3.611 \(\int (d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4)^2 \, dx\)

Optimal. Leaf size=104 \[ \frac{a^2 f^3 (d+e x)^4}{4 e}+\frac{f^3 \left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac{a b f^3 (d+e x)^6}{3 e}+\frac{b c f^3 (d+e x)^{10}}{5 e}+\frac{c^2 f^3 (d+e x)^{12}}{12 e} \]

[Out]

(a^2*f^3*(d + e*x)^4)/(4*e) + (a*b*f^3*(d + e*x)^6)/(3*e) + ((b^2 + 2*a*c)*f^3*(d + e*x)^8)/(8*e) + (b*c*f^3*(
d + e*x)^10)/(5*e) + (c^2*f^3*(d + e*x)^12)/(12*e)

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Rubi [A]  time = 0.164176, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1142, 1114, 631} \[ \frac{a^2 f^3 (d+e x)^4}{4 e}+\frac{f^3 \left (2 a c+b^2\right ) (d+e x)^8}{8 e}+\frac{a b f^3 (d+e x)^6}{3 e}+\frac{b c f^3 (d+e x)^{10}}{5 e}+\frac{c^2 f^3 (d+e x)^{12}}{12 e} \]

Antiderivative was successfully verified.

[In]

Int[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]

[Out]

(a^2*f^3*(d + e*x)^4)/(4*e) + (a*b*f^3*(d + e*x)^6)/(3*e) + ((b^2 + 2*a*c)*f^3*(d + e*x)^8)/(8*e) + (b*c*f^3*(
d + e*x)^10)/(5*e) + (c^2*f^3*(d + e*x)^12)/(12*e)

Rule 1142

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 631

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

Rubi steps

\begin{align*} \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )^2 \, dx &=\frac{f^3 \operatorname{Subst}\left (\int x^3 \left (a+b x^2+c x^4\right )^2 \, dx,x,d+e x\right )}{e}\\ &=\frac{f^3 \operatorname{Subst}\left (\int x \left (a+b x+c x^2\right )^2 \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac{f^3 \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+\left (b^2+2 a c\right ) x^3+2 b c x^4+c^2 x^5\right ) \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac{a^2 f^3 (d+e x)^4}{4 e}+\frac{a b f^3 (d+e x)^6}{3 e}+\frac{\left (b^2+2 a c\right ) f^3 (d+e x)^8}{8 e}+\frac{b c f^3 (d+e x)^{10}}{5 e}+\frac{c^2 f^3 (d+e x)^{12}}{12 e}\\ \end{align*}

Mathematica [B]  time = 0.0758172, size = 405, normalized size = 3.89 \[ f^3 \left (\frac{1}{4} e^3 x^4 \left (a^2+20 a b d^2+70 a c d^4+35 b^2 d^4+168 b c d^6+165 c^2 d^8\right )+\frac{1}{3} d e^2 x^3 \left (3 a^2+20 a b d^2+42 a c d^4+21 b^2 d^4+72 b c d^6+55 c^2 d^8\right )+\frac{1}{2} d^2 e x^2 \left (3 a^2+10 a b d^2+14 a c d^4+7 b^2 d^4+18 b c d^6+11 c^2 d^8\right )+\frac{1}{8} e^7 x^8 \left (2 a c+b^2+72 b c d^2+330 c^2 d^4\right )+d e^6 x^7 \left (2 a c+b^2+24 b c d^2+66 c^2 d^4\right )+\frac{1}{6} e^5 x^6 \left (2 a b+42 a c d^2+21 b^2 d^2+252 b c d^4+462 c^2 d^6\right )+\frac{1}{5} d e^4 x^5 \left (10 a b+70 a c d^2+35 b^2 d^2+252 b c d^4+330 c^2 d^6\right )+d^3 x \left (a+b d^2+c d^4\right )^2+\frac{1}{10} c e^9 x^{10} \left (2 b+55 c d^2\right )+\frac{1}{3} c d e^8 x^9 \left (6 b+55 c d^2\right )+c^2 d e^{10} x^{11}+\frac{1}{12} c^2 e^{11} x^{12}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*f + e*f*x)^3*(a + b*(d + e*x)^2 + c*(d + e*x)^4)^2,x]

