### 3.33 $$\int (d+e x)^2 (a+c x^2)^3 (A+B x+C x^2) \, dx$$

Optimal. Leaf size=289 $\frac{1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+a^3 A d^2 x+\frac{1}{2} a^2 c e x^6 (B e+2 C d)+\frac{1}{4} a^3 e x^4 (B e+2 C d)+\frac{1}{9} c^2 x^9 \left (3 a C e^2+c \left (e (A e+2 B d)+C d^2\right )\right )+\frac{1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac{1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac{d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}+\frac{3}{8} a c^2 e x^8 (B e+2 C d)+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11}$

[Out]

a^3*A*d^2*x + (a^2*(a*d*(C*d + 2*B*e) + A*(3*c*d^2 + a*e^2))*x^3)/3 + (a^3*e*(2*C*d + B*e)*x^4)/4 + (a*(3*A*c*
(c*d^2 + a*e^2) + a*(a*C*e^2 + 3*c*d*(C*d + 2*B*e)))*x^5)/5 + (a^2*c*e*(2*C*d + B*e)*x^6)/2 + (c*(A*c*(c*d^2 +
3*a*e^2) + 3*a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*x^7)/7 + (3*a*c^2*e*(2*C*d + B*e)*x^8)/8 + (c^2*(3*a*C*e^2 + c*
(C*d^2 + e*(2*B*d + A*e)))*x^9)/9 + (c^3*e*(2*C*d + B*e)*x^10)/10 + (c^3*C*e^2*x^11)/11 + (d*(B*d + 2*A*e)*(a
+ c*x^2)^4)/(8*c)

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Rubi [A]  time = 0.424141, antiderivative size = 288, normalized size of antiderivative = 1., 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} $\frac{1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+a^3 A d^2 x+\frac{1}{2} a^2 c e x^6 (B e+2 C d)+\frac{1}{4} a^3 e x^4 (B e+2 C d)+\frac{1}{9} c^2 x^9 \left (3 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac{1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac{1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac{d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}+\frac{3}{8} a c^2 e x^8 (B e+2 C d)+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^3*A*d^2*x + (a^2*(a*d*(C*d + 2*B*e) + A*(3*c*d^2 + a*e^2))*x^3)/3 + (a^3*e*(2*C*d + B*e)*x^4)/4 + (a*(3*A*c*
(c*d^2 + a*e^2) + a*(a*C*e^2 + 3*c*d*(C*d + 2*B*e)))*x^5)/5 + (a^2*c*e*(2*C*d + B*e)*x^6)/2 + (c*(A*c*(c*d^2 +
3*a*e^2) + 3*a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*x^7)/7 + (3*a*c^2*e*(2*C*d + B*e)*x^8)/8 + (c^2*(c*C*d^2 + 3*a*
C*e^2 + c*e*(2*B*d + A*e))*x^9)/9 + (c^3*e*(2*C*d + B*e)*x^10)/10 + (c^3*C*e^2*x^11)/11 + (d*(B*d + 2*A*e)*(a
