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

Optimal. Leaf size=217 \[ a^2 A d^2 x+\frac{1}{4} a^2 e x^4 (B e+2 C d)+\frac{1}{7} c x^7 \left (2 a C e^2+c \left (e (A e+2 B d)+C d^2\right )\right )+\frac{1}{5} x^5 \left (A c \left (2 a e^2+c d^2\right )+a \left (a C e^2+2 c d (2 B e+C d)\right )\right )+\frac{1}{3} a x^3 \left (A \left (a e^2+2 c d^2\right )+a d (2 B e+C d)\right )+\frac{d \left (a+c x^2\right )^3 (2 A e+B d)}{6 c}+\frac{1}{3} a c e x^6 (B e+2 C d)+\frac{1}{8} c^2 e x^8 (B e+2 C d)+\frac{1}{9} c^2 C e^2 x^9 \]

[Out]

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

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Rubi [A]  time = 0.313161, antiderivative size = 216, 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} \[ a^2 A d^2 x+\frac{1}{4} a^2 e x^4 (B e+2 C d)+\frac{1}{7} c x^7 \left (2 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac{1}{5} x^5 \left (A c \left (2 a e^2+c d^2\right )+a \left (a C e^2+2 c d (2 B e+C d)\right )\right )+\frac{1}{3} a x^3 \left (A \left (a e^2+2 c d^2\right )+a d (2 B e+C d)\right )+\frac{d \left (a+c x^2\right )^3 (2 A e+B d)}{6 c}+\frac{1}{3} a c e x^6 (B e+2 C d)+\frac{1}{8} c^2 e x^8 (B e+2 C d)+\frac{1}{9} c^2 C e^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.091812, size = 241, normalized size = 1.11 \[ \frac{1}{2} a^2 d x^2 (2 A e+B d)+a^2 A d^2 x+\frac{1}{7} c x^7 \left (2 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac{1}{6} c x^6 \left (2 a B e^2+4 a C d e+2 A c d e+B c d^2\right )+\frac{1}{5} x^5 \left (A c \left (2 a e^2+c d^2\right )+a \left (a C e^2+2 c d (2 B e+C d)\right )\right )+\frac{1}{4} a x^4 \left (a B e^2+2 a C d e+4 A c d e+2 B c d^2\right )+\frac{1}{3} a x^3 \left (A \left (a e^2+2 c d^2\right )+a d (2 B e+C d)\right )+\frac{1}{8} c^2 e x^8 (B e+2 C d)+\frac{1}{9} c^2 C e^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 268, normalized size = 1.2 \begin{align*}{\frac{{c}^{2}C{e}^{2}{x}^{9}}{9}}+{\frac{ \left ({c}^{2}{e}^{2}B+2\,{c}^{2}deC \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 2\,ac{e}^{2}+{c}^{2}{d}^{2} \right ) C+2\,{c}^{2}deB+{c}^{2}{e}^{2}A \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,acdeC+ \left ( 2\,ac{e}^{2}+{c}^{2}{d}^{2} \right ) B+2\,{c}^{2}deA \right ){x}^{6}}{6}}+{\frac{ \left ( \left ({a}^{2}{e}^{2}+2\,ac{d}^{2} \right ) C+4\,Bacde+ \left ( 2\,ac{e}^{2}+{c}^{2}{d}^{2} \right ) A \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,de{a}^{2}C+ \left ({a}^{2}{e}^{2}+2\,ac{d}^{2} \right ) B+4\,acdeA \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2}{d}^{2}C+2\,de{a}^{2}B+ \left ({a}^{2}{e}^{2}+2\,ac{d}^{2} \right ) A \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,de{a}^{2}A+{a}^{2}{d}^{2}B \right ){x}^{2}}{2}}+{a}^{2}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)^2*(C*x^2+B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.00544, size = 347, normalized size = 1.6 \begin{align*} \frac{1}{9} \, C c^{2} e^{2} x^{9} + \frac{1}{8} \,{\left (2 \, C c^{2} d e + B c^{2} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left (C c^{2} d^{2} + 2 \, B c^{2} d e +{\left (2 \, C a c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{2} + 2 \, B a c e^{2} + 2 \,{\left (2 \, C a c + A c^{2}\right )} d e\right )} x^{6} + A a^{2} d^{2} x + \frac{1}{5} \,{\left (4 \, B a c d e +{\left (2 \, C a c + A c^{2}\right )} d^{2} +{\left (C a^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, B a c d^{2} + B a^{2} e^{2} + 2 \,{\left (C a^{2} + 2 \, A a c\right )} d e\right )} x^{4} + \frac{1}{3} \,{\left (2 \, B a^{2} d e + A a^{2} e^{2} +{\left (C a^{2} + 2 \, A a c\right )} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{2} + 2 \, A a^{2} 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)^2*(C*x^2+B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.40483, size = 709, normalized size = 3.27 \begin{align*} \frac{1}{9} x^{9} e^{2} c^{2} C + \frac{1}{4} x^{8} e d c^{2} C + \frac{1}{8} x^{8} e^{2} c^{2} B + \frac{1}{7} x^{7} d^{2} c^{2} C + \frac{2}{7} x^{7} e^{2} c a C + \frac{2}{7} x^{7} e d c^{2} B + \frac{1}{7} x^{7} e^{2} c^{2} A + \frac{2}{3} x^{6} e d c a C + \frac{1}{6} x^{6} d^{2} c^{2} B + \frac{1}{3} x^{6} e^{2} c a B + \frac{1}{3} x^{6} e d c^{2} A + \frac{2}{5} x^{5} d^{2} c a C + \frac{1}{5} x^{5} e^{2} a^{2} C + \frac{4}{5} x^{5} e d c a B + \frac{1}{5} x^{5} d^{2} c^{2} A + \frac{2}{5} x^{5} e^{2} c a A + \frac{1}{2} x^{4} e d a^{2} C + \frac{1}{2} x^{4} d^{2} c a B + \frac{1}{4} x^{4} e^{2} a^{2} B + x^{4} e d c a A + \frac{1}{3} x^{3} d^{2} a^{2} C + \frac{2}{3} x^{3} e d a^{2} B + \frac{2}{3} x^{3} d^{2} c a A + \frac{1}{3} x^{3} e^{2} a^{2} A + \frac{1}{2} x^{2} d^{2} a^{2} B + x^{2} e d a^{2} A + x d^{2} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.104901, size = 311, normalized size = 1.43 \begin{align*} A a^{2} d^{2} x + \frac{C c^{2} e^{2} x^{9}}{9} + x^{8} \left (\frac{B c^{2} e^{2}}{8} + \frac{C c^{2} d e}{4}\right ) + x^{7} \left (\frac{A c^{2} e^{2}}{7} + \frac{2 B c^{2} d e}{7} + \frac{2 C a c e^{2}}{7} + \frac{C c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac{A c^{2} d e}{3} + \frac{B a c e^{2}}{3} + \frac{B c^{2} d^{2}}{6} + \frac{2 C a c d e}{3}\right ) + x^{5} \left (\frac{2 A a c e^{2}}{5} + \frac{A c^{2} d^{2}}{5} + \frac{4 B a c d e}{5} + \frac{C a^{2} e^{2}}{5} + \frac{2 C a c d^{2}}{5}\right ) + x^{4} \left (A a c d e + \frac{B a^{2} e^{2}}{4} + \frac{B a c d^{2}}{2} + \frac{C a^{2} d e}{2}\right ) + x^{3} \left (\frac{A a^{2} e^{2}}{3} + \frac{2 A a c d^{2}}{3} + \frac{2 B a^{2} d e}{3} + \frac{C a^{2} d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + \frac{B a^{2} 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)**2*(C*x**2+B*x+A),x)

