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

Optimal. Leaf size=149 \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{7} b x^7 (2 a g+b d)+\frac{1}{4} a x^4 (a g+2 b d)+\frac{1}{8} b x^8 (2 a h+b e)+\frac{1}{5} a x^5 (a h+2 b e)+\frac{f \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 c x^6+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11} \]

[Out]

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

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Rubi [A]  time = 0.242581, antiderivative size = 149, 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 \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{7} b x^7 (2 a g+b d)+\frac{1}{4} a x^4 (a g+2 b d)+\frac{1}{8} b x^8 (2 a h+b e)+\frac{1}{5} a x^5 (a h+2 b e)+\frac{f \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 c x^6+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11} \]

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,x]

[Out]

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

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

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,x)

[Out]

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

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Mathematica [A]  time = 0.113301, size = 154, normalized size = 1.03 \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{1}{6} b x^6 (2 a f+b c)+\frac{1}{3} a x^3 (a f+2 b c)+\frac{1}{7} b x^7 (2 a g+b d)+\frac{1}{4} a x^4 (a g+2 b d)+\frac{1}{8} b x^8 (2 a h+b e)+\frac{1}{5} a x^5 (a h+2 b e)+\frac{1}{9} b^2 f x^9+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11} \]

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,x]

[Out]

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

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Maple [A]  time = 0.004, size = 153, normalized size = 1. \[{\frac{{b}^{2}h{x}^{11}}{11}}+{\frac{{b}^{2}g{x}^{10}}{10}}+{\frac{{x}^{9}f{b}^{2}}{9}}+{\frac{{x}^{8}abh}{4}}+{\frac{{x}^{8}{b}^{2}e}{8}}+{\frac{2\,{x}^{7}abg}{7}}+{\frac{{b}^{2}d{x}^{7}}{7}}+{\frac{{x}^{6}abf}{3}}+{\frac{{b}^{2}c{x}^{6}}{6}}+{\frac{{x}^{5}{a}^{2}h}{5}}+{\frac{2\,{x}^{5}abe}{5}}+{\frac{{x}^{4}{a}^{2}g}{4}}+{\frac{{x}^{4}abd}{2}}+{\frac{{a}^{2}f{x}^{3}}{3}}+{\frac{2\,abc{x}^{3}}{3}}+{\frac{{a}^{2}e{x}^{2}}{2}}+{a}^{2}dx+{a}^{2}c\ln \left ( x \right ) \]

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,x)

[Out]

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

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Maxima [A]  time = 1.38052, size = 197, normalized size = 1.32 \[ \frac{1}{11} \, b^{2} h x^{11} + \frac{1}{10} \, b^{2} g x^{10} + \frac{1}{9} \, b^{2} f x^{9} + \frac{1}{8} \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + \frac{1}{7} \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + \frac{1}{6} \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + \frac{1}{5} \,{\left (2 \, a b e + a^{2} h\right )} x^{5} + \frac{1}{2} \, a^{2} e x^{2} + \frac{1}{4} \,{\left (2 \, a b d + a^{2} g\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b c + a^{2} f\right )} x^{3} + a^{2} c \log \left (x\right ) \]

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)^2/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.244357, size = 197, normalized size = 1.32 \[ \frac{1}{11} \, b^{2} h x^{11} + \frac{1}{10} \, b^{2} g x^{10} + \frac{1}{9} \, b^{2} f x^{9} + \frac{1}{8} \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + \frac{1}{7} \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + \frac{1}{6} \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + \frac{1}{5} \,{\left (2 \, a b e + a^{2} h\right )} x^{5} + \frac{1}{2} \, a^{2} e x^{2} + \frac{1}{4} \,{\left (2 \, a b d + a^{2} g\right )} x^{4} + a^{2} d x + \frac{1}{3} \,{\left (2 \, a b c + a^{2} f\right )} x^{3} + a^{2} c \log \left (x\right ) \]

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)^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.940603, size = 162, normalized size = 1.09 \[ a^{2} c \log{\left (x \right )} + a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{b^{2} f x^{9}}{9} + \frac{b^{2} g x^{10}}{10} + \frac{b^{2} h x^{11}}{11} + x^{8} \left (\frac{a b h}{4} + \frac{b^{2} e}{8}\right ) + x^{7} \left (\frac{2 a b g}{7} + \frac{b^{2} d}{7}\right ) + x^{6} \left (\frac{a b f}{3} + \frac{b^{2} c}{6}\right ) + x^{5} \left (\frac{a^{2} h}{5} + \frac{2 a b e}{5}\right ) + x^{4} \left (\frac{a^{2} g}{4} + \frac{a b d}{2}\right ) + x^{3} \left (\frac{a^{2} f}{3} + \frac{2 a b c}{3}\right ) \]

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,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.209234, size = 211, normalized size = 1.42 \[ \frac{1}{11} \, b^{2} h x^{11} + \frac{1}{10} \, b^{2} g x^{10} + \frac{1}{9} \, b^{2} f x^{9} + \frac{1}{4} \, a b h x^{8} + \frac{1}{8} \, b^{2} x^{8} e + \frac{1}{7} \, b^{2} d x^{7} + \frac{2}{7} \, a b g x^{7} + \frac{1}{6} \, b^{2} c x^{6} + \frac{1}{3} \, a b f x^{6} + \frac{1}{5} \, a^{2} h x^{5} + \frac{2}{5} \, a b x^{5} e + \frac{1}{2} \, a b d x^{4} + \frac{1}{4} \, a^{2} g x^{4} + \frac{2}{3} \, a b c x^{3} + \frac{1}{3} \, a^{2} f x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x + a^{2} c{\rm ln}\left ({\left | x \right |}\right ) \]

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)^2/x,x, algorithm="giac")

[Out]

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