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

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

[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 = 0.145609, 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, Rules used = {1583, 1820} \[ a^2 b c x^3+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{2} a b^2 c x^6+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{f \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 c x^9+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]

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]

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 )^3 \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 )^4}{12 b}+\int \frac{\left (a+b x^3\right )^3 \left (c+d x+e x^2+g x^4+h x^5\right )}{x} \, dx\\ &=\frac{f \left (a+b x^3\right )^4}{12 b}+\int \left (a^3 d+\frac{a^3 c}{x}+a^3 e x+3 a^2 b c x^2+a^2 (3 b d+a g) x^3+a^2 (3 b e+a h) x^4+3 a b^2 c x^5+3 a b (b d+a g) x^6+3 a b (b e+a h) x^7+b^3 c x^8+b^2 (b d+3 a g) x^9+b^2 (b e+3 a h) x^{10}+b^3 g x^{12}+b^3 h x^{13}\right ) \, dx\\ &=a^3 d x+\frac{1}{2} a^3 e x^2+a^2 b c x^3+\frac{1}{4} a^2 (3 b d+a g) x^4+\frac{1}{5} a^2 (3 b e+a h) x^5+\frac{1}{2} a b^2 c x^6+\frac{3}{7} a b (b d+a g) x^7+\frac{3}{8} a b (b e+a h) x^8+\frac{1}{9} b^3 c x^9+\frac{1}{10} b^2 (b d+3 a g) x^{10}+\frac{1}{11} b^2 (b e+3 a h) x^{11}+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14}+\frac{f \left (a+b x^3\right )^4}{12 b}+a^3 c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0777477, size = 214, normalized size = 1.07 \[ \frac{1}{3} a^2 x^3 (a f+3 b c)+\frac{1}{4} a^2 x^4 (a g+3 b d)+\frac{1}{5} a^2 x^5 (a h+3 b e)+a^3 c \log (x)+a^3 d x+\frac{1}{2} a^3 e x^2+\frac{1}{9} b^2 x^9 (3 a f+b c)+\frac{1}{10} b^2 x^{10} (3 a g+b d)+\frac{1}{11} b^2 x^{11} (3 a h+b e)+\frac{1}{2} a b x^6 (a f+b c)+\frac{3}{7} a b x^7 (a g+b d)+\frac{3}{8} a b x^8 (a h+b e)+\frac{1}{12} b^3 f x^{12}+\frac{1}{13} b^3 g x^{13}+\frac{1}{14} b^3 h x^{14} \]

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)/1
4 + a^3*c*Log[x]

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Maple [A]  time = 0.004, size = 224, normalized size = 1.1 \begin{align*}{\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{{b}^{3}e{x}^{11}}{11}}+{\frac{3\,{x}^{10}a{b}^{2}g}{10}}+{\frac{{b}^{3}d{x}^{10}}{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\,a{b}^{2}e{x}^{8}}{8}}+{\frac{3\,{x}^{7}{a}^{2}bg}{7}}+{\frac{3\,a{b}^{2}d{x}^{7}}{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\,{a}^{2}be{x}^{5}}{5}}+{\frac{{x}^{4}{a}^{3}g}{4}}+{\frac{3\,{a}^{2}bd{x}^{4}}{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 ) \end{align*}

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*b^3*e*x^11+3/10*x^10*a*b^2*g+1/10*b^3*d
*x^10+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*a*b^2*e*x^8+3/7*x^7*a^2*b*g+3/7*a*b^2*d*x^7+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*a^2*b*e*x^5+1/4*x^4*a^3*g+3/4*a^2*b*d*x^4+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 = 0.960838, size = 286, normalized size = 1.43 \begin{align*} \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} \end{align*}

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, 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 = 0.997019, size = 497, normalized size = 2.48 \begin{align*} \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} \end{align*}

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, 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 = 0.604787, size = 240, normalized size = 1.2 \begin{align*} 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 ) \end{align*}

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 [A]  time = 1.06867, size = 308, normalized size = 1.54 \begin{align*} \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 \log \left ({\left | x \right |}\right ) \end{align*}

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, 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*log(abs(x))