3.768 \(\int (d+e x)^m (f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\)

Optimal. Leaf size=343 \[ -\frac{6 (d+e x)^{m-1} (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac{6 g (d+e x)^m (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac{3 (f+g x)^2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac{(f+g x)^3 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (4-m)} \]

[Out]

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

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Rubi [A]  time = 0.449159, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {870, 794, 648} \[ -\frac{6 (d+e x)^{m-1} (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac{6 g (d+e x)^m (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac{3 (f+g x)^2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac{(f+g x)^3 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (4-m)} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^m*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]

[Out]

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

Rule 870

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
-Simp[(e*(d + e*x)^(m - 1)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(c*(m - n - 1)), x] - Dist[(n*(c*e*f + c*d*g
- b*e*g))/(c*e*(m - n - 1)), Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, p}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Integ
erQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g
, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

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

Mathematica [A]  time = 0.16738, size = 134, normalized size = 0.39 \[ \frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \left (\frac{3 g^2 (a e+c d x)^2 (a e g-c d f)}{m-3}-\frac{3 g (a e+c d x) (c d f-a e g)^2}{m-2}-\frac{(c d f-a e g)^3}{m-1}-\frac{g^3 (a e+c d x)^3}{m-4}\right )}{c^4 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^m*(f + g*x)^3)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^m,x]

[Out]

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

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Maple [A]  time = 0.051, size = 527, normalized size = 1.5 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{m} \left ({c}^{3}{d}^{3}{g}^{3}{m}^{3}{x}^{3}+3\,{c}^{3}{d}^{3}f{g}^{2}{m}^{3}{x}^{2}-6\,{c}^{3}{d}^{3}{g}^{3}{m}^{2}{x}^{3}+3\,a{c}^{2}{d}^{2}e{g}^{3}{m}^{2}{x}^{2}+3\,{c}^{3}{d}^{3}{f}^{2}g{m}^{3}x-21\,{c}^{3}{d}^{3}f{g}^{2}{m}^{2}{x}^{2}+11\,{c}^{3}{d}^{3}{g}^{3}m{x}^{3}+6\,a{c}^{2}{d}^{2}ef{g}^{2}{m}^{2}x-9\,a{c}^{2}{d}^{2}e{g}^{3}m{x}^{2}+{c}^{3}{d}^{3}{f}^{3}{m}^{3}-24\,{c}^{3}{d}^{3}{f}^{2}g{m}^{2}x+42\,{c}^{3}{d}^{3}f{g}^{2}m{x}^{2}-6\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+6\,{a}^{2}cd{e}^{2}{g}^{3}mx+3\,a{c}^{2}{d}^{2}e{f}^{2}g{m}^{2}-30\,a{c}^{2}{d}^{2}ef{g}^{2}mx+6\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-9\,{c}^{3}{d}^{3}{f}^{3}{m}^{2}+57\,{c}^{3}{d}^{3}{f}^{2}gmx-24\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}+6\,{a}^{2}cd{e}^{2}f{g}^{2}m-6\,{a}^{2}cd{e}^{2}{g}^{3}x-21\,a{c}^{2}{d}^{2}e{f}^{2}gm+24\,a{c}^{2}{d}^{2}ef{g}^{2}x+26\,{c}^{3}{d}^{3}{f}^{3}m-36\,{c}^{3}{d}^{3}{f}^{2}gx+6\,{a}^{3}{e}^{3}{g}^{3}-24\,{a}^{2}cd{e}^{2}f{g}^{2}+36\,a{c}^{2}{d}^{2}e{f}^{2}g-24\,{c}^{3}{d}^{3}{f}^{3} \right ) \left ( cdx+ae \right ) }{ \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}{c}^{4}{d}^{4} \left ({m}^{4}-10\,{m}^{3}+35\,{m}^{2}-50\,m+24 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x)

[Out]

