3.375 \(\int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx\)

Optimal. Leaf size=153 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{6} a^2 h x^6+\frac{1}{7} b x^7 (2 a f+b c)+\frac{1}{4} a x^4 (a f+2 b c)+\frac{1}{8} b x^8 (2 a g+b d)+\frac{1}{5} a x^5 (a g+2 b d)+\frac{e \left (a+b x^3\right )^3}{9 b}+\frac{2}{9} a b h x^9+\frac{1}{10} b^2 f x^{10}+\frac{1}{11} b^2 g x^{11}+\frac{1}{12} b^2 h x^{12} \]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a*(2*b*c + a*f)*x^4)/4 + (a*(2*b*d + a*g)*x^5)/5 + (a
^2*h*x^6)/6 + (b*(b*c + 2*a*f)*x^7)/7 + (b*(b*d + 2*a*g)*x^8)/8 + (2*a*b*h*x^9)/
9 + (b^2*f*x^10)/10 + (b^2*g*x^11)/11 + (b^2*h*x^12)/12 + (e*(a + b*x^3)^3)/(9*b
)

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Rubi [A]  time = 0.349921, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{6} a^2 h x^6+\frac{1}{7} b x^7 (2 a f+b c)+\frac{1}{4} a x^4 (a f+2 b c)+\frac{1}{8} b x^8 (2 a g+b d)+\frac{1}{5} a x^5 (a g+2 b d)+\frac{e \left (a+b x^3\right )^3}{9 b}+\frac{2}{9} a b h x^9+\frac{1}{10} b^2 f x^{10}+\frac{1}{11} b^2 g x^{11}+\frac{1}{12} b^2 h x^{12} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a*(2*b*c + a*f)*x^4)/4 + (a*(2*b*d + a*g)*x^5)/5 + (a
^2*h*x^6)/6 + (b*(b*c + 2*a*f)*x^7)/7 + (b*(b*d + 2*a*g)*x^8)/8 + (2*a*b*h*x^9)/
9 + (b^2*f*x^10)/10 + (b^2*g*x^11)/11 + (b^2*h*x^12)/12 + (e*(a + b*x^3)^3)/(9*b
)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} d \int x\, dx + \frac{a^{2} h x^{6}}{6} + a^{2} \int c\, dx + \frac{2 a b h x^{9}}{9} + \frac{a x^{5} \left (a g + 2 b d\right )}{5} + \frac{a x^{4} \left (a f + 2 b c\right )}{4} + \frac{b^{2} f x^{10}}{10} + \frac{b^{2} g x^{11}}{11} + \frac{b^{2} h x^{12}}{12} + \frac{b x^{8} \left (2 a g + b d\right )}{8} + \frac{b x^{7} \left (2 a f + b c\right )}{7} + \frac{e \left (a + b x^{3}\right )^{3}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)

[Out]

a**2*d*Integral(x, x) + a**2*h*x**6/6 + a**2*Integral(c, x) + 2*a*b*h*x**9/9 + a
*x**5*(a*g + 2*b*d)/5 + a*x**4*(a*f + 2*b*c)/4 + b**2*f*x**10/10 + b**2*g*x**11/
11 + b**2*h*x**12/12 + b*x**8*(2*a*g + b*d)/8 + b*x**7*(2*a*f + b*c)/7 + e*(a +
b*x**3)**3/(9*b)

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Mathematica [A]  time = 0.172723, size = 125, normalized size = 0.82 \[ \frac{462 a^2 x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+22 a b x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+b^2 x^7 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )}{27720} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

(b^2*x^7*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(66*f + 60*g*x + 55*h*x^2))) + 4
62*a^2*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))) + 22*a*b*x^
4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x))))))/27720

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Maple [A]  time = 0., size = 149, normalized size = 1. \[{\frac{{b}^{2}h{x}^{12}}{12}}+{\frac{{b}^{2}g{x}^{11}}{11}}+{\frac{{b}^{2}f{x}^{10}}{10}}+{\frac{ \left ( 2\,abh+{b}^{2}e \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,abg+{b}^{2}d \right ){x}^{8}}{8}}+{\frac{ \left ( 2\,abf+{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{2}h+2\,bea \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}g+2\,bda \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}f+2\,abc \right ){x}^{4}}{4}}+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{2}}{2}}+{a}^{2}cx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x)

[Out]

1/12*b^2*h*x^12+1/11*b^2*g*x^11+1/10*b^2*f*x^10+1/9*(2*a*b*h+b^2*e)*x^9+1/8*(2*a
*b*g+b^2*d)*x^8+1/7*(2*a*b*f+b^2*c)*x^7+1/6*(a^2*h+2*a*b*e)*x^6+1/5*(a^2*g+2*a*b
*d)*x^5+1/4*(a^2*f+2*a*b*c)*x^4+1/3*a^2*e*x^3+1/2*a^2*d*x^2+a^2*c*x

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Maxima [A]  time = 1.37888, size = 200, normalized size = 1.31 \[ \frac{1}{12} \, b^{2} h x^{12} + \frac{1}{11} \, b^{2} g x^{11} + \frac{1}{10} \, b^{2} f x^{10} + \frac{1}{9} \,{\left (b^{2} e + 2 \, a b h\right )} x^{9} + \frac{1}{8} \,{\left (b^{2} d + 2 \, a b g\right )} x^{8} + \frac{1}{7} \,{\left (b^{2} c + 2 \, a b f\right )} x^{7} + \frac{1}{6} \,{\left (2 \, a b e + a^{2} h\right )} x^{6} + \frac{1}{3} \, a^{2} e x^{3} + \frac{1}{5} \,{\left (2 \, a b d + a^{2} g\right )} x^{5} + \frac{1}{2} \, a^{2} d x^{2} + \frac{1}{4} \,{\left (2 \, a b c + a^{2} f\right )} x^{4} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

