3.16 \(\int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^2 \, dx\)

Optimal. Leaf size=193 \[ \frac{(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac{d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac{(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac{(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac{d^2 f^2 (a+b x)^7}{7 b^5} \]

[Out]

((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^3)/(3*b^5) + ((b*c - a*d)*(b*e - a*f)*(b*
d*e + b*c*f - 2*a*d*f)*(a + b*x)^4)/(2*b^5) + ((6*a^2*d^2*f^2 - 6*a*b*d*f*(d*e +
c*f) + b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*(a + b*x)^5)/(5*b^5) + (d*f*(b*d*e
+ b*c*f - 2*a*d*f)*(a + b*x)^6)/(3*b^5) + (d^2*f^2*(a + b*x)^7)/(7*b^5)

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Rubi [A]  time = 0.605631, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac{d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac{(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac{(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac{d^2 f^2 (a+b x)^7}{7 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3)^2,x]

[Out]

((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^3)/(3*b^5) + ((b*c - a*d)*(b*e - a*f)*(b*
d*e + b*c*f - 2*a*d*f)*(a + b*x)^4)/(2*b^5) + ((6*a^2*d^2*f^2 - 6*a*b*d*f*(d*e +
c*f) + b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*(a + b*x)^5)/(5*b^5) + (d*f*(b*d*e
+ b*c*f - 2*a*d*f)*(a + b*x)^6)/(3*b^5) + (d^2*f^2*(a + b*x)^7)/(7*b^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a c e \left (a c f + a d e + b c e\right ) \int x\, dx + \frac{b^{2} d^{2} f^{2} x^{7}}{7} + \frac{b d f x^{6} \left (a d f + b c f + b d e\right )}{3} + c^{2} e^{2} \int a^{2}\, dx + x^{5} \left (\frac{a^{2} d^{2} f^{2}}{5} + \frac{4 a b c d f^{2}}{5} + \frac{4 a b d^{2} e f}{5} + \frac{b^{2} c^{2} f^{2}}{5} + \frac{4 b^{2} c d e f}{5} + \frac{b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} c d f^{2}}{2} + \frac{a^{2} d^{2} e f}{2} + \frac{a b c^{2} f^{2}}{2} + 2 a b c d e f + \frac{a b d^{2} e^{2}}{2} + \frac{b^{2} c^{2} e f}{2} + \frac{b^{2} c d e^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} c^{2} f^{2}}{3} + \frac{4 a^{2} c d e f}{3} + \frac{a^{2} d^{2} e^{2}}{3} + \frac{4 a b c^{2} e f}{3} + \frac{4 a b c d e^{2}}{3} + \frac{b^{2} c^{2} e^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x**3)**2,x)

[Out]

2*a*c*e*(a*c*f + a*d*e + b*c*e)*Integral(x, x) + b**2*d**2*f**2*x**7/7 + b*d*f*x
**6*(a*d*f + b*c*f + b*d*e)/3 + c**2*e**2*Integral(a**2, x) + x**5*(a**2*d**2*f*
*2/5 + 4*a*b*c*d*f**2/5 + 4*a*b*d**2*e*f/5 + b**2*c**2*f**2/5 + 4*b**2*c*d*e*f/5
+ b**2*d**2*e**2/5) + x**4*(a**2*c*d*f**2/2 + a**2*d**2*e*f/2 + a*b*c**2*f**2/2
+ 2*a*b*c*d*e*f + a*b*d**2*e**2/2 + b**2*c**2*e*f/2 + b**2*c*d*e**2/2) + x**3*(
a**2*c**2*f**2/3 + 4*a**2*c*d*e*f/3 + a**2*d**2*e**2/3 + 4*a*b*c**2*e*f/3 + 4*a*
b*c*d*e**2/3 + b**2*c**2*e**2/3)

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Mathematica [A]  time = 0.146409, size = 241, normalized size = 1.25 \[ \frac{1}{5} x^5 \left (a^2 d^2 f^2+4 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac{1}{2} x^4 \left (a^2 d f (c f+d e)+a b \left (c^2 f^2+4 c d e f+d^2 e^2\right )+b^2 c e (c f+d e)\right )+\frac{1}{3} x^3 \left (a^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )+4 a b c e (c f+d e)+b^2 c^2 e^2\right )+a^2 c^2 e^2 x+\frac{1}{3} b d f x^6 (a d f+b c f+b d e)+a c e x^2 (a c f+a d e+b c e)+\frac{1}{7} b^2 d^2 f^2 x^7 \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d*f*x^3)^2,x]