[Out]

f^3*(d^3*(a + b*d^2 + c*d^4)^2*x + (d^2*(3*a^2 + 10*a*b*d^2 + 7*b^2*d^4 + 14*a*c*d^4 + 18*b*c*d^6 + 11*c^2*d^8
)*e*x^2)/2 + (d*(3*a^2 + 20*a*b*d^2 + 21*b^2*d^4 + 42*a*c*d^4 + 72*b*c*d^6 + 55*c^2*d^8)*e^2*x^3)/3 + ((a^2 +
20*a*b*d^2 + 35*b^2*d^4 + 70*a*c*d^4 + 168*b*c*d^6 + 165*c^2*d^8)*e^3*x^4)/4 + (d*(10*a*b + 35*b^2*d^2 + 70*a*
c*d^2 + 252*b*c*d^4 + 330*c^2*d^6)*e^4*x^5)/5 + ((2*a*b + 21*b^2*d^2 + 42*a*c*d^2 + 252*b*c*d^4 + 462*c^2*d^6)
*e^5*x^6)/6 + d*(b^2 + 2*a*c + 24*b*c*d^2 + 66*c^2*d^4)*e^6*x^7 + ((b^2 + 2*a*c + 72*b*c*d^2 + 330*c^2*d^4)*e^
7*x^8)/8 + (c*d*(6*b + 55*c*d^2)*e^8*x^9)/3 + (c*(2*b + 55*c*d^2)*e^9*x^10)/10 + c^2*d*e^10*x^11 + (c^2*e^11*x
^12)/12)

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Maple [B]  time = 0.001, size = 1413, normalized size = 13.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x)

[Out]

1/12*e^11*f^3*c^2*x^12+d*f^3*e^10*c^2*x^11+1/10*(27*d^2*f^3*e^9*c^2+e^3*f^3*(2*(6*c*d^2*e^2+b*e^2)*c*e^4+16*c^
2*d^2*e^6))*x^10+1/9*(25*d^3*f^3*c^2*e^8+3*d*f^3*e^2*(2*(6*c*d^2*e^2+b*e^2)*c*e^4+16*c^2*d^2*e^6)+e^3*f^3*(2*(
4*c*d^3*e+2*b*d*e)*c*e^4+8*(6*c*d^2*e^2+b*e^2)*c*d*e^3))*x^9+1/8*(8*d^4*f^3*c^2*e^7+3*d^2*f^3*e*(2*(6*c*d^2*e^
2+b*e^2)*c*e^4+16*c^2*d^2*e^6)+3*d*f^3*e^2*(2*(4*c*d^3*e+2*b*d*e)*c*e^4+8*(6*c*d^2*e^2+b*e^2)*c*d*e^3)+e^3*f^3
*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*e+2*b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2))*x^8+1/7*(d^3*f^3*(2*(6*c*d^2*e
^2+b*e^2)*c*e^4+16*c^2*d^2*e^6)+3*d^2*f^3*e*(2*(4*c*d^3*e+2*b*d*e)*c*e^4+8*(6*c*d^2*e^2+b*e^2)*c*d*e^3)+3*d*f^
3*e^2*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*e+2*b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2)+e^3*f^3*(8*(c*d^4+b*d^2+a)
*c*d*e^3+2*(4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2)))*x^7+1/6*(d^3*f^3*(2*(4*c*d^3*e+2*b*d*e)*c*e^4+8*(6*c*d^2*
e^2+b*e^2)*c*d*e^3)+3*d^2*f^3*e*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*e+2*b*d*e)*c*d*e^3+(6*c*d^2*e^2+b*e^2)^2)+
3*d*f^3*e^2*(8*(c*d^4+b*d^2+a)*c*d*e^3+2*(4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2))+e^3*f^3*(2*(c*d^4+b*d^2+a)*(
6*c*d^2*e^2+b*e^2)+(4*c*d^3*e+2*b*d*e)^2))*x^6+1/5*(d^3*f^3*(2*(c*d^4+b*d^2+a)*c*e^4+8*(4*c*d^3*e+2*b*d*e)*c*d
*e^3+(6*c*d^2*e^2+b*e^2)^2)+3*d^2*f^3*e*(8*(c*d^4+b*d^2+a)*c*d*e^3+2*(4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2))+
3*d*f^3*e^2*(2*(c*d^4+b*d^2+a)*(6*c*d^2*e^2+b*e^2)+(4*c*d^3*e+2*b*d*e)^2)+2*e^3*f^3*(c*d^4+b*d^2+a)*(4*c*d^3*e
+2*b*d*e))*x^5+1/4*(d^3*f^3*(8*(c*d^4+b*d^2+a)*c*d*e^3+2*(4*c*d^3*e+2*b*d*e)*(6*c*d^2*e^2+b*e^2))+3*d^2*f^3*e*
(2*(c*d^4+b*d^2+a)*(6*c*d^2*e^2+b*e^2)+(4*c*d^3*e+2*b*d*e)^2)+6*d*f^3*e^2*(c*d^4+b*d^2+a)*(4*c*d^3*e+2*b*d*e)+
e^3*f^3*(c*d^4+b*d^2+a)^2)*x^4+1/3*(d^3*f^3*(2*(c*d^4+b*d^2+a)*(6*c*d^2*e^2+b*e^2)+(4*c*d^3*e+2*b*d*e)^2)+6*d^
2*f^3*e*(c*d^4+b*d^2+a)*(4*c*d^3*e+2*b*d*e)+3*d*f^3*e^2*(c*d^4+b*d^2+a)^2)*x^3+1/2*(2*d^3*f^3*(c*d^4+b*d^2+a)*
(4*c*d^3*e+2*b*d*e)+3*d^2*f^3*e*(c*d^4+b*d^2+a)^2)*x^2+d^3*f^3*(c*d^4+b*d^2+a)^2*x