+ c*x^2)^4)/(8*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)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx &=\frac{d (B d+2 A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (-\left (B d^2+2 A d e\right ) x+(d+e x)^2 \left (A+B x+C x^2\right )\right ) \, dx\\ &=\frac{d (B d+2 A e) \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 A d^2+a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^2+a^3 e (2 C d+B e) x^3+a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^4+3 a^2 c e (2 C d+B e) x^5+c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^6+3 a c^2 e (2 C d+B e) x^7+c^2 \left (c C d^2+3 a C e^2+c e (2 B d+A e)\right ) x^8+c^3 e (2 C d+B e) x^9+c^3 C e^2 x^{10}\right ) \, dx\\ &=a^3 A d^2 x+\frac{1}{3} a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^3+\frac{1}{4} a^3 e (2 C d+B e) x^4+\frac{1}{5} a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^5+\frac{1}{2} a^2 c e (2 C d+B e) x^6+\frac{1}{7} c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^7+\frac{3}{8} a c^2 e (2 C d+B e) x^8+\frac{1}{9} c^2 \left (c C d^2+3 a C e^2+c e (2 B d+A e)\right ) x^9+\frac{1}{10} c^3 e (2 C d+B e) x^{10}+\frac{1}{11} c^3 C e^2 x^{11}+\frac{d (B d+2 A e) \left (a+c x^2\right )^4}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.131723, size = 329, normalized size = 1.14 $\frac{1}{4} a^2 x^4 \left (a B e^2+2 a C d e+6 A c d e+3 B c d^2\right )+\frac{1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+\frac{1}{2} a^3 d x^2 (2 A e+B d)+a^3 A d^2 x+\frac{1}{9} c^2 x^9 \left (3 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac{1}{8} c^2 x^8 \left (3 a B e^2+6 a C d e+2 A c d e+B c d^2\right )+\frac{1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac{1}{2} a c x^6 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+\frac{1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac{1}{10} c^3 e x^{10} (B e+2 C d)+\frac{1}{11} c^3 C e^2 x^{11}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^3*A*d^2*x + (a^3*d*(B*d + 2*A*e)*x^2)/2 + (a^2*(a*d*(C*d + 2*B*e) + A*(3*c*d^2 + a*e^2))*x^3)/3 + (a^2*(3*B*
c*d^2 + 6*A*c*d*e + 2*a*C*d*e + a*B*e^2)*x^4)/4 + (a*(3*A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + 3*c*d*(C*d + 2*B*e)
))*x^5)/5 + (a*c*(2*(A*c + a*C)*d*e + B*(c*d^2 + a*e^2))*x^6)/2 + (c*(A*c*(c*d^2 + 3*a*e^2) + 3*a*(a*C*e^2 + c
*d*(C*d + 2*B*e)))*x^7)/7 + (c^2*(B*c*d^2 + 2*A*c*d*e + 6*a*C*d*e + 3*a*B*e^2)*x^8)/8 + (c^2*(c*C*d^2 + 3*a*C*
e^2 + c*e*(2*B*d + A*e))*x^9)/9 + (c^3*e*(2*C*d + B*e)*x^10)/10 + (c^3*C*e^2*x^11)/11

