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

Optimal. Leaf size=212 \[ \frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{1}{6} a^3 g x^6+\frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{3} a^2 b g x^9+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{1}{4} a b^2 g x^{12}+\frac{3}{8} a b x^8 (a f+b c)+\frac{d \left (a+b x^3\right )^4}{12 b}+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16} \]

[Out]

(a^3*c*x^2)/2 + (a^3*e*x^4)/4 + (a^2*(3*b*c + a*f)*x^5)/5 + (a^3*g*x^6)/6 + (a^2
*(3*b*e + a*h)*x^7)/7 + (3*a*b*(b*c + a*f)*x^8)/8 + (a^2*b*g*x^9)/3 + (3*a*b*(b*
e + a*h)*x^10)/10 + (b^2*(b*c + 3*a*f)*x^11)/11 + (a*b^2*g*x^12)/4 + (b^2*(b*e +
3*a*h)*x^13)/13 + (b^3*f*x^14)/14 + (b^3*g*x^15)/15 + (b^3*h*x^16)/16 + (d*(a +
b*x^3)^4)/(12*b)

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Rubi [A]  time = 0.570306, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{1}{2} a^3 c x^2+\frac{1}{4} a^3 e x^4+\frac{1}{6} a^3 g x^6+\frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{3} a^2 b g x^9+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{1}{4} a b^2 g x^{12}+\frac{3}{8} a b x^8 (a f+b c)+\frac{d \left (a+b x^3\right )^4}{12 b}+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*c*x^2)/2 + (a^3*e*x^4)/4 + (a^2*(3*b*c + a*f)*x^5)/5 + (a^3*g*x^6)/6 + (a^2
*(3*b*e + a*h)*x^7)/7 + (3*a*b*(b*c + a*f)*x^8)/8 + (a^2*b*g*x^9)/3 + (3*a*b*(b*
e + a*h)*x^10)/10 + (b^2*(b*c + 3*a*f)*x^11)/11 + (a*b^2*g*x^12)/4 + (b^2*(b*e +
3*a*h)*x^13)/13 + (b^3*f*x^14)/14 + (b^3*g*x^15)/15 + (b^3*h*x^16)/16 + (d*(a +
b*x^3)^4)/(12*b)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{3} c \int x\, dx + \frac{a^{3} e x^{4}}{4} + \frac{a^{3} g x^{6}}{6} + \frac{a^{2} b g x^{9}}{3} + \frac{a^{2} x^{7} \left (a h + 3 b e\right )}{7} + \frac{a^{2} x^{5} \left (a f + 3 b c\right )}{5} + \frac{a b^{2} g x^{12}}{4} + \frac{3 a b x^{10} \left (a h + b e\right )}{10} + \frac{3 a b x^{8} \left (a f + b c\right )}{8} + \frac{b^{3} f x^{14}}{14} + \frac{b^{3} g x^{15}}{15} + \frac{b^{3} h x^{16}}{16} + \frac{b^{2} x^{13} \left (3 a h + b e\right )}{13} + \frac{b^{2} x^{11} \left (3 a f + b c\right )}{11} + \frac{d \left (a + b x^{3}\right )^{4}}{12 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*c*Integral(x, x) + a**3*e*x**4/4 + a**3*g*x**6/6 + a**2*b*g*x**9/3 + a**2*x
**7*(a*h + 3*b*e)/7 + a**2*x**5*(a*f + 3*b*c)/5 + a*b**2*g*x**12/4 + 3*a*b*x**10
*(a*h + b*e)/10 + 3*a*b*x**8*(a*f + b*c)/8 + b**3*f*x**14/14 + b**3*g*x**15/15 +
b**3*h*x**16/16 + b**2*x**13*(3*a*h + b*e)/13 + b**2*x**11*(3*a*f + b*c)/11 + d
*(a + b*x**3)**4/(12*b)

