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

Optimal. Leaf size=200 \[ a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+\frac{1}{2} a b^2 c x^6+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{f \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 c x^9+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]

[Out]

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

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Rubi [A]  time = 0.315798, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+\frac{1}{2} a b^2 c x^6+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{f \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 c x^9+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.195592, size = 214, normalized size = 1.07 \[ a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{3} a^2 x^3 (a f+3 b c)+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+\frac{1}{9} b^2 x^9 (3 a f+b c)+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{1}{2} a b x^6 (a f+b c)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{1}{12} b^3 f x^{12}+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 224, normalized size = 1.1 \[{\frac{{b}^{3}h{x}^{14}}{14}}+{\frac{{b}^{3}g{x}^{13}}{13}}+{\frac{f{x}^{12}{b}^{3}}{12}}+{\frac{3\,{x}^{11}a{b}^{2}h}{11}}+{\frac{{x}^{11}{b}^{3}e}{11}}+{\frac{3\,{x}^{10}a{b}^{2}g}{10}}+{\frac{{x}^{10}{b}^{3}d}{10}}+{\frac{{x}^{9}a{b}^{2}f}{3}}+{\frac{{b}^{3}c{x}^{9}}{9}}+{\frac{3\,{x}^{8}{a}^{2}bh}{8}}+{\frac{3\,{x}^{8}a{b}^{2}e}{8}}+{\frac{3\,{x}^{7}{a}^{2}bg}{7}}+{\frac{3\,{x}^{7}a{b}^{2}d}{7}}+{\frac{{x}^{6}{a}^{2}bf}{2}}+{\frac{a{b}^{2}c{x}^{6}}{2}}+{\frac{{x}^{5}{a}^{3}h}{5}}+{\frac{3\,{x}^{5}{a}^{2}be}{5}}+{\frac{{x}^{4}{a}^{3}g}{4}}+{\frac{3\,{x}^{4}{a}^{2}bd}{4}}+{\frac{{a}^{3}f{x}^{3}}{3}}+{a}^{2}bc{x}^{3}+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{3}dx+{a}^{3}c\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3949, size = 286, normalized size = 1.43 \[ \frac{1}{14} \, b^{3} h x^{14} + \frac{1}{13} \, b^{3} g x^{13} + \frac{1}{12} \, b^{3} f x^{12} + \frac{1}{11} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + \frac{1}{10} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + \frac{1}{9} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + \frac{3}{7} \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + \frac{1}{2} \, a^{3} e x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + a^{3} c \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \]

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="maxima")

[Out]

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

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Fricas [A]  time = 0.243257, size = 286, normalized size = 1.43 \[ \frac{1}{14} \, b^{3} h x^{14} + \frac{1}{13} \, b^{3} g x^{13} + \frac{1}{12} \, b^{3} f x^{12} + \frac{1}{11} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + \frac{1}{10} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + \frac{1}{9} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + \frac{3}{8} \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + \frac{3}{7} \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + \frac{1}{2} \, a^{3} e x^{2} + \frac{1}{5} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} + a^{3} d x + \frac{1}{4} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + a^{3} c \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \]

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]

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

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Sympy [A]  time = 1.25182, size = 240, normalized size = 1.2 \[ a^{3} c \log{\left (x \right )} + a^{3} d x + \frac{a^{3} e x^{2}}{2} + \frac{b^{3} f x^{12}}{12} + \frac{b^{3} g x^{13}}{13} + \frac{b^{3} h x^{14}}{14} + x^{11} \left (\frac{3 a b^{2} h}{11} + \frac{b^{3} e}{11}\right ) + x^{10} \left (\frac{3 a b^{2} g}{10} + \frac{b^{3} d}{10}\right ) + x^{9} \left (\frac{a b^{2} f}{3} + \frac{b^{3} c}{9}\right ) + x^{8} \left (\frac{3 a^{2} b h}{8} + \frac{3 a b^{2} e}{8}\right ) + x^{7} \left (\frac{3 a^{2} b g}{7} + \frac{3 a b^{2} d}{7}\right ) + x^{6} \left (\frac{a^{2} b f}{2} + \frac{a b^{2} c}{2}\right ) + x^{5} \left (\frac{a^{3} h}{5} + \frac{3 a^{2} b e}{5}\right ) + x^{4} \left (\frac{a^{3} g}{4} + \frac{3 a^{2} b d}{4}\right ) + x^{3} \left (\frac{a^{3} f}{3} + a^{2} b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220448, size = 308, normalized size = 1.54 \[ \frac{1}{14} \, b^{3} h x^{14} + \frac{1}{13} \, b^{3} g x^{13} + \frac{1}{12} \, b^{3} f x^{12} + \frac{3}{11} \, a b^{2} h x^{11} + \frac{1}{11} \, b^{3} x^{11} e + \frac{1}{10} \, b^{3} d x^{10} + \frac{3}{10} \, a b^{2} g x^{10} + \frac{1}{9} \, b^{3} c x^{9} + \frac{1}{3} \, a b^{2} f x^{9} + \frac{3}{8} \, a^{2} b h x^{8} + \frac{3}{8} \, a b^{2} x^{8} e + \frac{3}{7} \, a b^{2} d x^{7} + \frac{3}{7} \, a^{2} b g x^{7} + \frac{1}{2} \, a b^{2} c x^{6} + \frac{1}{2} \, a^{2} b f x^{6} + \frac{1}{5} \, a^{3} h x^{5} + \frac{3}{5} \, a^{2} b x^{5} e + \frac{3}{4} \, a^{2} b d x^{4} + \frac{1}{4} \, a^{3} g x^{4} + a^{2} b c x^{3} + \frac{1}{3} \, a^{3} f x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x + a^{3} c{\rm ln}\left ({\left | x \right |}\right ) \]

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]

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