3.8.68 \(\int (d+e x)^m (f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\) [768]

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

[Out]

-6*(-a*e*g+c*d*f)^2*(a*e^2*g+c*d*(d*g*(1-m)-e*f*(2-m)))*(e*x+d)^(-1+m)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1-m)
/c^4/d^4/e/(m^2-7*m+12)/(m^2-3*m+2)+6*g*(-a*e*g+c*d*f)^2*(e*x+d)^m*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1-m)/c^3
/d^3/e/(2-m)/(3-m)/(4-m)+3*(-a*e*g+c*d*f)*(e*x+d)^(-1+m)*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1-m)/c^2
/d^2/(3-m)/(4-m)+(e*x+d)^(-1+m)*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1-m)/c/d/(4-m)

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Rubi [A]
time = 0.43, antiderivative size = 343, normalized size of antiderivative = 1.00, 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 = {884, 808, 662} \begin {gather*} -\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)} \end {gather*}

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 662

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]

Rule 808

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 884

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] &&  !Int
egerQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

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*}

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Mathematica [A]
time = 0.27, size = 134, normalized size = 0.39 \begin {gather*} \frac {(d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m} \left (-\frac {(c d f-a e g)^3}{-1+m}-\frac {3 g (c d f-a e g)^2 (a e+c d x)}{-2+m}+\frac {3 g^2 (-c d f+a e g) (a e+c d x)^2}{-3+m}-\frac {g^3 (a e+c d x)^3}{-4+m}\right )}{c^4 d^4} \end {gather*}

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 = 0.17, size = 527, normalized size = 1.54

