3.25 \(\int (d+e x)^3 (a+c x^2)^2 (A+B x+C x^2) \, dx\)

Optimal. Leaf size=304 \[ \frac{1}{4} a^2 e x^4 \left (e (A e+3 B d)+3 C d^2\right )+a^2 A d^3 x+\frac{1}{8} c e x^8 \left (2 a C e^2+c \left (e (A e+3 B d)+3 C d^2\right )\right )+\frac{1}{7} c x^7 \left (2 a e^2 (B e+3 C d)+c d \left (3 e (A e+B d)+C d^2\right )\right )+\frac{1}{6} a e x^6 \left (a C e^2+2 c \left (e (A e+3 B d)+3 C d^2\right )\right )+\frac{1}{5} x^5 \left (A c d \left (6 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+2 c d^2 (3 B e+C d)\right )\right )+\frac{1}{3} a d x^3 \left (A \left (3 a e^2+2 c d^2\right )+a d (3 B e+C d)\right )+\frac{d^2 \left (a+c x^2\right )^3 (3 A e+B d)}{6 c}+\frac{1}{9} c^2 e^2 x^9 (B e+3 C d)+\frac{1}{10} c^2 C e^3 x^{10} \]

[Out]

a^2*A*d^3*x + (a*d*(a*d*(C*d + 3*B*e) + A*(2*c*d^2 + 3*a*e^2))*x^3)/3 + (a^2*e*(3*C*d^2 + e*(3*B*d + A*e))*x^4
)/4 + ((A*c*d*(c*d^2 + 6*a*e^2) + a*(a*e^2*(3*C*d + B*e) + 2*c*d^2*(C*d + 3*B*e)))*x^5)/5 + (a*e*(a*C*e^2 + 2*
c*(3*C*d^2 + e*(3*B*d + A*e)))*x^6)/6 + (c*(2*a*e^2*(3*C*d + B*e) + c*d*(C*d^2 + 3*e*(B*d + A*e)))*x^7)/7 + (c
*e*(2*a*C*e^2 + c*(3*C*d^2 + e*(3*B*d + A*e)))*x^8)/8 + (c^2*e^2*(3*C*d + B*e)*x^9)/9 + (c^2*C*e^3*x^10)/10 +
(d^2*(B*d + 3*A*e)*(a + c*x^2)^3)/(6*c)

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Rubi [A]  time = 0.534531, antiderivative size = 301, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1582, 1810} \[ \frac{1}{4} a^2 e x^4 \left (e (A e+3 B d)+3 C d^2\right )+a^2 A d^3 x+\frac{1}{8} c e x^8 \left (2 a C e^2+c e (A e+3 B d)+3 c C d^2\right )+\frac{1}{7} c x^7 \left (2 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )+\frac{1}{6} a e x^6 \left (a C e^2+2 c e (A e+3 B d)+6 c C d^2\right )+\frac{1}{5} x^5 \left (A c d \left (6 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+2 c d^2 (3 B e+C d)\right )\right )+\frac{1}{3} a d x^3 \left (A \left (3 a e^2+2 c d^2\right )+a d (3 B e+C d)\right )+\frac{d^2 \left (a+c x^2\right )^3 (3 A e+B d)}{6 c}+\frac{1}{9} c^2 e^2 x^9 (B e+3 C d)+\frac{1}{10} c^2 C e^3 x^{10} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + c*x^2)^2*(A + B*x + C*x^2),x]