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Maple [A]  time = 0.044, size = 388, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}C{e}^{2}{x}^{11}}{11}}+{\frac{ \left ({e}^{2}{c}^{3}B+2\,de{c}^{3}C \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 3\,a{c}^{2}{e}^{2}+{c}^{3}{d}^{2} \right ) C+2\,de{c}^{3}B+{e}^{2}{c}^{3}A \right ){x}^{9}}{9}}+{\frac{ \left ( 6\,a{c}^{2}deC+ \left ( 3\,a{c}^{2}{e}^{2}+{c}^{3}{d}^{2} \right ) B+2\,de{c}^{3}A \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 3\,{a}^{2}c{e}^{2}+3\,{d}^{2}a{c}^{2} \right ) C+6\,a{c}^{2}deB+ \left ( 3\,a{c}^{2}{e}^{2}+{c}^{3}{d}^{2} \right ) A \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,de{a}^{2}cC+ \left ( 3\,{a}^{2}c{e}^{2}+3\,{d}^{2}a{c}^{2} \right ) B+6\,a{c}^{2}deA \right ){x}^{6}}{6}}+{\frac{ \left ( \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ) C+6\,de{a}^{2}cB+ \left ( 3\,{a}^{2}c{e}^{2}+3\,{d}^{2}a{c}^{2} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,de{a}^{3}C+ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ) B+6\,de{a}^{2}cA \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}{d}^{2}C+2\,de{a}^{3}B+ \left ({e}^{2}{a}^{3}+3\,{a}^{2}c{d}^{2} \right ) A \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{3}A+{a}^{3}{d}^{2}B \right ){x}^{2}}{2}}+{a}^{3}A{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^3*C*e^2*x^11+1/10*(B*c^3*e^2+2*C*c^3*d*e)*x^10+1/9*((3*a*c^2*e^2+c^3*d^2)*C+2*d*e*c^3*B+e^2*c^3*A)*x^9+
1/8*(6*a*c^2*d*e*C+(3*a*c^2*e^2+c^3*d^2)*B+2*d*e*c^3*A)*x^8+1/7*((3*a^2*c*e^2+3*a*c^2*d^2)*C+6*a*c^2*d*e*B+(3*
a*c^2*e^2+c^3*d^2)*A)*x^7+1/6*(6*d*e*a^2*c*C+(3*a^2*c*e^2+3*a*c^2*d^2)*B+6*a*c^2*d*e*A)*x^6+1/5*((a^3*e^2+3*a^
2*c*d^2)*C+6*d*e*a^2*c*B+(3*a^2*c*e^2+3*a*c^2*d^2)*A)*x^5+1/4*(2*d*e*a^3*C+(a^3*e^2+3*a^2*c*d^2)*B+6*d*e*a^2*c
*A)*x^4+1/3*(a^3*d^2*C+2*d*e*a^3*B+(a^3*e^2+3*a^2*c*d^2)*A)*x^3+1/2*(2*A*a^3*d*e+B*a^3*d^2)*x^2+a^3*A*d^2*x

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Maxima [A]  time = 1.00264, size = 495, normalized size = 1.71 \begin{align*} \frac{1}{11} \, C c^{3} e^{2} x^{11} + \frac{1}{10} \,{\left (2 \, C c^{3} d e + B c^{3} e^{2}\right )} x^{10} + \frac{1}{9} \,{\left (C c^{3} d^{2} + 2 \, B c^{3} d e +{\left (3 \, C a c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{2} + 3 \, B a c^{2} e^{2} + 2 \,{\left (3 \, C a c^{2} + A c^{3}\right )} d e\right )} x^{8} + \frac{1}{7} \,{\left (6 \, B a c^{2} d e +{\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} + 3 \,{\left (C a^{2} c + A a c^{2}\right )} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac{1}{2} \,{\left (B a c^{2} d^{2} + B