[Out]

a^2*c^2*e^2*x + a*c*e*(b*c*e + a*d*e + a*c*f)*x^2 + ((b^2*c^2*e^2 + 4*a*b*c*e*(d
*e + c*f) + a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^3)/3 + ((b^2*c*e*(d*e + c*f)
+ a^2*d*f*(d*e + c*f) + a*b*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^4)/2 + ((a^2*d^2*
f^2 + 4*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + (b*d
*f*(b*d*e + b*c*f + a*d*f)*x^6)/3 + (b^2*d^2*f^2*x^7)/7

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Maple [A]  time = 0.001, size = 188, normalized size = 1. \[{\frac{{b}^{2}{d}^{2}{f}^{2}{x}^{7}}{7}}+{\frac{ \left ( adf+bcf+bde \right ) bdf{x}^{6}}{3}}+{\frac{ \left ( 2\, \left ( acf+ade+bce \right ) bdf+ \left ( adf+bcf+bde \right ) ^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,acebdf+2\, \left ( acf+ade+bce \right ) \left ( adf+bcf+bde \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,ace \left ( adf+bcf+bde \right ) + \left ( acf+ade+bce \right ) ^{2} \right ){x}^{3}}{3}}+ace \left ( acf+ade+bce \right ){x}^{2}+{a}^{2}{c}^{2}{e}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^2,x)

[Out]

1/7*b^2*d^2*f^2*x^7+1/3*(a*d*f+b*c*f+b*d*e)*b*d*f*x^6+1/5*(2*(a*c*f+a*d*e+b*c*e)
*b*d*f+(a*d*f+b*c*f+b*d*e)^2)*x^5+1/4*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+b*c*e)*(a*d*
f+b*c*f+b*d*e))*x^4+1/3*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b*c*e)^2)*x^3+
a*c*e*(a*c*f+a*d*e+b*c*e)*x^2+a^2*c^2*e^2*x

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Maxima [A]  time = 0.776301, size = 243, normalized size = 1.26 \[ \frac{1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac{1}{3} \,{\left (b d e + b c f + a d f\right )} b d f x^{6} + a^{2} c^{2} e^{2} x + \frac{1}{5} \,{\left (b d e + b c f + a d f\right )}^{2} x^{5} + \frac{1}{3} \,{\left (b c e + a d e + a c f\right )}^{2} x^{3} + \frac{1}{6} \,{\left (3 \, b d f x^{4} + 4 \,{\left (b d e + b c f + a d f\right )} x^{3} + 6 \,{\left (b c e + a d e + a c f\right )} x^{2}\right )} a c e + \frac{1}{10} \,{\left (4 \, b d f x^{5} + 5 \,{\left (b d e +{\left (b c + a d\right )} f\right )} x^{4}\right )}{\left (b c e + a d e + a c f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^2*d^2*f^2*x^7 + 1/3*(b*d*e + b*c*f + a*d*f)*b*d*f*x^6 + a^2*c^2*e^2*x + 1/
5*(b*d*e + b*c*f + a*d*f)^2*x^5 + 1/3*(b*c*e + a*d*e + a*c*f)^2*x^3 + 1/6*(3*b*d
*f*x^4 + 4*(b*d*e + b*c*f + a*d*f)*x^3 + 6*(b*c*e + a*d*e + a*c*f)*x^2)*a*c*e +
1/10*(4*b*d*f*x^5 + 5*(b*d*e + (b*c + a*d)*f)*x^4)*(b*c*e + a*d*e + a*c*f)