-(e*x+d)^m*(c^3*d^3*g^3*m^3*x^3+3*c^3*d^3*f*g^2*m^3*x^2-6*c^3*d^3*g^3*m^2*x^3+3*a*c^2*d^2*e*g^3*m^2*x^2+3*c^3*
d^3*f^2*g*m^3*x-21*c^3*d^3*f*g^2*m^2*x^2+11*c^3*d^3*g^3*m*x^3+6*a*c^2*d^2*e*f*g^2*m^2*x-9*a*c^2*d^2*e*g^3*m*x^
2+c^3*d^3*f^3*m^3-24*c^3*d^3*f^2*g*m^2*x+42*c^3*d^3*f*g^2*m*x^2-6*c^3*d^3*g^3*x^3+6*a^2*c*d*e^2*g^3*m*x+3*a*c^
2*d^2*e*f^2*g*m^2-30*a*c^2*d^2*e*f*g^2*m*x+6*a*c^2*d^2*e*g^3*x^2-9*c^3*d^3*f^3*m^2+57*c^3*d^3*f^2*g*m*x-24*c^3
*d^3*f*g^2*x^2+6*a^2*c*d*e^2*f*g^2*m-6*a^2*c*d*e^2*g^3*x-21*a*c^2*d^2*e*f^2*g*m+24*a*c^2*d^2*e*f*g^2*x+26*c^3*
d^3*f^3*m-36*c^3*d^3*f^2*g*x+6*a^3*e^3*g^3-24*a^2*c*d*e^2*f*g^2+36*a*c^2*d^2*e*f^2*g-24*c^3*d^3*f^3)*(c*d*x+a*
e)/((c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^m)/c^4/d^4/(m^4-10*m^3+35*m^2-50*m+24)

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Maxima [A]  time = 1.15708, size = 447, normalized size = 1.3 \begin{align*} -\frac{{\left (c d x + a e\right )} f^{3}}{{\left (c d x + a e\right )}^{m} c d{\left (m - 1\right )}} - \frac{3 \,{\left (c^{2} d^{2}{\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} f^{2} g}{{\left (m^{2} - 3 \, m + 2\right )}{\left (c d x + a e\right )}^{m} c^{2} d^{2}} - \frac{3 \,{\left ({\left (m^{2} - 3 \, m + 2\right )} c^{3} d^{3} x^{3} +{\left (m^{2} - m\right )} a c^{2} d^{2} e x^{2} + 2 \, a^{2} c d e^{2} m x + 2 \, a^{3} e^{3}\right )} f g^{2}}{{\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )}{\left (c d x + a e\right )}^{m} c^{3} d^{3}} - \frac{{\left ({\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )} c^{4} d^{4} x^{4} +{\left (m^{3} - 3 \, m^{2} + 2 \, m\right )} a c^{3} d^{3} e x^{3} + 3 \,{\left (m^{2} - m\right )} a^{2} c^{2} d^{2} e^{2} x^{2} + 6 \, a^{3} c d e^{3} m x + 6 \, a^{4} e^{4}\right )} g^{3}}{{\left (m^{4} - 10 \, m^{3} + 35 \, m^{2} - 50 \, m + 24\right )}{\left (c d x + a e\right )}^{m} c^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="maxima")

[Out]

-(c*d*x + a*e)*f^3/((c*d*x + a*e)^m*c*d*(m - 1)) - 3*(c^2*d^2*(m - 1)*x^2 + a*c*d*e*m*x + a^2*e^2)*f^2*g/((m^2
- 3*m + 2)*(c*d*x + a*e)^m*c^2*d^2) - 3*((m^2 - 3*m + 2)*c^3*d^3*x^3 + (m^2 - m)*a*c^2*d^2*e*x^2 + 2*a^2*c*d*
e^2*m*x + 2*a^3*e^3)*f*g^2/((m^3 - 6*m^2 + 11*m - 6)*(c*d*x + a*e)^m*c^3*d^3) - ((m^3 - 6*m^2 + 11*m - 6)*c^4*
d^4*x^4 + (m^3 - 3*m^2 + 2*m)*a*c^3*d^3*e*x^3 + 3*(m^2 - m)*a^2*c^2*d^2*e^2*x^2 + 6*a^3*c*d*e^3*m*x + 6*a^4*e^
4)*g^3/((m^4 - 10*m^3 + 35*m^2 - 50*m + 24)*(c*d*x + a*e)^m*c^4*d^4)