[Out]

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

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Giac [A]  time = 1.12811, size = 408, normalized size = 1.88 \begin{align*} \frac{1}{9} \, C c^{2} x^{9} e^{2} + \frac{1}{4} \, C c^{2} d x^{8} e + \frac{1}{7} \, C c^{2} d^{2} x^{7} + \frac{1}{8} \, B c^{2} x^{8} e^{2} + \frac{2}{7} \, B c^{2} d x^{7} e + \frac{1}{6} \, B c^{2} d^{2} x^{6} + \frac{2}{7} \, C a c x^{7} e^{2} + \frac{1}{7} \, A c^{2} x^{7} e^{2} + \frac{2}{3} \, C a c d x^{6} e + \frac{1}{3} \, A c^{2} d x^{6} e + \frac{2}{5} \, C a c d^{2} x^{5} + \frac{1}{5} \, A c^{2} d^{2} x^{5} + \frac{1}{3} \, B a c x^{6} e^{2} + \frac{4}{5} \, B a c d x^{5} e + \frac{1}{2} \, B a c d^{2} x^{4} + \frac{1}{5} \, C a^{2} x^{5} e^{2} + \frac{2}{5} \, A a c x^{5} e^{2} + \frac{1}{2} \, C a^{2} d x^{4} e + A a c d x^{4} e + \frac{1}{3} \, C a^{2} d^{2} x^{3} + \frac{2}{3} \, A a c d^{2} x^{3} + \frac{1}{4} \, B a^{2} x^{4} e^{2} + \frac{2}{3} \, B a^{2} d x^{3} e + \frac{1}{2} \, B a^{2} d^{2} x^{2} + \frac{1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} 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)^2*(C*x^2+B*x+A),x, algorithm="giac")

[Out]

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