[Out]

a^2*A*d^3*x + (a*d*(a*d*(C*d + 3*B*e) + A*(2*c*d^2 + 3*a*e^2))*x^3)/3 + (a^2*e*(3*C*d^2 + e*(3*B*d + A*e))*x^4
)/4 + ((A*c*d*(c*d^2 + 6*a*e^2) + a*(a*e^2*(3*C*d + B*e) + 2*c*d^2*(C*d + 3*B*e)))*x^5)/5 + (a*e*(6*c*C*d^2 +
a*C*e^2 + 2*c*e*(3*B*d + A*e))*x^6)/6 + (c*(c*C*d^3 + 3*c*d*e*(B*d + A*e) + 2*a*e^2*(3*C*d + B*e))*x^7)/7 + (c
*e*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(3*B*d + A*e))*x^8)/8 + (c^2*e^2*(3*C*d + B*e)*x^9)/9 + (c^2*C*e^3*x^10)/10 +
(d^2*(B*d + 3*A*e)*(a + c*x^2)^3)/(6*c)

Rule 1582

Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - 1]*(a + b*x^n)^(p + 1))/(b*n*(p +
1)), x] + Int[(Px - Coeff[Px, x, n - 1]*x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && I
GtQ[p, 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] &&  !MatchQ[P
x, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ[{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Co
eff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx &=\frac{d^2 (B d+3 A e) \left (a+c x^2\right )^3}{6 c}+\int \left (a+c x^2\right )^2 \left (-\left (B d^3+3 A d^2 e\right ) x+(d+e x)^3 \left (A+B x+C x^2\right )\right ) \, dx\\ &=\frac{d^2 (B d+3 A e) \left (a+c x^2\right )^3}{6 c}+\int \left (a^2 A d^3+a d \left (a d (C d+3 B e)+A \left (2 c d^2+3 a e^2\right )\right ) x^2+a^2 e \left (3 C d^2+e (3 B d+A e)\right ) x^3+\left (A c d \left (c d^2+6 a e^2\right )+a \left (a e^2 (3 C d+B e)+2 c d^2 (C d+3 B e)\right )\right ) x^4+a e \left (6 c C d^2+a C e^2+2 c e (3 B d+A e)\right ) x^5+c \left (c C d^3+3 c d e (B d+A e)+2 a e^2 (3 C d+B e)\right ) x^6+c e \left (3 c C d^2+2 a C e^2+c e (3 B d+A e)\right ) x^7+c^2 e^2 (3 C d+B e) x^8+c^2 C e^3 x^9\right ) \, dx\\ &=a^2 A d^3 x+\frac{1}{3} a d \left (a d (C d+3 B e)+A \left (2 c d^2+3 a e^2\right )\right ) x^3+\frac{1}{4} a^2 e \left (3 C d^2+e (3 B d+A e)\right ) x^4+\frac{1}{5} \left (A c d \left (c d^2+6 a e^2\right )+a \left (a e^2 (3 C d+B e)+2 c d^2 (C d+3 B e)\right )\right ) x^5+\frac{1}{6} a e \left (6 c C d^2+a C e^2+2 c e (3 B d+A e)\right ) x^6+\frac{1}{7} c \left (c C d^3+3 c d e (B d+A e)+2 a e^2 (3 C d+B e)\right ) x^7+\frac{1}{8} c e \left (3 c C d^2+2 a C e^2+c e (3 B d+A e)\right ) x^8+\frac{1}{9} c^2 e^2 (3 C d+B e) x^9+\frac{1}{10} c^2 C e^3 x^{10}+\frac{d^2 (B d+3 A e) \left (a+c x^2\right )^3}{6 c}\\ \end{align*}

Mathematica [A]  time = 0.134529, size = 335, normalized size = 1.1 \[ \frac{1}{2} a^2 d^2 x^2 (3 A e+B d)+a^2 A d^3 x+\frac{1}{8} c e x^8 \left (2 a C e^2+c e (A e+3 B d)+3 c C d^2\right )+\frac{1}{7} c x^7 \left (2 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )+\frac{1}{6} x^6 \left (A c e \left (2 a e^2+3 c d^2\right )+B c d \left (6 a e^2+c d^2\right )+a C e \left (a e^2+6 c d^2\right )\right )+\frac{1}{5} x^5 \left (A c d \left (6 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+2 c d^2 (3 B e+C d)\right )\right )+\frac{1}{4} a x^4 \left (a A e^3+3 a B d e^2+3 a C d^2 e+6 A c d^2 e+2 B c d^3\right )+\frac{1}{3} a d x^3 \left (A \left (3 a e^2+2 c d^2\right )+a d (3 B e+C d)\right )+\frac{1}{9} c^2 e^2 x^9 (B e+3 C d)+\frac{1}{10} c^2 C e^3 x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + c*x^2)^2*(A + B*x + C*x^2),x]

