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

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 = 0.931671, 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 \[ -\frac{6 (d+e x)^{m-1} (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^4 d^4 e (1-m) (2-m) (3-m) (4-m)}+\frac{6 g (d+e x)^m (c d f-a e g)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^3 d^3 e (2-m) (3-m) (4-m)}+\frac{3 (f+g x)^2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 (3-m) (4-m)}+\frac{(f+g x)^3 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (4-m)} \]

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))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.3017, size = 233, normalized size = 0.68 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \left (6 a^3 e^3 g^3+6 a^2 c d e^2 g^2 (f (m-4)+g (m-1) x)+3 a c^2 d^2 e g \left (f^2 \left (m^2-7 m+12\right )+2 f g \left (m^2-5 m+4\right ) x+g^2 \left (m^2-3 m+2\right ) x^2\right )+c^3 d^3 \left (f^3 \left (m^3-9 m^2+26 m-24\right )+3 f^2 g \left (m^3-8 m^2+19 m-12\right ) x+3 f g^2 \left (m^3-7 m^2+14 m-8\right ) x^2+g^3 \left (m^3-6 m^2+11 m-6\right ) x^3\right )\right )}{c^4 d^4 (m-4) (m-3) (m-2) (m-1)} \]

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)*(6*a^3*e^3*g^3 + 6*a^2*c
*d*e^2*g^2*(f*(-4 + m) + g*(-1 + m)*x) + 3*a*c^2*d^2*e*g*(f^2*(12 - 7*m + m^2) +
2*f*g*(4 - 5*m + m^2)*x + g^2*(2 - 3*m + m^2)*x^2) + c^3*d^3*(f^3*(-24 + 26*m -
9*m^2 + m^3) + 3*f^2*g*(-12 + 19*m - 8*m^2 + m^3)*x + 3*f*g^2*(-8 + 14*m - 7*m^
2 + m^3)*x^2 + g^3*(-6 + 11*m - 6*m^2 + m^3)*x^3)))/(c^4*d^4*(-4 + m)*(-3 + m)*(
-2 + m)*(-1 + m)))

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Maple [A]  time = 0.015, size = 527, normalized size = 1.5 \[ -{\frac{ \left ( ex+d \right ) ^{m} \left ({c}^{3}{d}^{3}{g}^{3}{m}^{3}{x}^{3}+3\,{c}^{3}{d}^{3}f{g}^{2}{m}^{3}{x}^{2}-6\,{c}^{3}{d}^{3}{g}^{3}{m}^{2}{x}^{3}+3\,a{c}^{2}{d}^{2}e{g}^{3}{m}^{2}{x}^{2}+3\,{c}^{3}{d}^{3}{f}^{2}g{m}^{3}x-21\,{c}^{3}{d}^{3}f{g}^{2}{m}^{2}{x}^{2}+11\,{c}^{3}{d}^{3}{g}^{3}m{x}^{3}+6\,a{c}^{2}{d}^{2}ef{g}^{2}{m}^{2}x-9\,a{c}^{2}{d}^{2}e{g}^{3}m{x}^{2}+{c}^{3}{d}^{3}{f}^{3}{m}^{3}-24\,{c}^{3}{d}^{3}{f}^{2}g{m}^{2}x+42\,{c}^{3}{d}^{3}f{g}^{2}m{x}^{2}-6\,{g}^{3}{x}^{3}{c}^{3}{d}^{3}+6\,{a}^{2}cd{e}^{2}{g}^{3}mx+3\,a{c}^{2}{d}^{2}e{f}^{2}g{m}^{2}-30\,a{c}^{2}{d}^{2}ef{g}^{2}mx+6\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-9\,{c}^{3}{d}^{3}{f}^{3}{m}^{2}+57\,{c}^{3}{d}^{3}{f}^{2}gmx-24\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}+6\,{a}^{2}cd{e}^{2}f{g}^{2}m-6\,{a}^{2}cd{e}^{2}{g}^{3}x-21\,a{c}^{2}{d}^{2}e{f}^{2}gm+24\,a{c}^{2}{d}^{2}ef{g}^{2}x+26\,{c}^{3}{d}^{3}{f}^{3}m-36\,{c}^{3}{d}^{3}{f}^{2}gx+6\,{a}^{3}{e}^{3}{g}^{3}-24\,{a}^{2}cd{e}^{2}f{g}^{2}+36\,a{c}^{2}{d}^{2}e{f}^{2}g-24\,{c}^{3}{d}^{3}{f}^{3} \right ) \left ( cdx+ae \right ) }{ \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}{c}^{4}{d}^{4} \left ({m}^{4}-10\,{m}^{3}+35\,{m}^{2}-50\,m+24 \right ) }} \]