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Maxima [B]  time = 0.982613, size = 593, normalized size = 5.7 \begin{align*} \frac{1}{12} \, c^{2} e^{11} f^{3} x^{12} + c^{2} d e^{10} f^{3} x^{11} + \frac{1}{10} \,{\left (55 \, c^{2} d^{2} + 2 \, b c\right )} e^{9} f^{3} x^{10} + \frac{1}{3} \,{\left (55 \, c^{2} d^{3} + 6 \, b c d\right )} e^{8} f^{3} x^{9} + \frac{1}{8} \,{\left (330 \, c^{2} d^{4} + 72 \, b c d^{2} + b^{2} + 2 \, a c\right )} e^{7} f^{3} x^{8} +{\left (66 \, c^{2} d^{5} + 24 \, b c d^{3} +{\left (b^{2} + 2 \, a c\right )} d\right )} e^{6} f^{3} x^{7} + \frac{1}{6} \,{\left (462 \, c^{2} d^{6} + 252 \, b c d^{4} + 21 \,{\left (b^{2} + 2 \, a c\right )} d^{2} + 2 \, a b\right )} e^{5} f^{3} x^{6} + \frac{1}{5} \,{\left (330 \, c^{2} d^{7} + 252 \, b c d^{5} + 35 \,{\left (b^{2} + 2 \, a c\right )} d^{3} + 10 \, a b d\right )} e^{4} f^{3} x^{5} + \frac{1}{4} \,{\left (165 \, c^{2} d^{8} + 168 \, b c d^{6} + 35 \,{\left (b^{2} + 2 \, a c\right )} d^{4} + 20 \, a b d^{2} + a^{2}\right )} e^{3} f^{3} x^{4} + \frac{1}{3} \,{\left (55 \, c^{2} d^{9} + 72 \, b c d^{7} + 21 \,{\left (b^{2} + 2 \, a c\right )} d^{5} + 20 \, a b d^{3} + 3 \, a^{2} d\right )} e^{2} f^{3} x^{3} + \frac{1}{2} \,{\left (11 \, c^{2} d^{10} + 18 \, b c d^{8} + 7 \,{\left (b^{2} + 2 \, a c\right )} d^{6} + 10 \, a b d^{4} + 3 \, a^{2} d^{2}\right )} e f^{3} x^{2} +{\left (c^{2} d^{11} + 2 \, b c d^{9} +{\left (b^{2} + 2 \, a c\right )} d^{7} + 2 \, a b d^{5} + a^{2} d^{3}\right )} f^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="maxima")

