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

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]

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

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Rubi [A]  time = 0.23, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2059, 88} \[ \frac {(a+b x)^5 \left (6 a^2 d^2 f^2-6 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )}{5 b^5}+\frac {d f (a+b x)^6 (-2 a d f+b c f+b d e)}{3 b^5}+\frac {(a+b x)^4 (b c-a d) (b e-a f) (-2 a d f+b c f+b d e)}{2 b^5}+\frac {(a+b x)^3 (b c-a d)^2 (b e-a f)^2}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5} \]

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)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2059

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rubi steps

\begin {align*} \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 &=\int (a+b x)^2 (c+d x)^2 (e+f x)^2 \, dx\\ &=\int \left (\frac {(b c-a d)^2 (b e-a f)^2 (a+b x)^2}{b^4}+\frac {2 (b c-a d) (b e-a f) (b d e+b c f-2 a d f) (a+b x)^3}{b^4}+\frac {\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^4}{b^4}+\frac {2 d f (b d e+b c f-2 a d f) (a+b x)^5}{b^4}+\frac {d^2 f^2 (a+b x)^6}{b^4}\right ) \, dx\\ &=\frac {(b c-a d)^2 (b e-a f)^2 (a+b x)^3}{3 b^5}+\frac {(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}+\frac {\left (6 a^2 d^2 f^2-6 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) (a+b x)^5}{5 b^5}+\frac {d f (b d e+b c f-2 a d f) (a+b x)^6}{3 b^5}+\frac {d^2 f^2 (a+b x)^7}{7 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.09, 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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fricas [A]  time = 0.56, size = 346, normalized size = 1.79 \[ \frac {1}{7} x^{7} f^{2} d^{2} b^{2} + \frac {1}{3} x^{6} f e d^{2} b^{2} + \frac {1}{3} x^{6} f^{2} d c b^{2} + \frac {1}{3} x^{6} f^{2} d^{2} b a + \frac {1}{5} x^{5} e^{2} d^{2} b^{2} + \frac {4}{5} x^{5} f e d c b^{2} + \frac {1}{5} x^{5} f^{2} c^{2} b^{2} + \frac {4}{5} x^{5} f e d^{2} b a + \frac {4}{5} x^{5} f^{2} d c b a + \frac {1}{5} x^{5} f^{2} d^{2} a^{2} + \frac {1}{2} x^{4} e^{2} d c b^{2} + \frac {1}{2} x^{4} f e c^{2} b^{2} + \frac {1}{2} x^{4} e^{2} d^{2} b a + 2 x^{4} f e d c b a + \frac {1}{2} x^{4} f^{2} c^{2} b a + \frac {1}{2} x^{4} f e d^{2} a^{2} + \frac {1}{2} x^{4} f^{2} d c a^{2} + \frac {1}{3} x^{3} e^{2} c^{2} b^{2} + \frac {4}{3} x^{3} e^{2} d c b a + \frac {4}{3} x^{3} f e c^{2} b a + \frac {1}{3} x^{3} e^{2} d^{2} a^{2} + \frac {4}{3} x^{3} f e d c a^{2} + \frac {1}{3} x^{3} f^{2} c^{2} a^{2} + x^{2} e^{2} c^{2} b a + x^{2} e^{2} d c a^{2} + x^{2} f e c^{2} a^{2} + x e^{2} c^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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, 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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giac [A]  time = 0.42, size = 346, normalized size = 1.79 \[ \frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac {1}{3} \, b^{2} c d f^{2} x^{6} + \frac {1}{3} \, a b d^{2} f^{2} x^{6} + \frac {1}{3} \, b^{2} d^{2} f x^{6} e + \frac {1}{5} \, b^{2} c^{2} f^{2} x^{5} + \frac {4}{5} \, a b c d f^{2} x^{5} + \frac {1}{5} \, a^{2} d^{2} f^{2} x^{5} + \frac {4}{5} \, b^{2} c d f x^{5} e + \frac {4}{5} \, a b d^{2} f x^{5} e + \frac {1}{2} \, a b c^{2} f^{2} x^{4} + \frac {1}{2} \, a^{2} c d f^{2} x^{4} + \frac {1}{5} \, b^{2} d^{2} x^{5} e^{2} + \frac {1}{2} \, b^{2} c^{2} f x^{4} e + 2 \, a b c d f x^{4} e + \frac {1}{2} \, a^{2} d^{2} f x^{4} e + \frac {1}{3} \, a^{2} c^{2} f^{2} x^{3} + \frac {1}{2} \, b^{2} c d x^{4} e^{2} + \frac {1}{2} \, a b d^{2} x^{4} e^{2} + \frac {4}{3} \, a b c^{2} f x^{3} e + \frac {4}{3} \, a^{2} c d f x^{3} e + \frac {1}{3} \, b^{2} c^{2} x^{3} e^{2} + \frac {4}{3} \, a b c d x^{3} e^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} e^{2} + a^{2} c^{2} f x^{2} e + a b c^{2} x^{2} e^{2} + a^{2} c d x^{2} e^{2} + a^{2} c^{2} x e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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, 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

