3.317 \(\int x \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx\)

Optimal. Leaf size=97 \[ \frac{1}{2} a^2 c x^2+\frac{1}{3} a^2 d x^3+\frac{1}{4} a^2 e x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{8} b^2 c x^8+\frac{1}{9} b^2 d x^9+\frac{1}{10} b^2 e x^{10} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.145058, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{1}{2} a^2 c x^2+\frac{1}{3} a^2 d x^3+\frac{1}{4} a^2 e x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{8} b^2 c x^8+\frac{1}{9} b^2 d x^9+\frac{1}{10} b^2 e x^{10} \]

Antiderivative was successfully verified.

[In]  Int[x*(c + d*x + e*x^2)*(a + b*x^3)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} c \int x\, dx + \frac{a^{2} e x^{4}}{4} + \frac{2 a b c x^{5}}{5} + \frac{2 a b e x^{7}}{7} + \frac{b^{2} c x^{8}}{8} + \frac{b^{2} e x^{10}}{10} + \frac{d \left (a + b x^{3}\right )^{3}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(e*x**2+d*x+c)*(b*x**3+a)**2,x)

[Out]

a**2*c*Integral(x, x) + a**2*e*x**4/4 + 2*a*b*c*x**5/5 + 2*a*b*e*x**7/7 + b**2*c
*x**8/8 + b**2*e*x**10/10 + d*(a + b*x**3)**3/(9*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00573634, size = 97, normalized size = 1. \[ \frac{1}{2} a^2 c x^2+\frac{1}{3} a^2 d x^3+\frac{1}{4} a^2 e x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{8} b^2 c x^8+\frac{1}{9} b^2 d x^9+\frac{1}{10} b^2 e x^{10} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(c + d*x + e*x^2)*(a + b*x^3)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 80, normalized size = 0.8 \[{\frac{{a}^{2}c{x}^{2}}{2}}+{\frac{{a}^{2}d{x}^{3}}{3}}+{\frac{{a}^{2}e{x}^{4}}{4}}+{\frac{2\,abc{x}^{5}}{5}}+{\frac{abd{x}^{6}}{3}}+{\frac{2\,abe{x}^{7}}{7}}+{\frac{{b}^{2}c{x}^{8}}{8}}+{\frac{{b}^{2}d{x}^{9}}{9}}+{\frac{{b}^{2}e{x}^{10}}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(e*x^2+d*x+c)*(b*x^3+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.37669, size = 107, normalized size = 1.1 \[ \frac{1}{10} \, b^{2} e x^{10} + \frac{1}{9} \, b^{2} d x^{9} + \frac{1}{8} \, b^{2} c x^{8} + \frac{2}{7} \, a b e x^{7} + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} e x^{4} + \frac{1}{3} \, a^{2} d x^{3} + \frac{1}{2} \, a^{2} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(e*x^2 + d*x + c)*x,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.188728, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} e b^{2} + \frac{1}{9} x^{9} d b^{2} + \frac{1}{8} x^{8} c b^{2} + \frac{2}{7} x^{7} e b a + \frac{1}{3} x^{6} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{4} x^{4} e a^{2} + \frac{1}{3} x^{3} d a^{2} + \frac{1}{2} x^{2} c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(e*x^2 + d*x + c)*x,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.069293, size = 94, normalized size = 0.97 \[ \frac{a^{2} c x^{2}}{2} + \frac{a^{2} d x^{3}}{3} + \frac{a^{2} e x^{4}}{4} + \frac{2 a b c x^{5}}{5} + \frac{a b d x^{6}}{3} + \frac{2 a b e x^{7}}{7} + \frac{b^{2} c x^{8}}{8} + \frac{b^{2} d x^{9}}{9} + \frac{b^{2} e x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(e*x**2+d*x+c)*(b*x**3+a)**2,x)

[Out]

a**2*c*x**2/2 + a**2*d*x**3/3 + a**2*e*x**4/4 + 2*a*b*c*x**5/5 + a*b*d*x**6/3 +
2*a*b*e*x**7/7 + b**2*c*x**8/8 + b**2*d*x**9/9 + b**2*e*x**10/10

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208943, size = 111, normalized size = 1.14 \[ \frac{1}{10} \, b^{2} x^{10} e + \frac{1}{9} \, b^{2} d x^{9} + \frac{1}{8} \, b^{2} c x^{8} + \frac{2}{7} \, a b x^{7} e + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} x^{4} e + \frac{1}{3} \, a^{2} d x^{3} + \frac{1}{2} \, a^{2} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(e*x^2 + d*x + c)*x,x, algorithm="giac")

[Out]

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