[Out]

a^2*A*d^3*x + (a^2*d^2*(B*d + 3*A*e)*x^2)/2 + (a*d*(a*d*(C*d + 3*B*e) + A*(2*c*d^2 + 3*a*e^2))*x^3)/3 + (a*(2*
B*c*d^3 + 6*A*c*d^2*e + 3*a*C*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + ((A*c*d*(c*d^2 + 6*a*e^2) + a*(a*e^2*(3*
C*d + B*e) + 2*c*d^2*(C*d + 3*B*e)))*x^5)/5 + ((a*C*e*(6*c*d^2 + a*e^2) + A*c*e*(3*c*d^2 + 2*a*e^2) + B*c*d*(c
*d^2 + 6*a*e^2))*x^6)/6 + (c*(c*C*d^3 + 3*c*d*e*(B*d + A*e) + 2*a*e^2*(3*C*d + B*e))*x^7)/7 + (c*e*(3*c*C*d^2
+ 2*a*C*e^2 + c*e*(3*B*d + A*e))*x^8)/8 + (c^2*e^2*(3*C*d + B*e)*x^9)/9 + (c^2*C*e^3*x^10)/10

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Maple [A]  time = 0.046, size = 385, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}C{e}^{3}{x}^{10}}{10}}+{\frac{ \left ({e}^{3}{c}^{2}B+3\,d{e}^{2}{c}^{2}C \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,ac{e}^{3}+3\,{c}^{2}{d}^{2}e \right ) C+3\,d{e}^{2}{c}^{2}B+{e}^{3}{c}^{2}A \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,ad{e}^{2}c+{c}^{2}{d}^{3} \right ) C+ \left ( 2\,ac{e}^{3}+3\,{c}^{2}{d}^{2}e \right ) B+3\,d{e}^{2}{c}^{2}A \right ){x}^{7}}{7}}+{\frac{ \left ( \left ({a}^{2}{e}^{3}+6\,ac{d}^{2}e \right ) C+ \left ( 6\,ad{e}^{2}c+{c}^{2}{d}^{3} \right ) B+ \left ( 2\,ac{e}^{3}+3\,{c}^{2}{d}^{2}e \right ) A \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,d{e}^{2}{a}^{2}+2\,{d}^{3}ac \right ) C+ \left ({a}^{2}{e}^{3}+6\,ac{d}^{2}e \right ) B+ \left ( 6\,ad{e}^{2}c+{c}^{2}{d}^{3} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}C+ \left ( 3\,d{e}^{2}{a}^{2}+2\,{d}^{3}ac \right ) B+ \left ({a}^{2}{e}^{3}+6\,ac{d}^{2}e \right ) A \right ){x}^{4}}{4}}+{\frac{ \left ({d}^{3}{a}^{2}C+3\,{d}^{2}e{a}^{2}B+ \left ( 3\,d{e}^{2}{a}^{2}+2\,{d}^{3}ac \right ) A \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{d}^{2}e{a}^{2}A+{d}^{3}{a}^{2}B \right ){x}^{2}}{2}}+{a}^{2}A{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*x^2+a)^2*(C*x^2+B*x+A),x)

[Out]