method	result	size
gosper	\(-\frac {\left (e x +d \right )^{m} \left (c^{3} d^{3} g^{3} m^{3} x^{3}+3 c^{3} d^{3} f \,g^{2} m^{3} x^{2}-6 c^{3} d^{3} g^{3} m^{2} x^{3}+3 a \,c^{2} d^{2} e \,g^{3} m^{2} x^{2}+3 c^{3} d^{3} f^{2} g \,m^{3} x -21 c^{3} d^{3} f \,g^{2} m^{2} x^{2}+11 c^{3} d^{3} g^{3} m \,x^{3}+6 a \,c^{2} d^{2} e f \,g^{2} m^{2} x -9 a \,c^{2} d^{2} e \,g^{3} m \,x^{2}+c^{3} d^{3} f^{3} m^{3}-24 c^{3} d^{3} f^{2} g \,m^{2} x +42 c^{3} d^{3} f \,g^{2} m \,x^{2}-6 g^{3} x^{3} c^{3} d^{3}+6 a^{2} c d \,e^{2} g^{3} m x +3 a \,c^{2} d^{2} e \,f^{2} g \,m^{2}-30 a \,c^{2} d^{2} e f \,g^{2} m x +6 a \,c^{2} d^{2} e \,g^{3} x^{2}-9 c^{3} d^{3} f^{3} m^{2}+57 c^{3} d^{3} f^{2} g m x -24 c^{3} d^{3} f \,g^{2} x^{2}+6 a^{2} c d \,e^{2} f \,g^{2} m -6 a^{2} c d \,e^{2} g^{3} x -21 a \,c^{2} d^{2} e \,f^{2} g m +24 a \,c^{2} d^{2} e f \,g^{2} x +26 c^{3} d^{3} f^{3} m -36 c^{3} d^{3} f^{2} g x +6 a^{3} e^{3} g^{3}-24 a^{2} c d \,e^{2} f \,g^{2}+36 a \,c^{2} d^{2} e \,f^{2} g -24 f^{3} c^{3} d^{3}\right ) \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{c^{4} d^{4} \left (m^{4}-10 m^{3}+35 m^{2}-50 m +24\right )}\)	\(527\)
risch	\(-\frac {\left (c^{4} d^{4} g^{3} m^{3} x^{4}+a \,c^{3} d^{3} e \,g^{3} m^{3} x^{3}+3 c^{4} d^{4} f \,g^{2} m^{3} x^{3}-6 c^{4} d^{4} g^{3} m^{2} x^{4}+3 a \,c^{3} d^{3} e f \,g^{2} m^{3} x^{2}-3 a \,c^{3} d^{3} e \,g^{3} m^{2} x^{3}+3 c^{4} d^{4} f^{2} g \,m^{3} x^{2}-21 c^{4} d^{4} f \,g^{2} m^{2} x^{3}+11 c^{4} d^{4} g^{3} m \,x^{4}+3 a^{2} c^{2} d^{2} e^{2} g^{3} m^{2} x^{2}+3 a \,c^{3} d^{3} e \,f^{2} g \,m^{3} x -15 a \,c^{3} d^{3} e f \,g^{2} m^{2} x^{2}+2 a \,c^{3} d^{3} e \,g^{3} m \,x^{3}+c^{4} d^{4} f^{3} m^{3} x -24 c^{4} d^{4} f^{2} g \,m^{2} x^{2}+42 c^{4} d^{4} f \,g^{2} m \,x^{3}-6 g^{3} x^{4} c^{4} d^{4}+6 a^{2} c^{2} d^{2} e^{2} f \,g^{2} m^{2} x -3 a^{2} c^{2} d^{2} e^{2} g^{3} m \,x^{2}+a \,c^{3} d^{3} e \,f^{3} m^{3}-21 a \,c^{3} d^{3} e \,f^{2} g \,m^{2} x +12 a \,c^{3} d^{3} e f \,g^{2} m \,x^{2}-9 c^{4} d^{4} f^{3} m^{2} x +57 c^{4} d^{4} f^{2} g m \,x^{2}-24 c^{4} d^{4} f \,g^{2} x^{3}+6 a^{3} c d \,e^{3} g^{3} m x +3 a^{2} c^{2} d^{2} e^{2} f^{2} g \,m^{2}-24 a^{2} c^{2} d^{2} e^{2} f \,g^{2} m x -9 a \,c^{3} d^{3} e \,f^{3} m^{2}+36 a \,c^{3} d^{3} e \,f^{2} g m x +26 c^{4} d^{4} f^{3} m x -36 c^{4} d^{4} f^{2} g \,x^{2}+6 a^{3} c d \,e^{3} f \,g^{2} m -21 a^{2} c^{2} d^{2} e^{2} f^{2} g m +26 a \,c^{3} d^{3} e \,f^{3} m -24 c^{4} d^{4} f^{3} x +6 g^{3} e^{4} a^{4}-24 a^{3} c d \,e^{3} f \,g^{2}+36 a^{2} c^{2} d^{2} e^{2} f^{2} g -24 a \,c^{3} d^{3} e \,f^{3}\right ) \left (e x +d \right )^{m} {\mathrm e}^{\frac {m \left (i \pi \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{3}-i \pi \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{2} \mathrm {csgn}\left (i \left (e x +d \right )\right )-i \pi \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{2} \mathrm {csgn}\left (i \left (c d x +a e \right )\right )+i \pi \,\mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right ) \mathrm {csgn}\left (i \left (e x +d \right )\right ) \mathrm {csgn}\left (i \left (c d x +a e \right )\right )-2 \ln \left (e x +d \right )-2 \ln \left (c d x +a e \right )\right )}{2}}}{\left (-3+m \right ) \left (m -4\right ) \left (m -2\right ) \left (-1+m \right ) c^{4} d^{4}}\)	\(864\)

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,method=_RETURNVERBOSE)

[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 = 0.32, size = 333, normalized size = 0.97 \begin {gather*} -\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 m x e + 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} x^{2} e + 2 \, a^{2} c d m x e^{2} + 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} x^{3} e + 3 \, {\left (m^{2} - m\right )} a^{2} c^{2} d^{2} x^{2} e^{2} + 6 \, a^{3} c d m x e^{3} + 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 {gather*}

