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

Optimal. Leaf size=198 \[ a^2 x (a f+3 b c)+\frac{1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{1}{7} b^2 x^7 (3 a f+b c)+\frac{1}{8} b^2 x^8 (3 a g+b d)+\frac{1}{2} a b^2 e x^6+\frac{3}{4} a b x^4 (a f+b c)+\frac{3}{5} a b x^5 (a g+b d)+\frac{h \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 e x^9+\frac{1}{10} b^3 f x^{10}+\frac{1}{11} b^3 g x^{11} \]

[Out]

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

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Rubi [A]  time = 0.196774, antiderivative size = 198, 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} \[ a^2 x (a f+3 b c)+\frac{1}{2} a^2 x^2 (a g+3 b d)+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{1}{7} b^2 x^7 (3 a f+b c)+\frac{1}{8} b^2 x^8 (3 a g+b d)+\frac{1}{2} a b^2 e x^6+\frac{3}{4} a b x^4 (a f+b c)+\frac{3}{5} a b x^5 (a g+b d)+\frac{h \left (a+b x^3\right )^4}{12 b}+\frac{1}{9} b^3 e x^9+\frac{1}{10} b^3 f x^{10}+\frac{1}{11} b^3 g x^{11} \]

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^3,x]

[Out]

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

Mathematica [A]  time = 0.117362, size = 174, normalized size = 0.88 \[ \frac{1}{20} a^2 b x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+\frac{a^3 \left (-3 c-6 d x+x^3 \left (6 f+3 g x+2 h x^2\right )\right )}{6 x^2}+a^3 e \log (x)+\frac{1}{840} a b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))+\frac{b^3 x^7 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )}{27720} \]

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^3,x]

[Out]

(a^3*(-3*c - 6*d*x + x^3*(6*f + 3*g*x + 2*h*x^2)))/(6*x^2) + (b^3*x^7*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(
66*f + 60*g*x + 55*h*x^2))))/27720 + (a^2*b*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))))/20
+ (a*b^2*x^4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x))))))/840 + a^3*e*Log[x]

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Maple [A]  time = 0.008, size = 222, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{12}}{12}}+{\frac{{b}^{3}g{x}^{11}}{11}}+{\frac{{b}^{3}f{x}^{10}}{10}}+{\frac{{x}^{9}a{b}^{2}h}{3}}+{\frac{{b}^{3}e{x}^{9}}{9}}+{\frac{3\,{x}^{8}a{b}^{2}g}{8}}+{\frac{{b}^{3}d{x}^{8}}{8}}+{\frac{3\,{x}^{7}a{b}^{2}f}{7}}+{\frac{{b}^{3}c{x}^{7}}{7}}+{\frac{{x}^{6}{a}^{2}bh}{2}}+{\frac{a{b}^{2}e{x}^{6}}{2}}+{\frac{3\,{x}^{5}{a}^{2}bg}{5}}+{\frac{3\,a{b}^{2}d{x}^{5}}{5}}+{\frac{3\,{x}^{4}{a}^{2}bf}{4}}+{\frac{3\,a{b}^{2}c{x}^{4}}{4}}+{\frac{{x}^{3}{a}^{3}h}{3}}+{a}^{2}be{x}^{3}+{\frac{{x}^{2}{a}^{3}g}{2}}+{\frac{3\,{a}^{2}bd{x}^{2}}{2}}+{a}^{3}fx+3\,{a}^{2}bcx+{a}^{3}e\ln \left ( x \right ) -{\frac{{a}^{3}c}{2\,{x}^{2}}}-{\frac{{a}^{3}d}{x}} \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^3,x)

[Out]