1/10*c^2*C*e^3*x^10+1/9*(B*c^2*e^3+3*C*c^2*d*e^2)*x^9+1/8*((2*a*c*e^3+3*c^2*d^2*e)*C+3*d*e^2*c^2*B+e^3*c^2*A)*
x^8+1/7*((6*a*c*d*e^2+c^2*d^3)*C+(2*a*c*e^3+3*c^2*d^2*e)*B+3*d*e^2*c^2*A)*x^7+1/6*((a^2*e^3+6*a*c*d^2*e)*C+(6*
a*c*d*e^2+c^2*d^3)*B+(2*a*c*e^3+3*c^2*d^2*e)*A)*x^6+1/5*((3*a^2*d*e^2+2*a*c*d^3)*C+(a^2*e^3+6*a*c*d^2*e)*B+(6*
a*c*d*e^2+c^2*d^3)*A)*x^5+1/4*(3*d^2*e*a^2*C+(3*a^2*d*e^2+2*a*c*d^3)*B+(a^2*e^3+6*a*c*d^2*e)*A)*x^4+1/3*(d^3*a
^2*C+3*d^2*e*a^2*B+(3*a^2*d*e^2+2*a*c*d^3)*A)*x^3+1/2*(3*A*a^2*d^2*e+B*a^2*d^3)*x^2+a^2*A*d^3*x

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Maxima [A]  time = 1.02582, size = 486, normalized size = 1.6 \begin{align*} \frac{1}{10} \, C c^{2} e^{3} x^{10} + \frac{1}{9} \,{\left (3 \, C c^{2} d e^{2} + B c^{2} e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (3 \, C c^{2} d^{2} e + 3 \, B c^{2} d e^{2} +{\left (2 \, C a c + A c^{2}\right )} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (C c^{2} d^{3} + 3 \, B c^{2} d^{2} e + 2 \, B a c e^{3} + 3 \,{\left (2 \, C a c + A c^{2}\right )} d e^{2}\right )} x^{7} + A a^{2} d^{3} x + \frac{1}{6} \,{\left (B c^{2} d^{3} + 6 \, B a c d e^{2} + 3 \,{\left (2 \, C a c + A c^{2}\right )} d^{2} e +{\left (C a^{2} + 2 \, A a c\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, B a c d^{2} e + B a^{2} e^{3} +{\left (2 \, C a c + A c^{2}\right )} d^{3} + 3 \,{\left (C a^{2} + 2 \, A a c\right )} d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, B a c d^{3} + 3 \, B a^{2} d e^{2} + A a^{2} e^{3} + 3 \,{\left (C a^{2} + 2 \, A a c\right )} d^{2} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \, B a^{2} d^{2} e + 3 \, A a^{2} d e^{2} +{\left (C a^{2} + 2 \, A a c\right )} d^{3}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{3} + 3 \, A a^{2} d^{2} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a)^2*(C*x^2+B*x+A),x, algorithm="maxima")

[Out]

1/10*C*c^2*e^3*x^10 + 1/9*(3*C*c^2*d*e^2 + B*c^2*e^3)*x^9 + 1/8*(3*C*c^2*d^2*e + 3*B*c^2*d*e^2 + (2*C*a*c + A*
c^2)*e^3)*x^8 + 1/7*(C*c^2*d^3 + 3*B*c^2*d^2*e + 2*B*a*c*e^3 + 3*(2*C*a*c + A*c^2)*d*e^2)*x^7 + A*a^2*d^3*x +
1/6*(B*c^2*d^3 + 6*B*a*c*d*e^2 + 3*(2*C*a*c + A*c^2)*d^2*e + (C*a^2 + 2*A*a*c)*e^3)*x^6 + 1/5*(6*B*a*c*d^2*e +
B*a^2*e^3 + (2*C*a*c + A*c^2)*d^3 + 3*(C*a^2 + 2*A*a*c)*d*e^2)*x^5 + 1/4*(2*B*a*c*d^3 + 3*B*a^2*d*e^2 + A*a^2
*e^3 + 3*(C*a^2 + 2*A*a*c)*d^2*e)*x^4 + 1/3*(3*B*a^2*d^2*e + 3*A*a^2*d*e^2 + (C*a^2 + 2*A*a*c)*d^3)*x^3 + 1/2*
(B*a^2*d^3 + 3*A*a^2*d^2*e)*x^2