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Mathematica [A]  time = 0.0910579, size = 223, normalized size = 1.05 \[ \frac{1}{2} a^3 c x^2+\frac{1}{3} a^3 d x^3+\frac{1}{4} a^3 e x^4+\frac{1}{5} a^2 x^5 (a f+3 b c)+\frac{1}{6} a^2 x^6 (a g+3 b d)+\frac{1}{7} a^2 x^7 (a h+3 b e)+\frac{1}{11} b^2 x^{11} (3 a f+b c)+\frac{1}{12} b^2 x^{12} (3 a g+b d)+\frac{1}{13} b^2 x^{13} (3 a h+b e)+\frac{3}{8} a b x^8 (a f+b c)+\frac{1}{3} a b x^9 (a g+b d)+\frac{3}{10} a b x^{10} (a h+b e)+\frac{1}{14} b^3 f x^{14}+\frac{1}{15} b^3 g x^{15}+\frac{1}{16} b^3 h x^{16} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*c*x^2)/2 + (a^3*d*x^3)/3 + (a^3*e*x^4)/4 + (a^2*(3*b*c + a*f)*x^5)/5 + (a^2
*(3*b*d + a*g)*x^6)/6 + (a^2*(3*b*e + a*h)*x^7)/7 + (3*a*b*(b*c + a*f)*x^8)/8 +
(a*b*(b*d + a*g)*x^9)/3 + (3*a*b*(b*e + a*h)*x^10)/10 + (b^2*(b*c + 3*a*f)*x^11)
/11 + (b^2*(b*d + 3*a*g)*x^12)/12 + (b^2*(b*e + 3*a*h)*x^13)/13 + (b^3*f*x^14)/1
4 + (b^3*g*x^15)/15 + (b^3*h*x^16)/16

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Maple [A]  time = 0.002, size = 224, normalized size = 1.1 \[{\frac{{b}^{3}h{x}^{16}}{16}}+{\frac{{b}^{3}g{x}^{15}}{15}}+{\frac{{b}^{3}f{x}^{14}}{14}}+{\frac{ \left ( 3\,a{b}^{2}h+{b}^{3}e \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,a{b}^{2}g+{b}^{3}d \right ){x}^{12}}{12}}+{\frac{ \left ( 3\,a{b}^{2}f+{b}^{3}c \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,{a}^{2}bh+3\,ae{b}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{a}^{2}bg+3\,a{b}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}bf+3\,ac{b}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ({a}^{3}h+3\,{a}^{2}be \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{3}g+3\,{a}^{2}bd \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{3}f+3\,{a}^{2}bc \right ){x}^{5}}{5}}+{\frac{{a}^{3}e{x}^{4}}{4}}+{\frac{{a}^{3}d{x}^{3}}{3}}+{\frac{{a}^{3}c{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b^3*h*x^16+1/15*b^3*g*x^15+1/14*b^3*f*x^14+1/13*(3*a*b^2*h+b^3*e)*x^13+1/12
*(3*a*b^2*g+b^3*d)*x^12+1/11*(3*a*b^2*f+b^3*c)*x^11+1/10*(3*a^2*b*h+3*a*b^2*e)*x
^10+1/9*(3*a^2*b*g+3*a*b^2*d)*x^9+1/8*(3*a^2*b*f+3*a*b^2*c)*x^8+1/7*(a^3*h+3*a^2
*b*e)*x^7+1/6*(a^3*g+3*a^2*b*d)*x^6+1/5*(a^3*f+3*a^2*b*c)*x^5+1/4*a^3*e*x^4+1/3*
a^3*d*x^3+1/2*a^3*c*x^2

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Maxima [A]  time = 1.37465, size = 293, normalized size = 1.38 \[ \frac{1}{16} \, b^{3} h x^{16} + \frac{1}{15} \, b^{3} g x^{15} + \frac{1}{14} \, b^{3} f x^{14} + \frac{1}{13} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{13} + \frac{1}{12} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{12} + \frac{1}{11} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{11} + \frac{3}{10} \,{\left (a b^{2} e + a^{2} b h\right )} x^{10} + \frac{1}{3} \,{\left (a b^{2} d + a^{2} b g\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} c + a^{2} b f\right )} x^{8} + \frac{1}{4} \, a^{3} e x^{4} + \frac{1}{7} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{7} + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{6} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{6} + \frac{1}{2} \, a^{3} c x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{5} \]