[Out]

1/12*c^2*e^11*f^3*x^12 + c^2*d*e^10*f^3*x^11 + 1/10*(55*c^2*d^2 + 2*b*c)*e^9*f^3*x^10 + 1/3*(55*c^2*d^3 + 6*b*
c*d)*e^8*f^3*x^9 + 1/8*(330*c^2*d^4 + 72*b*c*d^2 + b^2 + 2*a*c)*e^7*f^3*x^8 + (66*c^2*d^5 + 24*b*c*d^3 + (b^2
+ 2*a*c)*d)*e^6*f^3*x^7 + 1/6*(462*c^2*d^6 + 252*b*c*d^4 + 21*(b^2 + 2*a*c)*d^2 + 2*a*b)*e^5*f^3*x^6 + 1/5*(33
0*c^2*d^7 + 252*b*c*d^5 + 35*(b^2 + 2*a*c)*d^3 + 10*a*b*d)*e^4*f^3*x^5 + 1/4*(165*c^2*d^8 + 168*b*c*d^6 + 35*(
b^2 + 2*a*c)*d^4 + 20*a*b*d^2 + a^2)*e^3*f^3*x^4 + 1/3*(55*c^2*d^9 + 72*b*c*d^7 + 21*(b^2 + 2*a*c)*d^5 + 20*a*
b*d^3 + 3*a^2*d)*e^2*f^3*x^3 + 1/2*(11*c^2*d^10 + 18*b*c*d^8 + 7*(b^2 + 2*a*c)*d^6 + 10*a*b*d^4 + 3*a^2*d^2)*e
*f^3*x^2 + (c^2*d^11 + 2*b*c*d^9 + (b^2 + 2*a*c)*d^7 + 2*a*b*d^5 + a^2*d^3)*f^3*x

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Fricas [B]  time = 1.55417, size = 1517, normalized size = 14.59 \begin{align*} \frac{1}{12} x^{12} f^{3} e^{11} c^{2} + x^{11} f^{3} e^{10} d c^{2} + \frac{11}{2} x^{10} f^{3} e^{9} d^{2} c^{2} + \frac{55}{3} x^{9} f^{3} e^{8} d^{3} c^{2} + \frac{165}{4} x^{8} f^{3} e^{7} d^{4} c^{2} + \frac{1}{5} x^{10} f^{3} e^{9} c b + 66 x^{7} f^{3} e^{6} d^{5} c^{2} + 2 x^{9} f^{3} e^{8} d c b + 77 x^{6} f^{3} e^{5} d^{6} c^{2} + 9 x^{8} f^{3} e^{7} d^{2} c b + 66 x^{5} f^{3} e^{4} d^{7} c^{2} + 24 x^{7} f^{3} e^{6} d^{3} c b + \frac{165}{4} x^{4} f^{3} e^{3} d^{8} c^{2} + 42 x^{6} f^{3} e^{5} d^{4} c b + \frac{1}{8} x^{8} f^{3} e^{7} b^{2} + \frac{1}{4} x^{8} f^{3} e^{7} c a + \frac{55}{3} x^{3} f^{3} e^{2} d^{9} c^{2} + \frac{252}{5} x^{5} f^{3} e^{4} d^{5} c b + x^{7} f^{3} e^{6} d b^{2} + 2 x^{7} f^{3} e^{6} d c a + \frac{11}{2} x^{2} f^{3} e d^{10} c^{2} + 42 x^{4} f^{3} e^{3} d^{6} c b + \frac{7}{2} x^{6} f^{3} e^{5} d^{2} b^{2} + 7 x^{6} f^{3} e^{5} d^{2} c a + x f^{3} d^{11} c^{2} + 24 x^{3} f^{3} e^{2} d^{7} c b + 7 x^{5} f^{3} e^{4} d^{3} b^{2} + 14 x^{5} f^{3} e^{4} d^{3} c a + 9 x^{2} f^{3} e d^{8} c b + \frac{35}{4} x^{4} f^{3} e^{3} d^{4} b^{2} + \frac{35}{2} x^{4} f^{3} e^{3} d^{4} c a + \frac{1}{3} x^{6} f^{3} e^{5} b a + 2 x f^{3} d^{9} c b + 7 x^{3} f^{3} e^{2} d^{5} b^{2} + 14 x^{3} f^{3} e^{2} d^{5} c a + 2 x^{5} f^{3} e^{4} d b a + \frac{7}{2} x^{2} f^{3} e d^{6} b^{2} + 7 x^{2} f^{3} e d^{6} c a + 5 x^{4} f^{3} e^{3} d^{2} b a + x f^{3} d^{7} b^{2} + 2 x f^{3} d^{7} c a + \frac{20}{3} x^{3} f^{3} e^{2} d^{3} b a + 5 x^{2} f^{3} e d^{4} b a + \frac{1}{4} x^{4} f^{3} e^{3} a^{2} + 2 x f^{3} d^{5} b a + x^{3} f^{3} e^{2} d a^{2} + \frac{3}{2} x^{2} f^{3} e d^{2} a^{2} + x f^{3} d^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="fricas")

