### 3.34 $$\int (d+e x) (a+c x^2)^3 (A+B x+C x^2) \, dx$$

Optimal. Leaf size=169 $\frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+a^3 A d x+\frac{1}{2} a^2 c C e x^6+\frac{1}{4} a^3 C e x^4+\frac{1}{7} c^2 x^7 (3 a (B e+C d)+A c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{\left (a+c x^2\right )^4 (A e+B d)}{8 c}+\frac{3}{8} a c^2 C e x^8+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10}$

[Out]

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

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Rubi [A]  time = 0.187031, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.08, Rules used = {1582, 1810} $\frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+a^3 A d x+\frac{1}{2} a^2 c C e x^6+\frac{1}{4} a^3 C e x^4+\frac{1}{7} c^2 x^7 (3 a (B e+C d)+A c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{\left (a+c x^2\right )^4 (A e+B d)}{8 c}+\frac{3}{8} a c^2 C e x^8+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0751338, size = 196, normalized size = 1.16 $\frac{1}{4} a^2 x^4 (a C e+3 A c e+3 B c d)+\frac{1}{3} a^2 x^3 (a B e+a C d+3 A c d)+\frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{1}{8} c^2 x^8 (3 a C e+A c e+B c d)+\frac{1}{7} c^2 x^7 (3 a B e+3 a C d+A c d)+\frac{1}{2} a c x^6 (a C e+A c e+B c d)+\frac{3}{5} a c x^5 (a B e+a C d+A c d)+\frac{1}{9} c^3 x^9 (B e+C d)+\frac{1}{10} c^3 C e x^{10}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 223, normalized size = 1.3 \begin{align*}{\frac{{c}^{3}Ce{x}^{10}}{10}}+{\frac{ \left ({c}^{3}eB+{c}^{3}dC \right ){x}^{9}}{9}}+{\frac{ \left ({c}^{3}eA+{c}^{3}dB+3\,ea{c}^{2}C \right ){x}^{8}}{8}}+{\frac{ \left ({c}^{3}dA+3\,ea{c}^{2}B+3\,a{c}^{2}dC \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,ea{c}^{2}A+3\,a{c}^{2}dB+3\,e{a}^{2}cC \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,a{c}^{2}dA+3\,e{a}^{2}cB+3\,cd{a}^{2}C \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,e{a}^{2}cA+3\,cd{a}^{2}B+e{a}^{3}C \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,cd{a}^{2}A+e{a}^{3}B+d{a}^{3}C \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{3}A+d{a}^{3}B \right ){x}^{2}}{2}}+{a}^{3}Adx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.321, size = 300, normalized size = 1.78 \begin{align*} \frac{1}{10} \, C c^{3} e x^{10} + \frac{1}{9} \,{\left (C c^{3} d + B c^{3} e\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d +{\left (3 \, C a c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac{1}{7} \,{\left (3 \, B a c^{2} e +{\left (3 \, C a c^{2} + A c^{3}\right )} d\right )} x^{7} + \frac{1}{2} \,{\left (B a c^{2} d +{\left (C a^{2} c + A a c^{2}\right )} e\right )} x^{6} + A a^{3} d x + \frac{3}{5} \,{\left (B a^{2} c e +{\left (C a^{2} c + A a c^{2}\right )} d\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d +{\left (C a^{3} + 3 \, A a^{2} c\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (B a^{3} e +{\left (C a^{3} + 3 \, A a^{2} c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d + A a^{3} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45735, size = 603, normalized size = 3.57 \begin{align*} \frac{1}{10} x^{10} e c^{3} C + \frac{1}{9} x^{9} d c^{3} C + \frac{1}{9} x^{9} e c^{3} B + \frac{3}{8} x^{8} e c^{2} a C + \frac{1}{8} x^{8} d c^{3} B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} d c^{2} a C + \frac{3}{7} x^{7} e c^{2} a B + \frac{1}{7} x^{7} d c^{3} A + \frac{1}{2} x^{6} e c a^{2} C + \frac{1}{2} x^{6} d c^{2} a B + \frac{1}{2} x^{6} e c^{2} a A + \frac{3}{5} x^{5} d c a^{2} C + \frac{3}{5} x^{5} e c a^{2} B + \frac{3}{5} x^{5} d c^{2} a A + \frac{1}{4} x^{4} e a^{3} C + \frac{3}{4} x^{4} d c a^{2} B + \frac{3}{4} x^{4} e c a^{2} A + \frac{1}{3} x^{3} d a^{3} C + \frac{1}{3} x^{3} e a^{3} B + x^{3} d c a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.101808, size = 265, normalized size = 1.57 \begin{align*} A a^{3} d x + \frac{C c^{3} e x^{10}}{10} + x^{9} \left (\frac{B c^{3} e}{9} + \frac{C c^{3} d}{9}\right ) + x^{8} \left (\frac{A c^{3} e}{8} + \frac{B c^{3} d}{8} + \frac{3 C a c^{2} e}{8}\right ) + x^{7} \left (\frac{A c^{3} d}{7} + \frac{3 B a c^{2} e}{7} + \frac{3 C a c^{2} d}{7}\right ) + x^{6} \left (\frac{A a c^{2} e}{2} + \frac{B a c^{2} d}{2} + \frac{C a^{2} c e}{2}\right ) + x^{5} \left (\frac{3 A a c^{2} d}{5} + \frac{3 B a^{2} c e}{5} + \frac{3 C a^{2} c d}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c e}{4} + \frac{3 B a^{2} c d}{4} + \frac{C a^{3} e}{4}\right ) + x^{3} \left (A a^{2} c d + \frac{B a^{3} e}{3} + \frac{C a^{3} d}{3}\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{B a^{3} d}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14016, size = 352, normalized size = 2.08 \begin{align*} \frac{1}{10} \, C c^{3} x^{10} e + \frac{1}{9} \, C c^{3} d x^{9} + \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{3}{8} \, C a c^{2} x^{8} e + \frac{1}{8} \, A c^{3} x^{8} e + \frac{3}{7} \, C a c^{2} d x^{7} + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B a c^{2} x^{7} e + \frac{1}{2} \, B a c^{2} d x^{6} + \frac{1}{2} \, C a^{2} c x^{6} e + \frac{1}{2} \, A a c^{2} x^{6} e + \frac{3}{5} \, C a^{2} c d x^{5} + \frac{3}{5} \, A a c^{2} d x^{5} + \frac{3}{5} \, B a^{2} c x^{5} e + \frac{3}{4} \, B a^{2} c d x^{4} + \frac{1}{4} \, C a^{3} x^{4} e + \frac{3}{4} \, A a^{2} c x^{4} e + \frac{1}{3} \, C a^{3} d x^{3} + A a^{2} c d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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