a^{2} c e^{2} + 2 \,{\left (C a^{2} c + A a c^{2}\right )} d e\right )} x^{6} + \frac{1}{5} \,{\left (6 \, B a^{2} c d e + 3 \,{\left (C a^{2} c + A a c^{2}\right )} d^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d^{2} + B a^{3} e^{2} + 2 \,{\left (C a^{3} + 3 \, A a^{2} c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (2 \, B a^{3} d e + A a^{3} e^{2} +{\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d^{2} + 2 \, A a^{3} d e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*C*c^3*e^2*x^11 + 1/10*(2*C*c^3*d*e + B*c^3*e^2)*x^10 + 1/9*(C*c^3*d^2 + 2*B*c^3*d*e + (3*C*a*c^2 + A*c^3)
*e^2)*x^9 + 1/8*(B*c^3*d^2 + 3*B*a*c^2*e^2 + 2*(3*C*a*c^2 + A*c^3)*d*e)*x^8 + 1/7*(6*B*a*c^2*d*e + (3*C*a*c^2
+ A*c^3)*d^2 + 3*(C*a^2*c + A*a*c^2)*e^2)*x^7 + A*a^3*d^2*x + 1/2*(B*a*c^2*d^2 + B*a^2*c*e^2 + 2*(C*a^2*c + A*
a*c^2)*d*e)*x^6 + 1/5*(6*B*a^2*c*d*e + 3*(C*a^2*c + A*a*c^2)*d^2 + (C*a^3 + 3*A*a^2*c)*e^2)*x^5 + 1/4*(3*B*a^2
*c*d^2 + B*a^3*e^2 + 2*(C*a^3 + 3*A*a^2*c)*d*e)*x^4 + 1/3*(2*B*a^3*d*e + A*a^3*e^2 + (C*a^3 + 3*A*a^2*c)*d^2)*
x^3 + 1/2*(B*a^3*d^2 + 2*A*a^3*d*e)*x^2

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Fricas [A]  time = 1.39606, size = 996, normalized size = 3.45 \begin{align*} \frac{1}{11} x^{11} e^{2} c^{3} C + \frac{1}{5} x^{10} e d c^{3} C + \frac{1}{10} x^{10} e^{2} c^{3} B + \frac{1}{9} x^{9} d^{2} c^{3} C + \frac{1}{3} x^{9} e^{2} c^{2} a C + \frac{2}{9} x^{9} e d c^{3} B + \frac{1}{9} x^{9} e^{2} c^{3} A + \frac{3}{4} x^{8} e d c^{2} a C + \frac{1}{8} x^{8} d^{2} c^{3} B + \frac{3}{8} x^{8} e^{2} c^{2} a B + \frac{1}{4} x^{8} e d c^{3} A + \frac{3}{7} x^{7} d^{2} c^{2} a C + \frac{3}{7} x^{7} e^{2} c a^{2} C + \frac{6}{7} x^{7} e d c^{2} a B + \frac{1}{7} x^{7} d^{2} c^{3} A + \frac{3}{7} x^{7} e^{2} c^{2} a A + x^{6} e d c a^{2} C + \frac{1}{2} x^{6} d^{2} c^{2} a B + \frac{1}{2} x^{6} e^{2} c a^{2} B + x^{6} e d c^{2} a A + \frac{3}{5} x^{5} d^{2} c a^{2} C + \frac{1}{5} x^{5} e^{2} a^{3} C + \frac{6}{5} x^{5} e d c a^{2} B + \frac{3}{5} x^{5} d^{2} c^{2} a A + \frac{3}{5} x^{5} e^{2} c a^{2} A + \frac{1}{2} x^{4} e d a^{3} C + \frac{3}{4} x^{4} d^{2} c a^{2} B + \frac{1}{4} x^{4} e^{2} a^{3} B + \frac{3}{2} x^{4} e d c a^{2} A + \frac{1}{3} x^{3} d^{2} a^{3} C + \frac{2}{3} x^{3} e d a^{3} B + x^{3} d^{2} c a^{2} A + \frac{1}{3} x^{3} e^{2} a^{3} A + \frac{1}{2} x^{2} d^{2} a^{3} B + x^{2} e d a^{3} A + x d^{2} a^{3} A \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e^2*c^3*C + 1/5*x^10*e*d*c^3*C + 1/10*x^10*e^2*c^3*B + 1/9*x^9*d^2*c^3*C + 1/3*x^9*e^2*c^2*a*C + 2/9
