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

Optimal. Leaf size=86 \[ \frac{1}{4} x^4 (a C e+A c e+B c d)+\frac{1}{3} x^3 (a B e+a C d+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{5} c x^5 (B e+C d)+\frac{1}{6} c C e x^6 \]

[Out]

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

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Rubi [A]  time = 0.106489, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1628} \[ \frac{1}{4} x^4 (a C e+A c e+B c d)+\frac{1}{3} x^3 (a B e+a C d+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{5} c x^5 (B e+C d)+\frac{1}{6} c C e x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.0288402, size = 86, normalized size = 1. \[ \frac{1}{4} x^4 (a C e+A c e+B c d)+\frac{1}{3} x^3 (a B e+a C d+A c d)+\frac{1}{2} a x^2 (A e+B d)+a A d x+\frac{1}{5} c x^5 (B e+C d)+\frac{1}{6} c C e x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 79, normalized size = 0.9 \begin{align*}{\frac{cCe{x}^{6}}{6}}+{\frac{ \left ( ceB+cdC \right ){x}^{5}}{5}}+{\frac{ \left ( Ace+Bcd+aCe \right ){x}^{4}}{4}}+{\frac{ \left ( Acd+aBe+Cad \right ){x}^{3}}{3}}+{\frac{ \left ( aAe+adB \right ){x}^{2}}{2}}+aAdx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981304, size = 108, normalized size = 1.26 \begin{align*} \frac{1}{6} \, C c e x^{6} + \frac{1}{5} \,{\left (C c d + B c e\right )} x^{5} + \frac{1}{4} \,{\left (B c d +{\left (C a + A c\right )} e\right )} x^{4} + A a d x + \frac{1}{3} \,{\left (B a e +{\left (C a + A c\right )} d\right )} x^{3} + \frac{1}{2} \,{\left (B a d + A a 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)*(C*x^2+B*x+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46919, size = 250, normalized size = 2.91 \begin{align*} \frac{1}{6} x^{6} e c C + \frac{1}{5} x^{5} d c C + \frac{1}{5} x^{5} e c B + \frac{1}{4} x^{4} e a C + \frac{1}{4} x^{4} d c B + \frac{1}{4} x^{4} e c A + \frac{1}{3} x^{3} d a C + \frac{1}{3} x^{3} e a B + \frac{1}{3} x^{3} d c A + \frac{1}{2} x^{2} d a B + \frac{1}{2} x^{2} e a A + x d a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.070829, size = 97, normalized size = 1.13 \begin{align*} A a d x + \frac{C c e x^{6}}{6} + x^{5} \left (\frac{B c e}{5} + \frac{C c d}{5}\right ) + x^{4} \left (\frac{A c e}{4} + \frac{B c d}{4} + \frac{C a e}{4}\right ) + x^{3} \left (\frac{A c d}{3} + \frac{B a e}{3} + \frac{C a d}{3}\right ) + x^{2} \left (\frac{A a e}{2} + \frac{B a d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16253, size = 135, normalized size = 1.57 \begin{align*} \frac{1}{6} \, C c x^{6} e + \frac{1}{5} \, C c d x^{5} + \frac{1}{5} \, B c x^{5} e + \frac{1}{4} \, B c d x^{4} + \frac{1}{4} \, C a x^{4} e + \frac{1}{4} \, A c x^{4} e + \frac{1}{3} \, C a d x^{3} + \frac{1}{3} \, A c d x^{3} + \frac{1}{3} \, B a x^{3} e + \frac{1}{2} \, B a d x^{2} + \frac{1}{2} \, A a x^{2} e + A a d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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