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Fricas [B]  time = 1.4557, size = 1396, normalized size = 4.07 \begin{align*} -\frac{{\left (a c^{3} d^{3} e f^{3} m^{3} - 24 \, a c^{3} d^{3} e f^{3} + 36 \, a^{2} c^{2} d^{2} e^{2} f^{2} g - 24 \, a^{3} c d e^{3} f g^{2} + 6 \, a^{4} e^{4} g^{3} +{\left (c^{4} d^{4} g^{3} m^{3} - 6 \, c^{4} d^{4} g^{3} m^{2} + 11 \, c^{4} d^{4} g^{3} m - 6 \, c^{4} d^{4} g^{3}\right )} x^{4} -{\left (24 \, c^{4} d^{4} f g^{2} -{\left (3 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m^{3} + 3 \,{\left (7 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m^{2} - 2 \,{\left (21 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m\right )} x^{3} - 3 \,{\left (3 \, a c^{3} d^{3} e f^{3} - a^{2} c^{2} d^{2} e^{2} f^{2} g\right )} m^{2} - 3 \,{\left (12 \, c^{4} d^{4} f^{2} g -{\left (c^{4} d^{4} f^{2} g + a c^{3} d^{3} e f g^{2}\right )} m^{3} +{\left (8 \, c^{4} d^{4} f^{2} g + 5 \, a c^{3} d^{3} e f g^{2} - a^{2} c^{2} d^{2} e^{2} g^{3}\right )} m^{2} -{\left (19 \, c^{4} d^{4} f^{2} g + 4 \, a c^{3} d^{3} e f g^{2} - a^{2} c^{2} d^{2} e^{2} g^{3}\right )} m\right )} x^{2} +{\left (26 \, a c^{3} d^{3} e f^{3} - 21 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 6 \, a^{3} c d e^{3} f g^{2}\right )} m -{\left (24 \, c^{4} d^{4} f^{3} -{\left (c^{4} d^{4} f^{3} + 3 \, a c^{3} d^{3} e f^{2} g\right )} m^{3} + 3 \,{\left (3 \, c^{4} d^{4} f^{3} + 7 \, a c^{3} d^{3} e f^{2} g - 2 \, a^{2} c^{2} d^{2} e^{2} f g^{2}\right )} m^{2} - 2 \,{\left (13 \, c^{4} d^{4} f^{3} + 18 \, a c^{3} d^{3} e f^{2} g - 12 \, a^{2} c^{2} d^{2} e^{2} f g^{2} + 3 \, a^{3} c d e^{3} g^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (c^{4} d^{4} m^{4} - 10 \, c^{4} d^{4} m^{3} + 35 \, c^{4} d^{4} m^{2} - 50 \, c^{4} d^{4} m + 24 \, c^{4} d^{4}\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="fricas")

[Out]

-(a*c^3*d^3*e*f^3*m^3 - 24*a*c^3*d^3*e*f^3 + 36*a^2*c^2*d^2*e^2*f^2*g - 24*a^3*c*d*e^3*f*g^2 + 6*a^4*e^4*g^3 +
(c^4*d^4*g^3*m^3 - 6*c^4*d^4*g^3*m^2 + 11*c^4*d^4*g^3*m - 6*c^4*d^4*g^3)*x^4 - (24*c^4*d^4*f*g^2 - (3*c^4*d^4
*f*g^2 + a*c^3*d^3*e*g^3)*m^3 + 3*(7*c^4*d^4*f*g^2 + a*c^3*d^3*e*g^3)*m^2 - 2*(21*c^4*d^4*f*g^2 + a*c^3*d^3*e*
g^3)*m)*x^3 - 3*(3*a*c^3*d^3*e*f^3 - a^2*c^2*d^2*e^2*f^2*g)*m^2 - 3*(12*c^4*d^4*f^2*g - (c^4*d^4*f^2*g + a*c^3
*d^3*e*f*g^2)*m^3 + (8*c^4*d^4*f^2*g + 5*a*c^3*d^3*e*f*g^2 - a^2*c^2*d^2*e^2*g^3)*m^2 - (19*c^4*d^4*f^2*g + 4*
a*c^3*d^3*e*f*g^2 - a^2*c^2*d^2*e^2*g^3)*m)*x^2 + (26*a*c^3*d^3*e*f^3 - 21*a^2*c^2*d^2*e^2*f^2*g + 6*a^3*c*d*e
^3*f*g^2)*m - (24*c^4*d^4*f^3 - (c^4*d^4*f^3 + 3*a*c^3*d^3*e*f^2*g)*m^3 + 3*(3*c^4*d^4*f^3 + 7*a*c^3*d^3*e*f^2
*g - 2*a^2*c^2*d^2*e^2*f*g^2)*m^2 - 2*(13*c^4*d^4*f^3 + 18*a*c^3*d^3*e*f^2*g - 12*a^2*c^2*d^2*e^2*f*g^2 + 3*a^
3*c*d*e^3*g^3)*m)*x)*(e*x + d)^m/((c^4*d^4*m^4 - 10*c^4*d^4*m^3 + 35*c^4*d^4*m^2 - 50*c^4*d^4*m + 24*c^4*d^4)*
(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m)

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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((e*x+d)**m*(g*x+f)**3/((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**m),x)

[Out]

Timed out

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Giac [B]  time = 1.42207, size = 2732, normalized size = 7.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^3/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="giac")

[Out]