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Fricas [A]  time = 1.44218, size = 992, normalized size = 3.26 \begin{align*} \frac{1}{10} x^{10} e^{3} c^{2} C + \frac{1}{3} x^{9} e^{2} d c^{2} C + \frac{1}{9} x^{9} e^{3} c^{2} B + \frac{3}{8} x^{8} e d^{2} c^{2} C + \frac{1}{4} x^{8} e^{3} c a C + \frac{3}{8} x^{8} e^{2} d c^{2} B + \frac{1}{8} x^{8} e^{3} c^{2} A + \frac{1}{7} x^{7} d^{3} c^{2} C + \frac{6}{7} x^{7} e^{2} d c a C + \frac{3}{7} x^{7} e d^{2} c^{2} B + \frac{2}{7} x^{7} e^{3} c a B + \frac{3}{7} x^{7} e^{2} d c^{2} A + x^{6} e d^{2} c a C + \frac{1}{6} x^{6} e^{3} a^{2} C + \frac{1}{6} x^{6} d^{3} c^{2} B + x^{6} e^{2} d c a B + \frac{1}{2} x^{6} e d^{2} c^{2} A + \frac{1}{3} x^{6} e^{3} c a A + \frac{2}{5} x^{5} d^{3} c a C + \frac{3}{5} x^{5} e^{2} d a^{2} C + \frac{6}{5} x^{5} e d^{2} c a B + \frac{1}{5} x^{5} e^{3} a^{2} B + \frac{1}{5} x^{5} d^{3} c^{2} A + \frac{6}{5} x^{5} e^{2} d c a A + \frac{3}{4} x^{4} e d^{2} a^{2} C + \frac{1}{2} x^{4} d^{3} c a B + \frac{3}{4} x^{4} e^{2} d a^{2} B + \frac{3}{2} x^{4} e d^{2} c a A + \frac{1}{4} x^{4} e^{3} a^{2} A + \frac{1}{3} x^{3} d^{3} a^{2} C + x^{3} e d^{2} a^{2} B + \frac{2}{3} x^{3} d^{3} c a A + x^{3} e^{2} d a^{2} A + \frac{1}{2} x^{2} d^{3} a^{2} B + \frac{3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a)^2*(C*x^2+B*x+A),x, algorithm="fricas")

[Out]

1/10*x^10*e^3*c^2*C + 1/3*x^9*e^2*d*c^2*C + 1/9*x^9*e^3*c^2*B + 3/8*x^8*e*d^2*c^2*C + 1/4*x^8*e^3*c*a*C + 3/8*
x^8*e^2*d*c^2*B + 1/8*x^8*e^3*c^2*A + 1/7*x^7*d^3*c^2*C + 6/7*x^7*e^2*d*c*a*C + 3/7*x^7*e*d^2*c^2*B + 2/7*x^7*
e^3*c*a*B + 3/7*x^7*e^2*d*c^2*A + x^6*e*d^2*c*a*C + 1/6*x^6*e^3*a^2*C + 1/6*x^6*d^3*c^2*B + x^6*e^2*d*c*a*B +
1/2*x^6*e*d^2*c^2*A + 1/3*x^6*e^3*c*a*A + 2/5*x^5*d^3*c*a*C + 3/5*x^5*e^2*d*a^2*C + 6/5*x^5*e*d^2*c*a*B + 1/5*
x^5*e^3*a^2*B + 1/5*x^5*d^3*c^2*A + 6/5*x^5*e^2*d*c*a*A + 3/4*x^4*e*d^2*a^2*C + 1/2*x^4*d^3*c*a*B + 3/4*x^4*e^
2*d*a^2*B + 3/2*x^4*e*d^2*c*a*A + 1/4*x^4*e^3*a^2*A + 1/3*x^3*d^3*a^2*C + x^3*e*d^2*a^2*B + 2/3*x^3*d^3*c*a*A
+ x^3*e^2*d*a^2*A + 1/2*x^2*d^3*a^2*B + 3/2*x^2*e*d^2*a^2*A + x*d^3*a^2*A