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maple [A]  time = 0.00, size = 188, normalized size = 0.97 \[ \frac {b^{2} d^{2} f^{2} x^{7}}{7}+\frac {\left (a d f +b c f +b d e \right ) b d f \,x^{6}}{3}+a^{2} c^{2} e^{2} x +\left (a c f +a d e +b c e \right ) a c e \,x^{2}+\frac {\left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right ) x^{5}}{5}+\frac {\left (2 a b c d e f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right ) x^{4}}{4}+\frac {\left (2 \left (a d f +b c f +b d e \right ) a c e +\left (a c f +a d e +b c e \right )^{2}\right ) x^{3}}{3} \]

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*b*c*d*e*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*d*f+b*c*f+b*d*e)*a*c*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 = 0.62, size = 180, normalized size = 0.93 \[ \frac {1}{7} \, b^{2} d^{2} f^{2} x^{7} + \frac {1}{3} \, {\left (b d e + b c f + a d f\right )} b d f x^{6} + a^{2} c^{2} e^{2} x + \frac {1}{5} \, {\left (b d e + b c f + a d f\right )}^{2} x^{5} + \frac {1}{3} \, {\left (b c e + a d e + a c f\right )}^{2} x^{3} + \frac {1}{6} \, {\left (3 \, b d f x^{4} + 4 \, {\left (b d e + b c f + a d f\right )} x^{3} + 6 \, {\left (b c e + a d e + a c f\right )} x^{2}\right )} a c e + \frac {1}{10} \, {\left (4 \, b d f x^{5} + 5 \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{4}\right )} {\left (b c e + a d e + a c f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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, 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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mupad [B]  time = 0.08, size = 270, normalized size = 1.40 \[ x^4\,\left (\frac {a^2\,c\,d\,f^2}{2}+\frac {a^2\,d^2\,e\,f}{2}+\frac {a\,b\,c^2\,f^2}{2}+2\,a\,b\,c\,d\,e\,f+\frac {a\,b\,d^2\,e^2}{2}+\frac {b^2\,c^2\,e\,f}{2}+\frac {b^2\,c\,d\,e^2}{2}\right )+x^3\,\left (\frac {a^2\,c^2\,f^2}{3}+\frac {4\,a^2\,c\,d\,e\,f}{3}+\frac {a^2\,d^2\,e^2}{3}+\frac {4\,a\,b\,c^2\,e\,f}{3}+\frac {4\,a\,b\,c\,d\,e^2}{3}+\frac {b^2\,c^2\,e^2}{3}\right )+x^5\,\left (\frac {a^2\,d^2\,f^2}{5}+\frac {4\,a\,b\,c\,d\,f^2}{5}+\frac {4\,a\,b\,d^2\,e\,f}{5}+\frac {b^2\,c^2\,f^2}{5}+\frac {4\,b^2\,c\,d\,e\,f}{5}+\frac {b^2\,d^2\,e^2}{5}\right )+a^2\,c^2\,e^2\,x+\frac {b^2\,d^2\,f^2\,x^7}{7}+a\,c\,e\,x^2\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )+\frac {b\,d\,f\,x^6\,\left (a\,d\,f+b\,c\,f+b\,d\,e\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, 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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