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

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 = 0.105703, 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, Rules used = {1583, 1820} \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{7} b x^7 (2 a g+b d)+\frac{1}{4} a x^4 (a g+2 b d)+\frac{1}{8} b x^8 (2 a h+b e)+\frac{1}{5} a x^5 (a h+2 b e)+\frac{f \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 c x^6+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11} \]

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]

Rule 1583

Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(Coeff[Px, x, n - m - 1]*(a + b*x^n)^(p
+ 1))/(b*n*(p + 1)), x] + Int[(Px - Coeff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a,
b, m, n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x, n - m - 1], 0]

Rule 1820

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \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 &=\frac{f \left (a+b x^3\right )^3}{9 b}+\int \frac{\left (a+b x^3\right )^2 \left (c+d x+e x^2+g x^4+h x^5\right )}{x} \, dx\\ &=\frac{f \left (a+b x^3\right )^3}{9 b}+\int \left (a^2 d+\frac{a^2 c}{x}+a^2 e x+2 a b c x^2+a (2 b d+a g) x^3+a (2 b e+a h) x^4+b^2 c x^5+b (b d+2 a g) x^6+b (b e+2 a h) x^7+b^2 g x^9+b^2 h x^{10}\right ) \, dx\\ &=a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{4} a (2 b d+a g) x^4+\frac{1}{5} a (2 b e+a h) x^5+\frac{1}{6} b^2 c x^6+\frac{1}{7} b (b d+2 a g) x^7+\frac{1}{8} b (b e+2 a h) x^8+\frac{1}{10} b^2 g x^{10}+\frac{1}{11} b^2 h x^{11}+\frac{f \left (a+b x^3\right )^3}{9 b}+a^2 c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0438508, 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 = 0.003, size = 153, normalized size = 1. \begin{align*}{\frac{{b}^{2}h{x}^{11}}{11}}+{\frac{{b}^{2}g{x}^{10}}{10}}+{\frac{{x}^{9}{b}^{2}f}{9}}+{\frac{{x}^{8}abh}{4}}+{\frac{{b}^{2}e{x}^{8}}{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\,abe{x}^{5}}{5}}+{\frac{{x}^{4}{a}^{2}g}{4}}+{\frac{abd{x}^{4}}{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 ) \end{align*}

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*b^2*f+1/4*x^8*a*b*h+1/8*b^2*e*x^8+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*a*b*e*x^5+1/4*x^4*a^2*g+1/2*a*b*d*x^4+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 = 0.934238, size = 197, normalized size = 1.32 \begin{align*} \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 ) \end{align*}

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, 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 = 0.937602, size = 351, normalized size = 2.36 \begin{align*} \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 ) \end{align*}

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, 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 = 0.47338, size = 162, normalized size = 1.09 \begin{align*} 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 ) \end{align*}

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 [A]  time = 1.06798, size = 211, normalized size = 1.42 \begin{align*} \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 \log \left ({\left | x \right |}\right ) \end{align*}

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, 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*log(abs(x))