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Sympy [A]  time = 0.118716, size = 445, normalized size = 1.46 \begin{align*} A a^{2} d^{3} x + \frac{C c^{2} e^{3} x^{10}}{10} + x^{9} \left (\frac{B c^{2} e^{3}}{9} + \frac{C c^{2} d e^{2}}{3}\right ) + x^{8} \left (\frac{A c^{2} e^{3}}{8} + \frac{3 B c^{2} d e^{2}}{8} + \frac{C a c e^{3}}{4} + \frac{3 C c^{2} d^{2} e}{8}\right ) + x^{7} \left (\frac{3 A c^{2} d e^{2}}{7} + \frac{2 B a c e^{3}}{7} + \frac{3 B c^{2} d^{2} e}{7} + \frac{6 C a c d e^{2}}{7} + \frac{C c^{2} d^{3}}{7}\right ) + x^{6} \left (\frac{A a c e^{3}}{3} + \frac{A c^{2} d^{2} e}{2} + B a c d e^{2} + \frac{B c^{2} d^{3}}{6} + \frac{C a^{2} e^{3}}{6} + C a c d^{2} e\right ) + x^{5} \left (\frac{6 A a c d e^{2}}{5} + \frac{A c^{2} d^{3}}{5} + \frac{B a^{2} e^{3}}{5} + \frac{6 B a c d^{2} e}{5} + \frac{3 C a^{2} d e^{2}}{5} + \frac{2 C a c d^{3}}{5}\right ) + x^{4} \left (\frac{A a^{2} e^{3}}{4} + \frac{3 A a c d^{2} e}{2} + \frac{3 B a^{2} d e^{2}}{4} + \frac{B a c d^{3}}{2} + \frac{3 C a^{2} d^{2} e}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + \frac{2 A a c d^{3}}{3} + B a^{2} d^{2} e + \frac{C a^{2} d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a^{2} d^{2} e}{2} + \frac{B a^{2} d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(c*x**2+a)**2*(C*x**2+B*x+A),x)

[Out]

A*a**2*d**3*x + C*c**2*e**3*x**10/10 + x**9*(B*c**2*e**3/9 + C*c**2*d*e**2/3) + x**8*(A*c**2*e**3/8 + 3*B*c**2
*d*e**2/8 + C*a*c*e**3/4 + 3*C*c**2*d**2*e/8) + x**7*(3*A*c**2*d*e**2/7 + 2*B*a*c*e**3/7 + 3*B*c**2*d**2*e/7 +
6*C*a*c*d*e**2/7 + C*c**2*d**3/7) + x**6*(A*a*c*e**3/3 + A*c**2*d**2*e/2 + B*a*c*d*e**2 + B*c**2*d**3/6 + C*a
**2*e**3/6 + C*a*c*d**2*e) + x**5*(6*A*a*c*d*e**2/5 + A*c**2*d**3/5 + B*a**2*e**3/5 + 6*B*a*c*d**2*e/5 + 3*C*a
**2*d*e**2/5 + 2*C*a*c*d**3/5) + x**4*(A*a**2*e**3/4 + 3*A*a*c*d**2*e/2 + 3*B*a**2*d*e**2/4 + B*a*c*d**3/2 + 3
*C*a**2*d**2*e/4) + x**3*(A*a**2*d*e**2 + 2*A*a*c*d**3/3 + B*a**2*d**2*e + C*a**2*d**3/3) + x**2*(3*A*a**2*d**
2*e/2 + B*a**2*d**3/2)