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*m*x*e + 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*x^2*e + 2*a^2*c*d*
m*x*e^2 + 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*x^3*e + 3*(m^2 - m)*a^2*c^2*d^2*x^2*e^2 + 6*a^3*c*d*m*x*e^3 + 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] Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (330) = 660\).
time = 0.82, size = 699, normalized size = 2.04 \begin {gather*} -\frac {{\left (6 \, a^{4} g^{3} e^{4} + {\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} + 3 \, {\left (c^{4} d^{4} f g^{2} m^{3} - 7 \, c^{4} d^{4} f g^{2} m^{2} + 14 \, c^{4} d^{4} f g^{2} m - 8 \, c^{4} d^{4} f g^{2}\right )} x^{3} + 3 \, {\left (c^{4} d^{4} f^{2} g m^{3} - 8 \, c^{4} d^{4} f^{2} g m^{2} + 19 \, c^{4} d^{4} f^{2} g m - 12 \, c^{4} d^{4} f^{2} g\right )} x^{2} + {\left (c^{4} d^{4} f^{3} m^{3} - 9 \, c^{4} d^{4} f^{3} m^{2} + 26 \, c^{4} d^{4} f^{3} m - 24 \, c^{4} d^{4} f^{3}\right )} x + 6 \, {\left (a^{3} c d g^{3} m x + a^{3} c d f g^{2} m - 4 \, a^{3} c d f g^{2}\right )} e^{3} + 3 \, {\left (a^{2} c^{2} d^{2} f^{2} g m^{2} - 7 \, a^{2} c^{2} d^{2} f^{2} g m + 12 \, a^{2} c^{2} d^{2} f^{2} g + {\left (a^{2} c^{2} d^{2} g^{3} m^{2} - a^{2} c^{2} d^{2} g^{3} m\right )} x^{2} + 2 \, {\left (a^{2} c^{2} d^{2} f g^{2} m^{2} - 4 \, a^{2} c^{2} d^{2} f g^{2} m\right )} x\right )} e^{2} + {\left (a c^{3} d^{3} f^{3} m^{3} - 9 \, a c^{3} d^{3} f^{3} m^{2} + 26 \, a c^{3} d^{3} f^{3} m - 24 \, a c^{3} d^{3} f^{3} + {\left (a c^{3} d^{3} g^{3} m^{3} - 3 \, a c^{3} d^{3} g^{3} m^{2} + 2 \, a c^{3} d^{3} g^{3} m\right )} x^{3} + 3 \, {\left (a c^{3} d^{3} f g^{2} m^{3} - 5 \, a c^{3} d^{3} f g^{2} m^{2} + 4 \, a c^{3} d^{3} f g^{2} m\right )} x^{2} + 3 \, {\left (a c^{3} d^{3} f^{2} g m^{3} - 7 \, a c^{3} d^{3} f^{2} g m^{2} + 12 \, a c^{3} d^{3} f^{2} g m\right )} x\right )} e\right )} {\left (x e + 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^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )}^{m}} \end {gather*}

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]

-(6*a^4*g^3*e^4 + (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 + 3*(c^4*d^4*f*
g^2*m^3 - 7*c^4*d^4*f*g^2*m^2 + 14*c^4*d^4*f*g^2*m - 8*c^4*d^4*f*g^2)*x^3 + 3*(c^4*d^4*f^2*g*m^3 - 8*c^4*d^4*f
^2*g*m^2 + 19*c^4*d^4*f^2*g*m - 12*c^4*d^4*f^2*g)*x^2 + (c^4*d^4*f^3*m^3 - 9*c^4*d^4*f^3*m^2 + 26*c^4*d^4*f^3*
m - 24*c^4*d^4*f^3)*x + 6*(a^3*c*d*g^3*m*x + a^3*c*d*f*g^2*m - 4*a^3*c*d*f*g^2)*e^3 + 3*(a^2*c^2*d^2*f^2*g*m^2
- 7*a^2*c^2*d^2*f^2*g*m + 12*a^2*c^2*d^2*f^2*g + (a^2*c^2*d^2*g^3*m^2 - a^2*c^2*d^2*g^3*m)*x^2 + 2*(a^2*c^2*d
^2*f*g^2*m^2 - 4*a^2*c^2*d^2*f*g^2*m)*x)*e^2 + (a*c^3*d^3*f^3*m^3 - 9*a*c^3*d^3*f^3*m^2 + 26*a*c^3*d^3*f^3*m -
24*a*c^3*d^3*f^3 + (a*c^3*d^3*g^3*m^3 - 3*a*c^3*d^3*g^3*m^2 + 2*a*c^3*d^3*g^3*m)*x^3 + 3*(a*c^3*d^3*f*g^2*m^3
- 5*a*c^3*d^3*f*g^2*m^2 + 4*a*c^3*d^3*f*g^2*m)*x^2 + 3*(a*c^3*d^3*f^2*g*m^3 - 7*a*c^3*d^3*f^2*g*m^2 + 12*a*c^
3*d^3*f^2*g*m)*x)*e)*(x*e + 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^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)^m)

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Sympy [F(-1)] Timed out
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((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] Leaf count of result is larger than twice the leaf count of optimal. 2024 vs. \(2 (330) = 660\).
time = 3.34, size = 2024, normalized size = 5.90 \begin {gather*} \text {Too large to display} \end {gather*}

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)