[Out]

1/12*x^12*f^3*e^11*c^2 + x^11*f^3*e^10*d*c^2 + 11/2*x^10*f^3*e^9*d^2*c^2 + 55/3*x^9*f^3*e^8*d^3*c^2 + 165/4*x^
8*f^3*e^7*d^4*c^2 + 1/5*x^10*f^3*e^9*c*b + 66*x^7*f^3*e^6*d^5*c^2 + 2*x^9*f^3*e^8*d*c*b + 77*x^6*f^3*e^5*d^6*c
^2 + 9*x^8*f^3*e^7*d^2*c*b + 66*x^5*f^3*e^4*d^7*c^2 + 24*x^7*f^3*e^6*d^3*c*b + 165/4*x^4*f^3*e^3*d^8*c^2 + 42*
x^6*f^3*e^5*d^4*c*b + 1/8*x^8*f^3*e^7*b^2 + 1/4*x^8*f^3*e^7*c*a + 55/3*x^3*f^3*e^2*d^9*c^2 + 252/5*x^5*f^3*e^4
*d^5*c*b + x^7*f^3*e^6*d*b^2 + 2*x^7*f^3*e^6*d*c*a + 11/2*x^2*f^3*e*d^10*c^2 + 42*x^4*f^3*e^3*d^6*c*b + 7/2*x^
6*f^3*e^5*d^2*b^2 + 7*x^6*f^3*e^5*d^2*c*a + x*f^3*d^11*c^2 + 24*x^3*f^3*e^2*d^7*c*b + 7*x^5*f^3*e^4*d^3*b^2 +
14*x^5*f^3*e^4*d^3*c*a + 9*x^2*f^3*e*d^8*c*b + 35/4*x^4*f^3*e^3*d^4*b^2 + 35/2*x^4*f^3*e^3*d^4*c*a + 1/3*x^6*f
^3*e^5*b*a + 2*x*f^3*d^9*c*b + 7*x^3*f^3*e^2*d^5*b^2 + 14*x^3*f^3*e^2*d^5*c*a + 2*x^5*f^3*e^4*d*b*a + 7/2*x^2*
f^3*e*d^6*b^2 + 7*x^2*f^3*e*d^6*c*a + 5*x^4*f^3*e^3*d^2*b*a + x*f^3*d^7*b^2 + 2*x*f^3*d^7*c*a + 20/3*x^3*f^3*e
^2*d^3*b*a + 5*x^2*f^3*e*d^4*b*a + 1/4*x^4*f^3*e^3*a^2 + 2*x*f^3*d^5*b*a + x^3*f^3*e^2*d*a^2 + 3/2*x^2*f^3*e*d
^2*a^2 + x*f^3*d^3*a^2