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)^3*x,x, algorithm="maxima")

[Out]

1/16*b^3*h*x^16 + 1/15*b^3*g*x^15 + 1/14*b^3*f*x^14 + 1/13*(b^3*e + 3*a*b^2*h)*x
^13 + 1/12*(b^3*d + 3*a*b^2*g)*x^12 + 1/11*(b^3*c + 3*a*b^2*f)*x^11 + 3/10*(a*b^
2*e + a^2*b*h)*x^10 + 1/3*(a*b^2*d + a^2*b*g)*x^9 + 3/8*(a*b^2*c + a^2*b*f)*x^8
+ 1/4*a^3*e*x^4 + 1/7*(3*a^2*b*e + a^3*h)*x^7 + 1/3*a^3*d*x^3 + 1/6*(3*a^2*b*d +
a^3*g)*x^6 + 1/2*a^3*c*x^2 + 1/5*(3*a^2*b*c + a^3*f)*x^5

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Fricas [A]  time = 0.221621, size = 1, normalized size = 0. \[ \frac{1}{16} x^{16} h b^{3} + \frac{1}{15} x^{15} g b^{3} + \frac{1}{14} x^{14} f b^{3} + \frac{1}{13} x^{13} e b^{3} + \frac{3}{13} x^{13} h b^{2} a + \frac{1}{12} x^{12} d b^{3} + \frac{1}{4} x^{12} g b^{2} a + \frac{1}{11} x^{11} c b^{3} + \frac{3}{11} x^{11} f b^{2} a + \frac{3}{10} x^{10} e b^{2} a + \frac{3}{10} x^{10} h b a^{2} + \frac{1}{3} x^{9} d b^{2} a + \frac{1}{3} x^{9} g b a^{2} + \frac{3}{8} x^{8} c b^{2} a + \frac{3}{8} x^{8} f b a^{2} + \frac{3}{7} x^{7} e b a^{2} + \frac{1}{7} x^{7} h a^{3} + \frac{1}{2} x^{6} d b a^{2} + \frac{1}{6} x^{6} g a^{3} + \frac{3}{5} x^{5} c b a^{2} + \frac{1}{5} x^{5} f a^{3} + \frac{1}{4} x^{4} e a^{3} + \frac{1}{3} x^{3} d a^{3} + \frac{1}{2} x^{2} c a^{3} \]

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)^3*x,x, algorithm="fricas")

[Out]

1/16*x^16*h*b^3 + 1/15*x^15*g*b^3 + 1/14*x^14*f*b^3 + 1/13*x^13*e*b^3 + 3/13*x^1
3*h*b^2*a + 1/12*x^12*d*b^3 + 1/4*x^12*g*b^2*a + 1/11*x^11*c*b^3 + 3/11*x^11*f*b
^2*a + 3/10*x^10*e*b^2*a + 3/10*x^10*h*b*a^2 + 1/3*x^9*d*b^2*a + 1/3*x^9*g*b*a^2
+ 3/8*x^8*c*b^2*a + 3/8*x^8*f*b*a^2 + 3/7*x^7*e*b*a^2 + 1/7*x^7*h*a^3 + 1/2*x^6
*d*b*a^2 + 1/6*x^6*g*a^3 + 3/5*x^5*c*b*a^2 + 1/5*x^5*f*a^3 + 1/4*x^4*e*a^3 + 1/3
*x^3*d*a^3 + 1/2*x^2*c*a^3