1/12*b^2*h*x^12 + 1/11*b^2*g*x^11 + 1/10*b^2*f*x^10 + 1/9*(b^2*e + 2*a*b*h)*x^9
+ 1/8*(b^2*d + 2*a*b*g)*x^8 + 1/7*(b^2*c + 2*a*b*f)*x^7 + 1/6*(2*a*b*e + a^2*h)*
x^6 + 1/3*a^2*e*x^3 + 1/5*(2*a*b*d + a^2*g)*x^5 + 1/2*a^2*d*x^2 + 1/4*(2*a*b*c +
a^2*f)*x^4 + a^2*c*x

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Fricas [A]  time = 0.216842, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} h b^{2} + \frac{1}{11} x^{11} g b^{2} + \frac{1}{10} x^{10} f b^{2} + \frac{1}{9} x^{9} e b^{2} + \frac{2}{9} x^{9} h b a + \frac{1}{8} x^{8} d b^{2} + \frac{1}{4} x^{8} g b a + \frac{1}{7} x^{7} c b^{2} + \frac{2}{7} x^{7} f b a + \frac{1}{3} x^{6} e b a + \frac{1}{6} x^{6} h a^{2} + \frac{2}{5} x^{5} d b a + \frac{1}{5} x^{5} g a^{2} + \frac{1}{2} x^{4} c b a + \frac{1}{4} x^{4} f a^{2} + \frac{1}{3} x^{3} e a^{2} + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/12*x^12*h*b^2 + 1/11*x^11*g*b^2 + 1/10*x^10*f*b^2 + 1/9*x^9*e*b^2 + 2/9*x^9*h*
b*a + 1/8*x^8*d*b^2 + 1/4*x^8*g*b*a + 1/7*x^7*c*b^2 + 2/7*x^7*f*b*a + 1/3*x^6*e*
b*a + 1/6*x^6*h*a^2 + 2/5*x^5*d*b*a + 1/5*x^5*g*a^2 + 1/2*x^4*c*b*a + 1/4*x^4*f*
a^2 + 1/3*x^3*e*a^2 + 1/2*x^2*d*a^2 + x*c*a^2

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Sympy [A]  time = 0.091981, size = 163, normalized size = 1.07 \[ a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e x^{3}}{3} + \frac{b^{2} f x^{10}}{10} + \frac{b^{2} g x^{11}}{11} + \frac{b^{2} h x^{12}}{12} + x^{9} \left (\frac{2 a b h}{9} + \frac{b^{2} e}{9}\right ) + x^{8} \left (\frac{a b g}{4} + \frac{b^{2} d}{8}\right ) + x^{7} \left (\frac{2 a b f}{7} + \frac{b^{2} c}{7}\right ) + x^{6} \left (\frac{a^{2} h}{6} + \frac{a b e}{3}\right ) + x^{5} \left (\frac{a^{2} g}{5} + \frac{2 a b d}{5}\right ) + x^{4} \left (\frac{a^{2} f}{4} + \frac{a b c}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)

[Out]

a**2*c*x + a**2*d*x**2/2 + a**2*e*x**3/3 + b**2*f*x**10/10 + b**2*g*x**11/11 + b
**2*h*x**12/12 + x**9*(2*a*b*h/9 + b**2*e/9) + x**8*(a*b*g/4 + b**2*d/8) + x**7*
(2*a*b*f/7 + b**2*c/7) + x**6*(a**2*h/6 + a*b*e/3) + x**5*(a**2*g/5 + 2*a*b*d/5)
+ x**4*(a**2*f/4 + a*b*c/2)

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GIAC/XCAS [A]  time = 0.2079, size = 212, normalized size = 1.39 \[ \frac{1}{12} \, b^{2} h x^{12} + \frac{1}{11} \, b^{2} g x^{11} + \frac{1}{10} \, b^{2} f x^{10} + \frac{2}{9} \, a b h x^{9} + \frac{1}{9} \, b^{2} x^{9} e + \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{4} \, a b g x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{7} \, a b f x^{7} + \frac{1}{6} \, a^{2} h x^{6} + \frac{1}{3} \, a b x^{6} e + \frac{2}{5} \, a b d x^{5} + \frac{1}{5} \, a^{2} g x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} x^{3} e + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((h*x^5 + g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^2,x, algorithm="giac")

[Out]

1/12*b^2*h*x^12 + 1/11*b^2*g*x^11 + 1/10*b^2*f*x^10 + 2/9*a*b*h*x^9 + 1/9*b^2*x^
9*e + 1/8*b^2*d*x^8 + 1/4*a*b*g*x^8 + 1/7*b^2*c*x^7 + 2/7*a*b*f*x^7 + 1/6*a^2*h*
x^6 + 1/3*a*b*x^6*e + 2/5*a*b*d*x^5 + 1/5*a^2*g*x^5 + 1/2*a*b*c*x^4 + 1/4*a^2*f*
x^4 + 1/3*a^2*x^3*e + 1/2*a^2*d*x^2 + a^2*c*x