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Sympy [B]  time = 0.182221, size = 722, normalized size = 6.94 \begin{align*} c^{2} d e^{10} f^{3} x^{11} + \frac{c^{2} e^{11} f^{3} x^{12}}{12} + x^{10} \left (\frac{b c e^{9} f^{3}}{5} + \frac{11 c^{2} d^{2} e^{9} f^{3}}{2}\right ) + x^{9} \left (2 b c d e^{8} f^{3} + \frac{55 c^{2} d^{3} e^{8} f^{3}}{3}\right ) + x^{8} \left (\frac{a c e^{7} f^{3}}{4} + \frac{b^{2} e^{7} f^{3}}{8} + 9 b c d^{2} e^{7} f^{3} + \frac{165 c^{2} d^{4} e^{7} f^{3}}{4}\right ) + x^{7} \left (2 a c d e^{6} f^{3} + b^{2} d e^{6} f^{3} + 24 b c d^{3} e^{6} f^{3} + 66 c^{2} d^{5} e^{6} f^{3}\right ) + x^{6} \left (\frac{a b e^{5} f^{3}}{3} + 7 a c d^{2} e^{5} f^{3} + \frac{7 b^{2} d^{2} e^{5} f^{3}}{2} + 42 b c d^{4} e^{5} f^{3} + 77 c^{2} d^{6} e^{5} f^{3}\right ) + x^{5} \left (2 a b d e^{4} f^{3} + 14 a c d^{3} e^{4} f^{3} + 7 b^{2} d^{3} e^{4} f^{3} + \frac{252 b c d^{5} e^{4} f^{3}}{5} + 66 c^{2} d^{7} e^{4} f^{3}\right ) + x^{4} \left (\frac{a^{2} e^{3} f^{3}}{4} + 5 a b d^{2} e^{3} f^{3} + \frac{35 a c d^{4} e^{3} f^{3}}{2} + \frac{35 b^{2} d^{4} e^{3} f^{3}}{4} + 42 b c d^{6} e^{3} f^{3} + \frac{165 c^{2} d^{8} e^{3} f^{3}}{4}\right ) + x^{3} \left (a^{2} d e^{2} f^{3} + \frac{20 a b d^{3} e^{2} f^{3}}{3} + 14 a c d^{5} e^{2} f^{3} + 7 b^{2} d^{5} e^{2} f^{3} + 24 b c d^{7} e^{2} f^{3} + \frac{55 c^{2} d^{9} e^{2} f^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} d^{2} e f^{3}}{2} + 5 a b d^{4} e f^{3} + 7 a c d^{6} e f^{3} + \frac{7 b^{2} d^{6} e f^{3}}{2} + 9 b c d^{8} e f^{3} + \frac{11 c^{2} d^{10} e f^{3}}{2}\right ) + x \left (a^{2} d^{3} f^{3} + 2 a b d^{5} f^{3} + 2 a c d^{7} f^{3} + b^{2} d^{7} f^{3} + 2 b c d^{9} f^{3} + c^{2} d^{11} f^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)**3*(a+b*(e*x+d)**2+c*(e*x+d)**4)**2,x)

[Out]