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Sympy [A]  time = 0.113549, size = 246, normalized size = 1.16 \[ \frac{a^{3} c x^{2}}{2} + \frac{a^{3} d x^{3}}{3} + \frac{a^{3} e x^{4}}{4} + \frac{b^{3} f x^{14}}{14} + \frac{b^{3} g x^{15}}{15} + \frac{b^{3} h x^{16}}{16} + x^{13} \left (\frac{3 a b^{2} h}{13} + \frac{b^{3} e}{13}\right ) + x^{12} \left (\frac{a b^{2} g}{4} + \frac{b^{3} d}{12}\right ) + x^{11} \left (\frac{3 a b^{2} f}{11} + \frac{b^{3} c}{11}\right ) + x^{10} \left (\frac{3 a^{2} b h}{10} + \frac{3 a b^{2} e}{10}\right ) + x^{9} \left (\frac{a^{2} b g}{3} + \frac{a b^{2} d}{3}\right ) + x^{8} \left (\frac{3 a^{2} b f}{8} + \frac{3 a b^{2} c}{8}\right ) + x^{7} \left (\frac{a^{3} h}{7} + \frac{3 a^{2} b e}{7}\right ) + x^{6} \left (\frac{a^{3} g}{6} + \frac{a^{2} b d}{2}\right ) + x^{5} \left (\frac{a^{3} f}{5} + \frac{3 a^{2} b c}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*c*x**2/2 + a**3*d*x**3/3 + a**3*e*x**4/4 + b**3*f*x**14/14 + b**3*g*x**15/1
5 + b**3*h*x**16/16 + x**13*(3*a*b**2*h/13 + b**3*e/13) + x**12*(a*b**2*g/4 + b*
*3*d/12) + x**11*(3*a*b**2*f/11 + b**3*c/11) + x**10*(3*a**2*b*h/10 + 3*a*b**2*e
/10) + x**9*(a**2*b*g/3 + a*b**2*d/3) + x**8*(3*a**2*b*f/8 + 3*a*b**2*c/8) + x**
7*(a**3*h/7 + 3*a**2*b*e/7) + x**6*(a**3*g/6 + a**2*b*d/2) + x**5*(a**3*f/5 + 3*
a**2*b*c/5)

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GIAC/XCAS [A]  time = 0.216325, size = 315, normalized size = 1.49 \[ \frac{1}{16} \, b^{3} h x^{16} + \frac{1}{15} \, b^{3} g x^{15} + \frac{1}{14} \, b^{3} f x^{14} + \frac{3}{13} \, a b^{2} h x^{13} + \frac{1}{13} \, b^{3} x^{13} e + \frac{1}{12} \, b^{3} d x^{12} + \frac{1}{4} \, a b^{2} g x^{12} + \frac{1}{11} \, b^{3} c x^{11} + \frac{3}{11} \, a b^{2} f x^{11} + \frac{3}{10} \, a^{2} b h x^{10} + \frac{3}{10} \, a b^{2} x^{10} e + \frac{1}{3} \, a b^{2} d x^{9} + \frac{1}{3} \, a^{2} b g x^{9} + \frac{3}{8} \, a b^{2} c x^{8} + \frac{3}{8} \, a^{2} b f x^{8} + \frac{1}{7} \, a^{3} h x^{7} + \frac{3}{7} \, a^{2} b x^{7} e + \frac{1}{2} \, a^{2} b d x^{6} + \frac{1}{6} \, a^{3} g x^{6} + \frac{3}{5} \, a^{2} b c x^{5} + \frac{1}{5} \, a^{3} f x^{5} + \frac{1}{4} \, a^{3} x^{4} e + \frac{1}{3} \, a^{3} d x^{3} + \frac{1}{2} \, a^{3} c x^{2} \]

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)^3*x,x, algorithm="giac")

[Out]

1/16*b^3*h*x^16 + 1/15*b^3*g*x^15 + 1/14*b^3*f*x^14 + 3/13*a*b^2*h*x^13 + 1/13*b
^3*x^13*e + 1/12*b^3*d*x^12 + 1/4*a*b^2*g*x^12 + 1/11*b^3*c*x^11 + 3/11*a*b^2*f*
x^11 + 3/10*a^2*b*h*x^10 + 3/10*a*b^2*x^10*e + 1/3*a*b^2*d*x^9 + 1/3*a^2*b*g*x^9
+ 3/8*a*b^2*c*x^8 + 3/8*a^2*b*f*x^8 + 1/7*a^3*h*x^7 + 3/7*a^2*b*x^7*e + 1/2*a^2
*b*d*x^6 + 1/6*a^3*g*x^6 + 3/5*a^2*b*c*x^5 + 1/5*a^3*f*x^5 + 1/4*a^3*x^4*e + 1/3
*a^3*d*x^3 + 1/2*a^3*c*x^2