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Fricas [A]  time = 0.267591, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} f^{2} d^{2} b^{2} + \frac{1}{3} x^{6} f e d^{2} b^{2} + \frac{1}{3} x^{6} f^{2} d c b^{2} + \frac{1}{3} x^{6} f^{2} d^{2} b a + \frac{1}{5} x^{5} e^{2} d^{2} b^{2} + \frac{4}{5} x^{5} f e d c b^{2} + \frac{1}{5} x^{5} f^{2} c^{2} b^{2} + \frac{4}{5} x^{5} f e d^{2} b a + \frac{4}{5} x^{5} f^{2} d c b a + \frac{1}{5} x^{5} f^{2} d^{2} a^{2} + \frac{1}{2} x^{4} e^{2} d c b^{2} + \frac{1}{2} x^{4} f e c^{2} b^{2} + \frac{1}{2} x^{4} e^{2} d^{2} b a + 2 x^{4} f e d c b a + \frac{1}{2} x^{4} f^{2} c^{2} b a + \frac{1}{2} x^{4} f e d^{2} a^{2} + \frac{1}{2} x^{4} f^{2} d c a^{2} + \frac{1}{3} x^{3} e^{2} c^{2} b^{2} + \frac{4}{3} x^{3} e^{2} d c b a + \frac{4}{3} x^{3} f e c^{2} b a + \frac{1}{3} x^{3} e^{2} d^{2} a^{2} + \frac{4}{3} x^{3} f e d c a^{2} + \frac{1}{3} x^{3} f^{2} c^{2} a^{2} + x^{2} e^{2} c^{2} b a + x^{2} e^{2} d c a^{2} + x^{2} f e c^{2} a^{2} + x e^{2} c^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e + a*d*e + a*c*f)*x)^2,x, algorithm="fricas")

[Out]

1/7*x^7*f^2*d^2*b^2 + 1/3*x^6*f*e*d^2*b^2 + 1/3*x^6*f^2*d*c*b^2 + 1/3*x^6*f^2*d^
2*b*a + 1/5*x^5*e^2*d^2*b^2 + 4/5*x^5*f*e*d*c*b^2 + 1/5*x^5*f^2*c^2*b^2 + 4/5*x^
5*f*e*d^2*b*a + 4/5*x^5*f^2*d*c*b*a + 1/5*x^5*f^2*d^2*a^2 + 1/2*x^4*e^2*d*c*b^2
+ 1/2*x^4*f*e*c^2*b^2 + 1/2*x^4*e^2*d^2*b*a + 2*x^4*f*e*d*c*b*a + 1/2*x^4*f^2*c^
2*b*a + 1/2*x^4*f*e*d^2*a^2 + 1/2*x^4*f^2*d*c*a^2 + 1/3*x^3*e^2*c^2*b^2 + 4/3*x^
3*e^2*d*c*b*a + 4/3*x^3*f*e*c^2*b*a + 1/3*x^3*e^2*d^2*a^2 + 4/3*x^3*f*e*d*c*a^2
+ 1/3*x^3*f^2*c^2*a^2 + x^2*e^2*c^2*b*a + x^2*e^2*d*c*a^2 + x^2*f*e*c^2*a^2 + x*
e^2*c^2*a^2

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Sympy [A]  time = 0.309959, size = 345, normalized size = 1.79 \[ a^{2} c^{2} e^{2} x + \frac{b^{2} d^{2} f^{2} x^{7}}{7} + x^{6} \left (\frac{a b d^{2} f^{2}}{3} + \frac{b^{2} c d f^{2}}{3} + \frac{b^{2} d^{2} e f}{3}\right ) + x^{5} \left (\frac{a^{2} d^{2} f^{2}}{5} + \frac{4 a b c d f^{2}}{5} + \frac{4 a b d^{2} e f}{5} + \frac{b^{2} c^{2} f^{2}}{5} + \frac{4 b^{2} c d e f}{5} + \frac{b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} c d f^{2}}{2} + \frac{a^{2} d^{2} e f}{2} + \frac{a b c^{2} f^{2}}{2} + 2 a b c d e f + \frac{a b d^{2} e^{2}}{2} + \frac{b^{2} c^{2} e f}{2} + \frac{b^{2} c d e^{2}}{2}\right ) + x^{3} \left (\frac{a^{2} c^{2} f^{2}}{3} + \frac{4 a^{2} c d e f}{3} + \frac{a^{2} d^{2} e^{2}}{3} + \frac{4 a b c^{2} e f}{3} + \frac{4 a b c d e^{2}}{3} + \frac{b^{2} c^{2} e^{2}}{3}\right ) + x^{2} \left (a^{2} c^{2} e f + a^{2} c d e^{2} + a b c^{2} e^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x**3)**2,x)