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Mupad [B]
time = 3.75, size = 615, normalized size = 1.79 \begin {gather*} -\frac {\frac {g^3\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3-6\,m^2+11\,m-6\right )}{m^4-10\,m^3+35\,m^2-50\,m+24}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (6\,a^3\,c\,d\,e^3\,g^3\,m+6\,a^2\,c^2\,d^2\,e^2\,f\,g^2\,m^2-24\,a^2\,c^2\,d^2\,e^2\,f\,g^2\,m+3\,a\,c^3\,d^3\,e\,f^2\,g\,m^3-21\,a\,c^3\,d^3\,e\,f^2\,g\,m^2+36\,a\,c^3\,d^3\,e\,f^2\,g\,m+c^4\,d^4\,f^3\,m^3-9\,c^4\,d^4\,f^3\,m^2+26\,c^4\,d^4\,f^3\,m-24\,c^4\,d^4\,f^3\right )}{c^4\,d^4\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}+\frac {a\,e\,{\left (d+e\,x\right )}^m\,\left (6\,a^3\,e^3\,g^3+6\,a^2\,c\,d\,e^2\,f\,g^2\,m-24\,a^2\,c\,d\,e^2\,f\,g^2+3\,a\,c^2\,d^2\,e\,f^2\,g\,m^2-21\,a\,c^2\,d^2\,e\,f^2\,g\,m+36\,a\,c^2\,d^2\,e\,f^2\,g+c^3\,d^3\,f^3\,m^3-9\,c^3\,d^3\,f^3\,m^2+26\,c^3\,d^3\,f^3\,m-24\,c^3\,d^3\,f^3\right )}{c^4\,d^4\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}+\frac {3\,g\,x^2\,\left (m-1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^2\,g^2\,m+a\,c\,d\,e\,f\,g\,m^2-4\,a\,c\,d\,e\,f\,g\,m+c^2\,d^2\,f^2\,m^2-7\,c^2\,d^2\,f^2\,m+12\,c^2\,d^2\,f^2\right )}{c^2\,d^2\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}+\frac {g^2\,x^3\,{\left (d+e\,x\right )}^m\,\left (a\,e\,g\,m-12\,c\,d\,f+3\,c\,d\,f\,m\right )\,\left (m^2-3\,m+2\right )}{c\,d\,\left (m^4-10\,m^3+35\,m^2-50\,m+24\right )}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((g^3*x^4*(d + e*x)^m*(11*m - 6*m^2 + m^3 - 6))/(35*m^2 - 50*m - 10*m^3 + m^4 + 24) + (x*(d + e*x)^m*(26*c^4*
d^4*f^3*m - 24*c^4*d^4*f^3 - 9*c^4*d^4*f^3*m^2 + c^4*d^4*f^3*m^3 + 6*a^3*c*d*e^3*g^3*m - 24*a^2*c^2*d^2*e^2*f*
g^2*m + 36*a*c^3*d^3*e*f^2*g*m + 6*a^2*c^2*d^2*e^2*f*g^2*m^2 - 21*a*c^3*d^3*e*f^2*g*m^2 + 3*a*c^3*d^3*e*f^2*g*
m^3))/(c^4*d^4*(35*m^2 - 50*m - 10*m^3 + m^4 + 24)) + (a*e*(d + e*x)^m*(6*a^3*e^3*g^3 - 24*c^3*d^3*f^3 + 26*c^
3*d^3*f^3*m - 9*c^3*d^3*f^3*m^2 + c^3*d^3*f^3*m^3 + 36*a*c^2*d^2*e*f^2*g - 24*a^2*c*d*e^2*f*g^2 - 21*a*c^2*d^2
*e*f^2*g*m + 6*a^2*c*d*e^2*f*g^2*m + 3*a*c^2*d^2*e*f^2*g*m^2))/(c^4*d^4*(35*m^2 - 50*m - 10*m^3 + m^4 + 24)) +
(3*g*x^2*(m - 1)*(d + e*x)^m*(12*c^2*d^2*f^2 + a^2*e^2*g^2*m - 7*c^2*d^2*f^2*m + c^2*d^2*f^2*m^2 - 4*a*c*d*e*
f*g*m + a*c*d*e*f*g*m^2))/(c^2*d^2*(35*m^2 - 50*m - 10*m^3 + m^4 + 24)) + (g^2*x^3*(d + e*x)^m*(a*e*g*m - 12*c
*d*f + 3*c*d*f*m)*(m^2 - 3*m + 2))/(c*d*(35*m^2 - 50*m - 10*m^3 + m^4 + 24)))/(x*(a*e^2 + c*d^2) + a*d*e + c*d
*e*x^2)^m

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