-((x*e + d)^m*c^4*d^4*g^3*m^3*x^4*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 3*(x*e + d)^m*c^4*d^4*f*g^2*m^3*x
^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) - 6*(x*e + d)^m*c^4*d^4*g^3*m^2*x^4*e^(-m*log(c*d*x + a*e) - m*log
(x*e + d)) + (x*e + d)^m*a*c^3*d^3*g^3*m^3*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + 3*(x*e + d)^m*c^
4*d^4*f^2*g*m^3*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) - 21*(x*e + d)^m*c^4*d^4*f*g^2*m^2*x^3*e^(-m*log(
c*d*x + a*e) - m*log(x*e + d)) + 11*(x*e + d)^m*c^4*d^4*g^3*m*x^4*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 3
*(x*e + d)^m*a*c^3*d^3*f*g^2*m^3*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - 3*(x*e + d)^m*a*c^3*d^3*g^
3*m^2*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + (x*e + d)^m*c^4*d^4*f^3*m^3*x*e^(-m*log(c*d*x + a*e)
- m*log(x*e + d)) - 24*(x*e + d)^m*c^4*d^4*f^2*g*m^2*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 42*(x*e +
d)^m*c^4*d^4*f*g^2*m*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) - 6*(x*e + d)^m*c^4*d^4*g^3*x^4*e^(-m*log(c*
d*x + a*e) - m*log(x*e + d)) + 3*(x*e + d)^m*a*c^3*d^3*f^2*g*m^3*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1
) - 15*(x*e + d)^m*a*c^3*d^3*f*g^2*m^2*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + 2*(x*e + d)^m*a*c^3*
d^3*g^3*m*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - 9*(x*e + d)^m*c^4*d^4*f^3*m^2*x*e^(-m*log(c*d*x +
a*e) - m*log(x*e + d)) + 57*(x*e + d)^m*c^4*d^4*f^2*g*m*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) - 24*(x*
e + d)^m*c^4*d^4*f*g^2*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 3*(x*e + d)^m*a^2*c^2*d^2*g^3*m^2*x^2*e^
(-m*log(c*d*x + a*e) - m*log(x*e + d) + 2) + (x*e + d)^m*a*c^3*d^3*f^3*m^3*e^(-m*log(c*d*x + a*e) - m*log(x*e
+ d) + 1) - 21*(x*e + d)^m*a*c^3*d^3*f^2*g*m^2*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + 12*(x*e + d)^m
*a*c^3*d^3*f*g^2*m*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + 26*(x*e + d)^m*c^4*d^4*f^3*m*x*e^(-m*log
(c*d*x + a*e) - m*log(x*e + d)) - 36*(x*e + d)^m*c^4*d^4*f^2*g*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) +
6*(x*e + d)^m*a^2*c^2*d^2*f*g^2*m^2*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 2) - 3*(x*e + d)^m*a^2*c^2*d^2
*g^3*m*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 2) - 9*(x*e + d)^m*a*c^3*d^3*f^3*m^2*e^(-m*log(c*d*x + a*
e) - m*log(x*e + d) + 1) + 36*(x*e + d)^m*a*c^3*d^3*f^2*g*m*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - 2
4*(x*e + d)^m*c^4*d^4*f^3*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 3*(x*e + d)^m*a^2*c^2*d^2*f^2*g*m^2*e^(
-m*log(c*d*x + a*e) - m*log(x*e + d) + 2) - 24*(x*e + d)^m*a^2*c^2*d^2*f*g^2*m*x*e^(-m*log(c*d*x + a*e) - m*lo
g(x*e + d) + 2) + 26*(x*e + d)^m*a*c^3*d^3*f^3*m*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + 6*(x*e + d)^m*
a^3*c*d*g^3*m*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 3) - 21*(x*e + d)^m*a^2*c^2*d^2*f^2*g*m*e^(-m*log(c*
d*x + a*e) - m*log(x*e + d) + 2) - 24*(x*e + d)^m*a*c^3*d^3*f^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) +
6*(x*e + d)^m*a^3*c*d*f*g^2*m*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 3) + 36*(x*e + d)^m*a^2*c^2*d^2*f^2*g
*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 2) - 24*(x*e + d)^m*a^3*c*d*f*g^2*e^(-m*log(c*d*x + a*e) - m*log(x*
e + d) + 3) + 6*(x*e + d)^m*a^4*g^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 4))/(c^4*d^4*m^4 - 10*c^4*d^4*m^
3 + 35*c^4*d^4*m^2 - 50*c^4*d^4*m + 24*c^4*d^4)