\(\int (d+e x) (a+c x^2) (A+B x+C x^2) \, dx\) [20]

Optimal result

Integrand size = 23, antiderivative size = 86 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=a A d x+\frac {1}{2} a (B d+A e) x^2+\frac {1}{3} (A c d+a C d+a B e) x^3+\frac {1}{4} (B c d+A c e+a C e) x^4+\frac {1}{5} c (C d+B e) x^5+\frac {1}{6} c C e x^6 \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1642} \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{4} x^4 (a C e+A c e+B c d)+\frac {1}{3} x^3 (a B e+a C d+A c d)+\frac {1}{2} a x^2 (A e+B d)+a A d x+\frac {1}{5} c x^5 (B e+C d)+\frac {1}{6} c C e x^6 \]

[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 1642

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=a A d x+\frac {1}{2} a (B d+A e) x^2+\frac {1}{3} (A c d+a C d+a B e) x^3+\frac {1}{4} (B c d+A c e+a C e) x^4+\frac {1}{5} c (C d+B e) x^5+\frac {1}{6} c C e x^6 \]

[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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {c C e \,x^{6}}{6}+\frac {\left (B c e +c d C \right ) x^{5}}{5}+\frac {\left (A c e +B c d +C a e \right ) x^{4}}{4}+\frac {\left (A c d +B a e +C a d \right ) x^{3}}{3}+\frac {\left (a A e +B a d \right ) x^{2}}{2}+a A d x\)	\(79\)
norman	\(\frac {c C e \,x^{6}}{6}+\left (\frac {1}{5} B c e +\frac {1}{5} c d C \right ) x^{5}+\left (\frac {1}{4} A c e +\frac {1}{4} B c d +\frac {1}{4} C a e \right ) x^{4}+\left (\frac {1}{3} A c d +\frac {1}{3} B a e +\frac {1}{3} C a d \right ) x^{3}+\left (\frac {1}{2} a A e +\frac {1}{2} B a d \right ) x^{2}+a A d x\)	\(85\)
gosper	\(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} B c e \,x^{5}+\frac {1}{5} x^{5} c d C +\frac {1}{4} x^{4} A c e +\frac {1}{4} x^{4} B c d +\frac {1}{4} x^{4} C a e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} a A e +\frac {1}{2} x^{2} B a d +a A d x\)	\(95\)
risch	\(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} B c e \,x^{5}+\frac {1}{5} x^{5} c d C +\frac {1}{4} x^{4} A c e +\frac {1}{4} x^{4} B c d +\frac {1}{4} x^{4} C a e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} a A e +\frac {1}{2} x^{2} B a d +a A d x\)	\(95\)
parallelrisch	\(\frac {1}{6} c C e \,x^{6}+\frac {1}{5} B c e \,x^{5}+\frac {1}{5} x^{5} c d C +\frac {1}{4} x^{4} A c e +\frac {1}{4} x^{4} B c d +\frac {1}{4} x^{4} C a e +\frac {1}{3} x^{3} A c d +\frac {1}{3} x^{3} B a e +\frac {1}{3} x^{3} C a d +\frac {1}{2} x^{2} a A e +\frac {1}{2} x^{2} B a d +a A d x\)	\(95\)

[In]

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, {\left (C c d + B c e\right )} x^{5} + \frac {1}{4} \, {\left (B c d + {\left (C a + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left (B a e + {\left (C a + A c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (B a d + A a e\right )} x^{2} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=A a d x + \frac {C c e x^{6}}{6} + x^{5} \left (\frac {B c e}{5} + \frac {C c d}{5}\right ) + x^{4} \left (\frac {A c e}{4} + \frac {B c d}{4} + \frac {C a e}{4}\right ) + x^{3} \left (\frac {A c d}{3} + \frac {B a e}{3} + \frac {C a d}{3}\right ) + x^{2} \left (\frac {A a e}{2} + \frac {B a d}{2}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, {\left (C c d + B c e\right )} x^{5} + \frac {1}{4} \, {\left (B c d + {\left (C a + A c\right )} e\right )} x^{4} + A a d x + \frac {1}{3} \, {\left (B a e + {\left (C a + A c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (B a d + A a e\right )} x^{2} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {1}{6} \, C c e x^{6} + \frac {1}{5} \, C c d x^{5} + \frac {1}{5} \, B c e x^{5} + \frac {1}{4} \, B c d x^{4} + \frac {1}{4} \, C a e x^{4} + \frac {1}{4} \, A c e x^{4} + \frac {1}{3} \, C a d x^{3} + \frac {1}{3} \, A c d x^{3} + \frac {1}{3} \, B a e x^{3} + \frac {1}{2} \, B a d x^{2} + \frac {1}{2} \, A a e x^{2} + A a d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.59 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+c x^2\right ) \left (A+B x+C x^2\right ) \, dx=\frac {C\,c\,e\,x^6}{6}+\frac {c\,\left (B\,e+C\,d\right )\,x^5}{5}+\left (\frac {A\,c\,e}{4}+\frac {B\,c\,d}{4}+\frac {C\,a\,e}{4}\right )\,x^4+\left (\frac {A\,c\,d}{3}+\frac {B\,a\,e}{3}+\frac {C\,a\,d}{3}\right )\,x^3+\frac {a\,\left (A\,e+B\,d\right )\,x^2}{2}+A\,a\,d\,x \]

[In]

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

[Out]

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