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

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Maxima [A]  time = 0.951091, size = 286, normalized size = 1.44 \begin{align*} \frac{1}{12} \, b^{3} h x^{12} + \frac{1}{11} \, b^{3} g x^{11} + \frac{1}{10} \, b^{3} f x^{10} + \frac{1}{9} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{9} + \frac{1}{8} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{8} + \frac{1}{7} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{7} + \frac{1}{2} \,{\left (a b^{2} e + a^{2} b h\right )} x^{6} + \frac{3}{5} \,{\left (a b^{2} d + a^{2} b g\right )} x^{5} + \frac{3}{4} \,{\left (a b^{2} c + a^{2} b f\right )} x^{4} + a^{3} e \log \left (x\right ) + \frac{1}{3} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{2} +{\left (3 \, a^{2} b c + a^{3} f\right )} x - \frac{2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.01098, size = 551, normalized size = 2.78 \begin{align*} \frac{2310 \, b^{3} h x^{14} + 2520 \, b^{3} g x^{13} + 2772 \, b^{3} f x^{12} + 3080 \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 3465 \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 3960 \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 13860 \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + 16632 \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + 20790 \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} + 27720 \, a^{3} e x^{2} \log \left (x\right ) + 9240 \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} - 27720 \, a^{3} d x + 13860 \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} - 13860 \, a^{3} c + 27720 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3}}{27720 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/27720*(2310*b^3*h*x^14 + 2520*b^3*g*x^13 + 2772*b^3*f*x^12 + 3080*(b^3*e + 3*a*b^2*h)*x^11 + 3465*(b^3*d + 3
*a*b^2*g)*x^10 + 3960*(b^3*c + 3*a*b^2*f)*x^9 + 13860*(a*b^2*e + a^2*b*h)*x^8 + 16632*(a*b^2*d + a^2*b*g)*x^7
+ 20790*(a*b^2*c + a^2*b*f)*x^6 + 27720*a^3*e*x^2*log(x) + 9240*(3*a^2*b*e + a^3*h)*x^5 - 27720*a^3*d*x + 1386
0*(3*a^2*b*d + a^3*g)*x^4 - 13860*a^3*c + 27720*(3*a^2*b*c + a^3*f)*x^3)/x^2

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Sympy [A]  time = 0.716463, size = 236, normalized size = 1.19 \begin{align*} a^{3} e \log{\left (x \right )} + \frac{b^{3} f x^{10}}{10} + \frac{b^{3} g x^{11}}{11} + \frac{b^{3} h x^{12}}{12} + x^{9} \left (\frac{a b^{2} h}{3} + \frac{b^{3} e}{9}\right ) + x^{8} \left (\frac{3 a b^{2} g}{8} + \frac{b^{3} d}{8}\right ) + x^{7} \left (\frac{3 a b^{2} f}{7} + \frac{b^{3} c}{7}\right ) + x^{6} \left (\frac{a^{2} b h}{2} + \frac{a b^{2} e}{2}\right ) + x^{5} \left (\frac{3 a^{2} b g}{5} + \frac{3 a b^{2} d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b f}{4} + \frac{3 a b^{2} c}{4}\right ) + x^{3} \left (\frac{a^{3} h}{3} + a^{2} b e\right ) + x^{2} \left (\frac{a^{3} g}{2} + \frac{3 a^{2} b d}{2}\right ) + x \left (a^{3} f + 3 a^{2} b c\right ) - \frac{a^{3} c + 2 a^{3} d x}{2 x^{2}} \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**3,x)

[Out]

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

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Giac [A]  time = 1.06294, size = 305, normalized size = 1.54 \begin{align*} \frac{1}{12} \, b^{3} h x^{12} + \frac{1}{11} \, b^{3} g x^{11} + \frac{1}{10} \, b^{3} f x^{10} + \frac{1}{3} \, a b^{2} h x^{9} + \frac{1}{9} \, b^{3} x^{9} e + \frac{1}{8} \, b^{3} d x^{8} + \frac{3}{8} \, a b^{2} g x^{8} + \frac{1}{7} \, b^{3} c x^{7} + \frac{3}{7} \, a b^{2} f x^{7} + \frac{1}{2} \, a^{2} b h x^{6} + \frac{1}{2} \, a b^{2} x^{6} e + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{5} \, a^{2} b g x^{5} + \frac{3}{4} \, a b^{2} c x^{4} + \frac{3}{4} \, a^{2} b f x^{4} + \frac{1}{3} \, a^{3} h x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} g x^{2} + 3 \, a^{2} b c x + a^{3} f x + a^{3} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{3} d x + a^{3} c}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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