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Giac [A]  time = 1.14872, size = 571, normalized size = 1.88 \begin{align*} \frac{1}{10} \, C c^{2} x^{10} e^{3} + \frac{1}{3} \, C c^{2} d x^{9} e^{2} + \frac{3}{8} \, C c^{2} d^{2} x^{8} e + \frac{1}{7} \, C c^{2} d^{3} x^{7} + \frac{1}{9} \, B c^{2} x^{9} e^{3} + \frac{3}{8} \, B c^{2} d x^{8} e^{2} + \frac{3}{7} \, B c^{2} d^{2} x^{7} e + \frac{1}{6} \, B c^{2} d^{3} x^{6} + \frac{1}{4} \, C a c x^{8} e^{3} + \frac{1}{8} \, A c^{2} x^{8} e^{3} + \frac{6}{7} \, C a c d x^{7} e^{2} + \frac{3}{7} \, A c^{2} d x^{7} e^{2} + C a c d^{2} x^{6} e + \frac{1}{2} \, A c^{2} d^{2} x^{6} e + \frac{2}{5} \, C a c d^{3} x^{5} + \frac{1}{5} \, A c^{2} d^{3} x^{5} + \frac{2}{7} \, B a c x^{7} e^{3} + B a c d x^{6} e^{2} + \frac{6}{5} \, B a c d^{2} x^{5} e + \frac{1}{2} \, B a c d^{3} x^{4} + \frac{1}{6} \, C a^{2} x^{6} e^{3} + \frac{1}{3} \, A a c x^{6} e^{3} + \frac{3}{5} \, C a^{2} d x^{5} e^{2} + \frac{6}{5} \, A a c d x^{5} e^{2} + \frac{3}{4} \, C a^{2} d^{2} x^{4} e + \frac{3}{2} \, A a c d^{2} x^{4} e + \frac{1}{3} \, C a^{2} d^{3} x^{3} + \frac{2}{3} \, A a c d^{3} x^{3} + \frac{1}{5} \, B a^{2} x^{5} e^{3} + \frac{3}{4} \, B a^{2} d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + \frac{1}{2} \, B a^{2} d^{3} x^{2} + \frac{1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac{3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(c*x^2+a)^2*(C*x^2+B*x+A),x, algorithm="giac")

[Out]

1/10*C*c^2*x^10*e^3 + 1/3*C*c^2*d*x^9*e^2 + 3/8*C*c^2*d^2*x^8*e + 1/7*C*c^2*d^3*x^7 + 1/9*B*c^2*x^9*e^3 + 3/8*
B*c^2*d*x^8*e^2 + 3/7*B*c^2*d^2*x^7*e + 1/6*B*c^2*d^3*x^6 + 1/4*C*a*c*x^8*e^3 + 1/8*A*c^2*x^8*e^3 + 6/7*C*a*c*
d*x^7*e^2 + 3/7*A*c^2*d*x^7*e^2 + C*a*c*d^2*x^6*e + 1/2*A*c^2*d^2*x^6*e + 2/5*C*a*c*d^3*x^5 + 1/5*A*c^2*d^3*x^
5 + 2/7*B*a*c*x^7*e^3 + B*a*c*d*x^6*e^2 + 6/5*B*a*c*d^2*x^5*e + 1/2*B*a*c*d^3*x^4 + 1/6*C*a^2*x^6*e^3 + 1/3*A*
a*c*x^6*e^3 + 3/5*C*a^2*d*x^5*e^2 + 6/5*A*a*c*d*x^5*e^2 + 3/4*C*a^2*d^2*x^4*e + 3/2*A*a*c*d^2*x^4*e + 1/3*C*a^
2*d^3*x^3 + 2/3*A*a*c*d^3*x^3 + 1/5*B*a^2*x^5*e^3 + 3/4*B*a^2*d*x^4*e^2 + B*a^2*d^2*x^3*e + 1/2*B*a^2*d^3*x^2
+ 1/4*A*a^2*x^4*e^3 + A*a^2*d*x^3*e^2 + 3/2*A*a^2*d^2*x^2*e + A*a^2*d^3*x