
2*f*g + 6*c^2*d*e*f^2*g - 12*c^2*d*e^2*f*h + 5*c*d^2*e^3*h*m + 3*c^3*e*f^2*h*m + c*d^2*e^3*h*m^2 + c^3*e*f^2*h
*m^2 - 7*c*d^2*e^2*f*g*m - c^2*d*e*f^2*g*m - 8*c^2*d*e^2*f*h*m - c*d^2*e^2*f*g*m^2 - c^2*d*e*f^2*g*m^2 - 2*c^2
*d*e^2*f*h*m^2))/((c*f - d*e)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) + (c*e*(e + f*x)^m*(6*c^2*f^2*g +
2*d^2*e^2*g + c^2*f^2*g*m^2 + d^2*e^2*g*m^2 + c*d*e^2*h - 3*c^2*e*f*h + 5*c^2*f^2*g*m + 3*d^2*e^2*g*m - 6*c*d*
e*f*g + c*d*e^2*h*m - c^2*e*f*h*m - 2*c*d*e*f*g*m^2 - 8*c*d*e*f*g*m))/((c*f - d*e)^3*(c + d*x)^(m + 4)*(11*m +
6*m^2 + m^3 + 6)) + (d^2*f^2*x^4*(e + f*x)^m*(c*f*h - 3*d*e*h + 2*d*f*g + c*f*h*m - d*e*h*m))/((c*f - d*e)^3*
(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6)) + (d*f*x^3*(e + f*x)^m*(4*c*f + c*f*m - d*e*m)*(c*f*h - 3*d*e*h +
2*d*f*g + c*f*h*m - d*e*h*m))/((c*f - d*e)^3*(c + d*x)^(m + 4)*(11*m + 6*m^2 + m^3 + 6))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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