*x^9*e*d*c^3*B + 1/9*x^9*e^2*c^3*A + 3/4*x^8*e*d*c^2*a*C + 1/8*x^8*d^2*c^3*B + 3/8*x^8*e^2*c^2*a*B + 1/4*x^8*e
*d*c^3*A + 3/7*x^7*d^2*c^2*a*C + 3/7*x^7*e^2*c*a^2*C + 6/7*x^7*e*d*c^2*a*B + 1/7*x^7*d^2*c^3*A + 3/7*x^7*e^2*c
^2*a*A + x^6*e*d*c*a^2*C + 1/2*x^6*d^2*c^2*a*B + 1/2*x^6*e^2*c*a^2*B + x^6*e*d*c^2*a*A + 3/5*x^5*d^2*c*a^2*C +
1/5*x^5*e^2*a^3*C + 6/5*x^5*e*d*c*a^2*B + 3/5*x^5*d^2*c^2*a*A + 3/5*x^5*e^2*c*a^2*A + 1/2*x^4*e*d*a^3*C + 3/4
*x^4*d^2*c*a^2*B + 1/4*x^4*e^2*a^3*B + 3/2*x^4*e*d*c*a^2*A + 1/3*x^3*d^2*a^3*C + 2/3*x^3*e*d*a^3*B + x^3*d^2*c
*a^2*A + 1/3*x^3*e^2*a^3*A + 1/2*x^2*d^2*a^3*B + x^2*e*d*a^3*A + x*d^2*a^3*A

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Sympy [A]  time = 0.121491, size = 447, normalized size = 1.55 \begin{align*} A a^{3} d^{2} x + \frac{C c^{3} e^{2} x^{11}}{11} + x^{10} \left (\frac{B c^{3} e^{2}}{10} + \frac{C c^{3} d e}{5}\right ) + x^{9} \left (\frac{A c^{3} e^{2}}{9} + \frac{2 B c^{3} d e}{9} + \frac{C a c^{2} e^{2}}{3} + \frac{C c^{3} d^{2}}{9}\right ) + x^{8} \left (\frac{A c^{3} d e}{4} + \frac{3 B a c^{2} e^{2}}{8} + \frac{B c^{3} d^{2}}{8} + \frac{3 C a c^{2} d e}{4}\right ) + x^{7} \left (\frac{3 A a c^{2} e^{2}}{7} + \frac{A c^{3} d^{2}}{7} + \frac{6 B a c^{2} d e}{7} + \frac{3 C a^{2} c e^{2}}{7} + \frac{3 C a c^{2} d^{2}}{7}\right ) + x^{6} \left (A a c^{2} d e + \frac{B a^{2} c e^{2}}{2} + \frac{B a c^{2} d^{2}}{2} + C a^{2} c d e\right ) + x^{5} \left (\frac{3 A a^{2} c e^{2}}{5} + \frac{3 A a c^{2} d^{2}}{5} + \frac{6 B a^{2} c d e}{5} + \frac{C a^{3} e^{2}}{5} + \frac{3 C a^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c d e}{2} + \frac{B a^{3} e^{2}}{4} + \frac{3 B a^{2} c d^{2}}{4} + \frac{C a^{3} d e}{2}\right ) + x^{3} \left (\frac{A a^{3} e^{2}}{3} + A a^{2} c d^{2} + \frac{2 B a^{3} d e}{3} + \frac{C a^{3} d^{2}}{3}\right ) + x^{2} \left (A a^{3} d e + \frac{B a^{3} d^{2}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*d**2*x + C*c**3*e**2*x**11/11 + x**10*(B*c**3*e**2/10 + C*c**3*d*e/5) + x**9*(A*c**3*e**2/9 + 2*B*c**3*
d*e/9 + C*a*c**2*e**2/3 + C*c**3*d**2/9) + x**8*(A*c**3*d*e/4 + 3*B*a*c**2*e**2/8 + B*c**3*d**2/8 + 3*C*a*c**2
*d*e/4) + x**7*(3*A*a*c**2*e**2/7 + A*c**3*d**2/7 + 6*B*a*c**2*d*e/7 + 3*C*a**2*c*e**2/7 + 3*C*a*c**2*d**2/7)
+ x**6*(A*a*c**2*d*e + B*a**2*c*e**2/2 + B*a*c**2*d**2/2 + C*a**2*c*d*e) + x**5*(3*A*a**2*c*e**2/5 + 3*A*a*c**