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 = 0.7568, size = 447, normalized size = 1.3 \[ -\frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{-m} f^{3}}{c d{\left (m - 1\right )}} - \frac{3 \,{\left (c^{2} d^{2}{\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )}{\left (c d x + a e\right )}^{-m} f^{2} g}{{\left (m^{2} - 3 \, m + 2\right )} c^{2} d^{2}} - \frac{3 \,{\left ({\left (m^{2} - 3 \, m + 2\right )} c^{3} d^{3} x^{3} +{\left (m^{2} - m\right )} a c^{2} d^{2} e x^{2} + 2 \, a^{2} c d e^{2} m x + 2 \, a^{3} e^{3}\right )}{\left (c d x + a e\right )}^{-m} f g^{2}}{{\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )} c^{3} d^{3}} - \frac{{\left ({\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )} c^{4} d^{4} x^{4} +{\left (m^{3} - 3 \, m^{2} + 2 \, m\right )} a c^{3} d^{3} e x^{3} + 3 \,{\left (m^{2} - m\right )} a^{2} c^{2} d^{2} e^{2} x^{2} + 6 \, a^{3} c d e^{3} m x + 6 \, a^{4} e^{4}\right )}{\left (c d x + a e\right )}^{-m} g^{3}}{{\left (m^{4} - 10 \, m^{3} + 35 \, m^{2} - 50 \, m + 24\right )} c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.287794, size = 952, normalized size = 2.78 \[ -\frac{{\left (a c^{3} d^{3} e f^{3} m^{3} - 24 \, a c^{3} d^{3} e f^{3} + 36 \, a^{2} c^{2} d^{2} e^{2} f^{2} g - 24 \, a^{3} c d e^{3} f g^{2} + 6 \, a^{4} e^{4} g^{3} +{\left (c^{4} d^{4} g^{3} m^{3} - 6 \, c^{4} d^{4} g^{3} m^{2} + 11 \, c^{4} d^{4} g^{3} m - 6 \, c^{4} d^{4} g^{3}\right )} x^{4} -{\left (24 \, c^{4} d^{4} f g^{2} -{\left (3 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m^{3} + 3 \,{\left (7 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m^{2} - 2 \,{\left (21 \, c^{4} d^{4} f g^{2} + a c^{3} d^{3} e g^{3}\right )} m\right )} x^{3} - 3 \,{\left (3 \, a c^{3} d^{3} e f^{3} - a^{2} c^{2} d^{2} e^{2} f^{2} g\right )} m^{2} - 3 \,{\left (12 \, c^{4} d^{4} f^{2} g -{\left (c^{4} d^{4} f^{2} g + a c^{3} d^{3} e f g^{2}\right )} m^{3} +{\left (8 \, c^{4} d^{4} f^{2} g + 5 \, a c^{3} d^{3} e f g^{2} - a^{2} c^{2} d^{2} e^{2} g^{3}\right )} m^{2} -{\left (19 \, c^{4} d^{4} f^{2} g + 4 \, a c^{3} d^{3} e f g^{2} - a^{2} c^{2} d^{2} e^{2} g^{3}\right )} m\right )} x^{2} +{\left (26 \, a c^{3} d^{3} e f^{3} - 21 \, a^{2} c^{2} d^{2} e^{2} f^{2} g + 6 \, a^{3} c d e^{3} f g^{2}\right )} m -{\left (24 \, c^{4} d^{4} f^{3} -{\left (c^{4} d^{4} f^{3} + 3 \, a c^{3} d^{3} e f^{2} g\right )} m^{3} + 3 \,{\left (3 \, c^{4} d^{4} f^{3} + 7 \, a c^{3} d^{3} e f^{2} g - 2 \, a^{2} c^{2} d^{2} e^{2} f g^{2}\right )} m^{2} - 2 \,{\left (13 \, c^{4} d^{4} f^{3} + 18 \, a c^{3} d^{3} e f^{2} g - 12 \, a^{2} c^{2} d^{2} e^{2} f g^{2} + 3 \, a^{3} c d e^{3} g^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (c^{4} d^{4} m^{4} - 10 \, c^{4} d^{4} m^{3} + 35 \, c^{4} d^{4} m^{2} - 50 \, c^{4} d^{4} m + 24 \, c^{4} d^{4}\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^3*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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*(1
2*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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.276912, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done