c**2*d*e**10*f**3*x**11 + c**2*e**11*f**3*x**12/12 + x**10*(b*c*e**9*f**3/5 + 11*c**2*d**2*e**9*f**3/2) + x**9
*(2*b*c*d*e**8*f**3 + 55*c**2*d**3*e**8*f**3/3) + x**8*(a*c*e**7*f**3/4 + b**2*e**7*f**3/8 + 9*b*c*d**2*e**7*f
**3 + 165*c**2*d**4*e**7*f**3/4) + x**7*(2*a*c*d*e**6*f**3 + b**2*d*e**6*f**3 + 24*b*c*d**3*e**6*f**3 + 66*c**
2*d**5*e**6*f**3) + x**6*(a*b*e**5*f**3/3 + 7*a*c*d**2*e**5*f**3 + 7*b**2*d**2*e**5*f**3/2 + 42*b*c*d**4*e**5*
f**3 + 77*c**2*d**6*e**5*f**3) + x**5*(2*a*b*d*e**4*f**3 + 14*a*c*d**3*e**4*f**3 + 7*b**2*d**3*e**4*f**3 + 252
*b*c*d**5*e**4*f**3/5 + 66*c**2*d**7*e**4*f**3) + x**4*(a**2*e**3*f**3/4 + 5*a*b*d**2*e**3*f**3 + 35*a*c*d**4*
e**3*f**3/2 + 35*b**2*d**4*e**3*f**3/4 + 42*b*c*d**6*e**3*f**3 + 165*c**2*d**8*e**3*f**3/4) + x**3*(a**2*d*e**
2*f**3 + 20*a*b*d**3*e**2*f**3/3 + 14*a*c*d**5*e**2*f**3 + 7*b**2*d**5*e**2*f**3 + 24*b*c*d**7*e**2*f**3 + 55*
c**2*d**9*e**2*f**3/3) + x**2*(3*a**2*d**2*e*f**3/2 + 5*a*b*d**4*e*f**3 + 7*a*c*d**6*e*f**3 + 7*b**2*d**6*e*f*
*3/2 + 9*b*c*d**8*e*f**3 + 11*c**2*d**10*e*f**3/2) + x*(a**2*d**3*f**3 + 2*a*b*d**5*f**3 + 2*a*c*d**7*f**3 + b
**2*d**7*f**3 + 2*b*c*d**9*f**3 + c**2*d**11*f**3)

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Giac [B]  time = 1.1143, size = 925, normalized size = 8.89 \begin{align*} \frac{1}{12} \, c^{2} f^{3} x^{12} e^{11} + c^{2} d f^{3} x^{11} e^{10} + \frac{11}{2} \, c^{2} d^{2} f^{3} x^{10} e^{9} + \frac{55}{3} \, c^{2} d^{3} f^{3} x^{9} e^{8} + \frac{165}{4} \, c^{2} d^{4} f^{3} x^{8} e^{7} + 66 \, c^{2} d^{5} f^{3} x^{7} e^{6} + 77 \, c^{2} d^{6} f^{3} x^{6} e^{5} + 66 \, c^{2} d^{7} f^{3} x^{5} e^{4} + \frac{165}{4} \, c^{2} d^{8} f^{3} x^{4} e^{3} + \frac{55}{3} \, c^{2} d^{9} f^{3} x^{3} e^{2} + \frac{11}{2} \, c^{2} d^{10} f^{3} x^{2} e + c^{2} d^{11} f^{3} x + \frac{1}{5} \, b c f^{3} x^{10} e^{9} + 2 \, b c d f^{3} x^{9} e^{8} + 9 \, b c d^{2} f^{3} x^{8} e^{7} + 24 \, b c d^{3} f^{3} x^{7} e^{6} + 42 \, b c d^{4} f^{3} x^{6} e^{5} + \frac{252}{5} \, b c d^{5} f^{3} x^{5} e^{4} + 42 \, b c d^{6} f^{3} x^{4} e^{3} + 24 \, b c d^{7} f^{3} x^{3} e^{2} + 9 \, b c d^{8} f^{3} x^{2} e + 2 \, b c d^{9} f^{3} x + \frac{1}{8} \, b^{2} f^{3} x^{8} e^{7} + \frac{1}{4} \, a c f^{3} x^{8} e^{7} + b^{2} d f^{3} x^{7} e^{6} + 2 \, a c d f^{3} x^{7} e^{6} + \frac{7}{2} \, b^{2} d^{2} f^{3} x^{6} e^{5} + 7 \, a c d^{2} f^{3} x^{6} e^{5} + 7 \, b^{2} d^{3} f^{3} x^{5} e^{4} + 14 \, a c d^{3} f^{3} x^{5} e^{4} + \frac{35}{4} \, b^{2} d^{4} f^{3} x^{4} e^{3} + \frac{35}{2} \, a c d^{4} f^{3} x^{4} e^{3} + 7 \, b^{2} d^{5} f^{3} x^{3} e^{2} + 14 \, a c d^{5} f^{3} x^{3} e^{2} + \frac{7}{2} \, b^{2} d^{6} f^{3} x^{2} e + 7 \, a c d^{6} f^{3} x^{2} e + b^{2} d^{7} f^{3} x + 2 \, a c d^{7} f^{3} x + \frac{1}{3} \, a b f^{3} x^{6} e^{5} + 2 \, a b d f^{3} x^{5} e^{4} + 5 \, a b d^{2} f^{3} x^{4} e^{3} + \frac{20}{3} \, a b d^{3} f^{3} x^{3} e^{2} + 5 \, a b d^{4} f^{3} x^{2} e + 2 \, a b d^{5} f^{3} x + \frac{1}{4} \, a^{2} f^{3} x^{4} e^{3} + a^{2} d f^{3} x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} f^{3} x^{2} e + a^{2} d^{3} f^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*f*x+d*f)^3*(a+b*(e*x+d)^2+c*(e*x+d)^4)^2,x, algorithm="giac")