2*d**2/5 + 6*B*a**2*c*d*e/5 + C*a**3*e**2/5 + 3*C*a**2*c*d**2/5) + x**4*(3*A*a**2*c*d*e/2 + B*a**3*e**2/4 + 3*
B*a**2*c*d**2/4 + C*a**3*d*e/2) + x**3*(A*a**3*e**2/3 + A*a**2*c*d**2 + 2*B*a**3*d*e/3 + C*a**3*d**2/3) + x**2
*(A*a**3*d*e + B*a**3*d**2/2)

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Giac [A]  time = 1.16747, size = 583, normalized size = 2.02 \begin{align*} \frac{1}{11} \, C c^{3} x^{11} e^{2} + \frac{1}{5} \, C c^{3} d x^{10} e + \frac{1}{9} \, C c^{3} d^{2} x^{9} + \frac{1}{10} \, B c^{3} x^{10} e^{2} + \frac{2}{9} \, B c^{3} d x^{9} e + \frac{1}{8} \, B c^{3} d^{2} x^{8} + \frac{1}{3} \, C a c^{2} x^{9} e^{2} + \frac{1}{9} \, A c^{3} x^{9} e^{2} + \frac{3}{4} \, C a c^{2} d x^{8} e + \frac{1}{4} \, A c^{3} d x^{8} e + \frac{3}{7} \, C a c^{2} d^{2} x^{7} + \frac{1}{7} \, A c^{3} d^{2} x^{7} + \frac{3}{8} \, B a c^{2} x^{8} e^{2} + \frac{6}{7} \, B a c^{2} d x^{7} e + \frac{1}{2} \, B a c^{2} d^{2} x^{6} + \frac{3}{7} \, C a^{2} c x^{7} e^{2} + \frac{3}{7} \, A a c^{2} x^{7} e^{2} + C a^{2} c d x^{6} e + A a c^{2} d x^{6} e + \frac{3}{5} \, C a^{2} c d^{2} x^{5} + \frac{3}{5} \, A a c^{2} d^{2} x^{5} + \frac{1}{2} \, B a^{2} c x^{6} e^{2} + \frac{6}{5} \, B a^{2} c d x^{5} e + \frac{3}{4} \, B a^{2} c d^{2} x^{4} + \frac{1}{5} \, C a^{3} x^{5} e^{2} + \frac{3}{5} \, A a^{2} c x^{5} e^{2} + \frac{1}{2} \, C a^{3} d x^{4} e + \frac{3}{2} \, A a^{2} c d x^{4} e + \frac{1}{3} \, C a^{3} d^{2} x^{3} + A a^{2} c d^{2} x^{3} + \frac{1}{4} \, B a^{3} x^{4} e^{2} + \frac{2}{3} \, B a^{3} d x^{3} e + \frac{1}{2} \, B a^{3} d^{2} x^{2} + \frac{1}{3} \, A a^{3} x^{3} e^{2} + A a^{3} d x^{2} e + A a^{3} d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*C*c^3*x^11*e^2 + 1/5*C*c^3*d*x^10*e + 1/9*C*c^3*d^2*x^9 + 1/10*B*c^3*x^10*e^2 + 2/9*B*c^3*d*x^9*e + 1/8*B
*c^3*d^2*x^8 + 1/3*C*a*c^2*x^9*e^2 + 1/9*A*c^3*x^9*e^2 + 3/4*C*a*c^2*d*x^8*e + 1/4*A*c^3*d*x^8*e + 3/7*C*a*c^2
*d^2*x^7 + 1/7*A*c^3*d^2*x^7 + 3/8*B*a*c^2*x^8*e^2 + 6/7*B*a*c^2*d*x^7*e + 1/2*B*a*c^2*d^2*x^6 + 3/7*C*a^2*c*x
^7*e^2 + 3/7*A*a*c^2*x^7*e^2 + C*a^2*c*d*x^6*e + A*a*c^2*d*x^6*e + 3/5*C*a^2*c*d^2*x^5 + 3/5*A*a*c^2*d^2*x^5 +
1/2*B*a^2*c*x^6*e^2 + 6/5*B*a^2*c*d*x^5*e + 3/4*B*a^2*c*d^2*x^4 + 1/5*C*a^3*x^5*e^2 + 3/5*A*a^2*c*x^5*e^2 + 1
/2*C*a^3*d*x^4*e + 3/2*A*a^2*c*d*x^4*e + 1/3*C*a^3*d^2*x^3 + A*a^2*c*d^2*x^3 + 1/4*B*a^3*x^4*e^2 + 2/3*B*a^3*d
*x^3*e + 1/2*B*a^3*d^2*x^2 + 1/3*A*a^3*x^3*e^2 + A*a^3*d*x^2*e + A*a^3*d^2*x