[Out]

a**2*c**2*e**2*x + b**2*d**2*f**2*x**7/7 + x**6*(a*b*d**2*f**2/3 + b**2*c*d*f**2
/3 + b**2*d**2*e*f/3) + x**5*(a**2*d**2*f**2/5 + 4*a*b*c*d*f**2/5 + 4*a*b*d**2*e
*f/5 + b**2*c**2*f**2/5 + 4*b**2*c*d*e*f/5 + b**2*d**2*e**2/5) + x**4*(a**2*c*d*
f**2/2 + a**2*d**2*e*f/2 + a*b*c**2*f**2/2 + 2*a*b*c*d*e*f + a*b*d**2*e**2/2 + b
**2*c**2*e*f/2 + b**2*c*d*e**2/2) + x**3*(a**2*c**2*f**2/3 + 4*a**2*c*d*e*f/3 +
a**2*d**2*e**2/3 + 4*a*b*c**2*e*f/3 + 4*a*b*c*d*e**2/3 + b**2*c**2*e**2/3) + x**
2*(a**2*c**2*e*f + a**2*c*d*e**2 + a*b*c**2*e**2)

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GIAC/XCAS [A]  time = 0.26279, size = 467, normalized size = 2.42 \[ \frac{1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac{1}{3} \, b^{2} c d f^{2} x^{6} + \frac{1}{3} \, a b d^{2} f^{2} x^{6} + \frac{1}{3} \, b^{2} d^{2} f x^{6} e + \frac{1}{5} \, b^{2} c^{2} f^{2} x^{5} + \frac{4}{5} \, a b c d f^{2} x^{5} + \frac{1}{5} \, a^{2} d^{2} f^{2} x^{5} + \frac{4}{5} \, b^{2} c d f x^{5} e + \frac{4}{5} \, a b d^{2} f x^{5} e + \frac{1}{2} \, a b c^{2} f^{2} x^{4} + \frac{1}{2} \, a^{2} c d f^{2} x^{4} + \frac{1}{5} \, b^{2} d^{2} x^{5} e^{2} + \frac{1}{2} \, b^{2} c^{2} f x^{4} e + 2 \, a b c d f x^{4} e + \frac{1}{2} \, a^{2} d^{2} f x^{4} e + \frac{1}{3} \, a^{2} c^{2} f^{2} x^{3} + \frac{1}{2} \, b^{2} c d x^{4} e^{2} + \frac{1}{2} \, a b d^{2} x^{4} e^{2} + \frac{4}{3} \, a b c^{2} f x^{3} e + \frac{4}{3} \, a^{2} c d f x^{3} e + \frac{1}{3} \, b^{2} c^{2} x^{3} e^{2} + \frac{4}{3} \, a b c d x^{3} e^{2} + \frac{1}{3} \, a^{2} d^{2} x^{3} e^{2} + a^{2} c^{2} f x^{2} e + a b c^{2} x^{2} e^{2} + a^{2} c d x^{2} e^{2} + a^{2} c^{2} x e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^2*d^2*f^2*x^7 + 1/3*b^2*c*d*f^2*x^6 + 1/3*a*b*d^2*f^2*x^6 + 1/3*b^2*d^2*f*
x^6*e + 1/5*b^2*c^2*f^2*x^5 + 4/5*a*b*c*d*f^2*x^5 + 1/5*a^2*d^2*f^2*x^5 + 4/5*b^
2*c*d*f*x^5*e + 4/5*a*b*d^2*f*x^5*e + 1/2*a*b*c^2*f^2*x^4 + 1/2*a^2*c*d*f^2*x^4
+ 1/5*b^2*d^2*x^5*e^2 + 1/2*b^2*c^2*f*x^4*e + 2*a*b*c*d*f*x^4*e + 1/2*a^2*d^2*f*
x^4*e + 1/3*a^2*c^2*f^2*x^3 + 1/2*b^2*c*d*x^4*e^2 + 1/2*a*b*d^2*x^4*e^2 + 4/3*a*
b*c^2*f*x^3*e + 4/3*a^2*c*d*f*x^3*e + 1/3*b^2*c^2*x^3*e^2 + 4/3*a*b*c*d*x^3*e^2
+ 1/3*a^2*d^2*x^3*e^2 + a^2*c^2*f*x^2*e + a*b*c^2*x^2*e^2 + a^2*c*d*x^2*e^2 + a^
2*c^2*x*e^2