[Out]

1/12*c^2*f^3*x^12*e^11 + c^2*d*f^3*x^11*e^10 + 11/2*c^2*d^2*f^3*x^10*e^9 + 55/3*c^2*d^3*f^3*x^9*e^8 + 165/4*c^
2*d^4*f^3*x^8*e^7 + 66*c^2*d^5*f^3*x^7*e^6 + 77*c^2*d^6*f^3*x^6*e^5 + 66*c^2*d^7*f^3*x^5*e^4 + 165/4*c^2*d^8*f
^3*x^4*e^3 + 55/3*c^2*d^9*f^3*x^3*e^2 + 11/2*c^2*d^10*f^3*x^2*e + c^2*d^11*f^3*x + 1/5*b*c*f^3*x^10*e^9 + 2*b*
c*d*f^3*x^9*e^8 + 9*b*c*d^2*f^3*x^8*e^7 + 24*b*c*d^3*f^3*x^7*e^6 + 42*b*c*d^4*f^3*x^6*e^5 + 252/5*b*c*d^5*f^3*
x^5*e^4 + 42*b*c*d^6*f^3*x^4*e^3 + 24*b*c*d^7*f^3*x^3*e^2 + 9*b*c*d^8*f^3*x^2*e + 2*b*c*d^9*f^3*x + 1/8*b^2*f^
3*x^8*e^7 + 1/4*a*c*f^3*x^8*e^7 + b^2*d*f^3*x^7*e^6 + 2*a*c*d*f^3*x^7*e^6 + 7/2*b^2*d^2*f^3*x^6*e^5 + 7*a*c*d^
2*f^3*x^6*e^5 + 7*b^2*d^3*f^3*x^5*e^4 + 14*a*c*d^3*f^3*x^5*e^4 + 35/4*b^2*d^4*f^3*x^4*e^3 + 35/2*a*c*d^4*f^3*x
^4*e^3 + 7*b^2*d^5*f^3*x^3*e^2 + 14*a*c*d^5*f^3*x^3*e^2 + 7/2*b^2*d^6*f^3*x^2*e + 7*a*c*d^6*f^3*x^2*e + b^2*d^
7*f^3*x + 2*a*c*d^7*f^3*x + 1/3*a*b*f^3*x^6*e^5 + 2*a*b*d*f^3*x^5*e^4 + 5*a*b*d^2*f^3*x^4*e^3 + 20/3*a*b*d^3*f
^3*x^3*e^2 + 5*a*b*d^4*f^3*x^2*e + 2*a*b*d^5*f^3*x + 1/4*a^2*f^3*x^4*e^3 + a^2*d*f^3*x^3*e^2 + 3/2*a^2*d^2*f^3
*x^2*e + a^2*d^3*f^3*x