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

Optimal. Leaf size=209 \[ a^2 \log (x) (a f+3 b c)+a^2 x (a g+3 b d)+\frac{1}{2} a^2 x^2 (a h+3 b e)-\frac{a^3 c}{3 x^3}-\frac{a^3 d}{2 x^2}-\frac{a^3 e}{x}+\frac{1}{6} b^2 x^6 (3 a f+b c)+\frac{1}{7} b^2 x^7 (3 a g+b d)+\frac{1}{8} b^2 x^8 (3 a h+b e)+a b x^3 (a f+b c)+\frac{3}{4} a b x^4 (a g+b d)+\frac{3}{5} a b x^5 (a h+b e)+\frac{1}{9} b^3 f x^9+\frac{1}{10} b^3 g x^{10}+\frac{1}{11} b^3 h x^{11} \]

[Out]

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

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Rubi [A]  time = 0.179548, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {1820} \[ a^2 \log (x) (a f+3 b c)+a^2 x (a g+3 b d)+\frac{1}{2} a^2 x^2 (a h+3 b e)-\frac{a^3 c}{3 x^3}-\frac{a^3 d}{2 x^2}-\frac{a^3 e}{x}+\frac{1}{6} b^2 x^6 (3 a f+b c)+\frac{1}{7} b^2 x^7 (3 a g+b d)+\frac{1}{8} b^2 x^8 (3 a h+b e)+a b x^3 (a f+b c)+\frac{3}{4} a b x^4 (a g+b d)+\frac{3}{5} a b x^5 (a h+b e)+\frac{1}{9} b^3 f x^9+\frac{1}{10} b^3 g x^{10}+\frac{1}{11} b^3 h x^{11} \]

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^4,x]

[Out]

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

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

Mathematica [A]  time = 0.126123, size = 172, normalized size = 0.82 \[ a^2 \log (x) (a f+3 b c)+\frac{1}{20} a^2 b x \left (60 d+x \left (30 e+x \left (20 f+15 g x+12 h x^2\right )\right )\right )-\frac{a^3 \left (2 c+3 x \left (d+2 e x+x^3 (-(2 g+h x))\right )\right )}{6 x^3}+\frac{1}{280} a b^2 x^3 \left (280 c+x \left (210 d+x \left (168 e+140 f x+120 g x^2+105 h x^3\right )\right )\right )+\frac{b^3 x^6 \left (4620 c+x \left (3960 d+7 x \left (495 e+4 x \left (110 f+99 g x+90 h x^2\right )\right )\right )\right )}{27720} \]

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^4,x]

[Out]

-(a^3*(2*c + 3*x*(d + 2*e*x - x^3*(2*g + h*x))))/(6*x^3) + (a^2*b*x*(60*d + x*(30*e + x*(20*f + 15*g*x + 12*h*
x^2))))/20 + (a*b^2*x^3*(280*c + x*(210*d + x*(168*e + 140*f*x + 120*g*x^2 + 105*h*x^3))))/280 + (b^3*x^6*(462
0*c + x*(3960*d + 7*x*(495*e + 4*x*(110*f + 99*g*x + 90*h*x^2)))))/27720 + a^2*(3*b*c + a*f)*Log[x]

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Maple [A]  time = 0.006, size = 220, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}h{x}^{11}}{11}}+{\frac{{b}^{3}g{x}^{10}}{10}}+{\frac{{b}^{3}f{x}^{9}}{9}}+{\frac{3\,{x}^{8}a{b}^{2}h}{8}}+{\frac{{x}^{8}{b}^{3}e}{8}}+{\frac{3\,{x}^{7}a{b}^{2}g}{7}}+{\frac{{x}^{7}{b}^{3}d}{7}}+{\frac{{x}^{6}a{b}^{2}f}{2}}+{\frac{{x}^{6}{b}^{3}c}{6}}+{\frac{3\,{x}^{5}{a}^{2}bh}{5}}+{\frac{3\,{x}^{5}a{b}^{2}e}{5}}+{\frac{3\,{x}^{4}{a}^{2}bg}{4}}+{\frac{3\,a{b}^{2}d{x}^{4}}{4}}+{x}^{3}{a}^{2}bf+{x}^{3}a{b}^{2}c+{\frac{{x}^{2}{a}^{3}h}{2}}+{\frac{3\,{a}^{2}be{x}^{2}}{2}}+{a}^{3}gx+3\,{a}^{2}bdx+\ln \left ( x \right ){a}^{3}f+3\,\ln \left ( x \right ){a}^{2}bc-{\frac{{a}^{3}c}{3\,{x}^{3}}}-{\frac{{a}^{3}d}{2\,{x}^{2}}}-{\frac{{a}^{3}e}{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^4,x)

[Out]

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

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Maxima [A]  time = 0.946772, size = 286, normalized size = 1.37 \begin{align*} \frac{1}{11} \, b^{3} h x^{11} + \frac{1}{10} \, b^{3} g x^{10} + \frac{1}{9} \, b^{3} f x^{9} + \frac{1}{8} \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{8} + \frac{1}{7} \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{7} + \frac{1}{6} \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{6} + \frac{3}{5} \,{\left (a b^{2} e + a^{2} b h\right )} x^{5} + \frac{3}{4} \,{\left (a b^{2} d + a^{2} b g\right )} x^{4} +{\left (a b^{2} c + a^{2} b f\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{2} +{\left (3 \, a^{2} b d + a^{3} g\right )} x +{\left (3 \, a^{2} b c + a^{3} f\right )} \log \left (x\right ) - \frac{6 \, a^{3} e x^{2} + 3 \, a^{3} d x + 2 \, a^{3} c}{6 \, 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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.00583, size = 551, normalized size = 2.64 \begin{align*} \frac{2520 \, b^{3} h x^{14} + 2772 \, b^{3} g x^{13} + 3080 \, b^{3} f x^{12} + 3465 \,{\left (b^{3} e + 3 \, a b^{2} h\right )} x^{11} + 3960 \,{\left (b^{3} d + 3 \, a b^{2} g\right )} x^{10} + 4620 \,{\left (b^{3} c + 3 \, a b^{2} f\right )} x^{9} + 16632 \,{\left (a b^{2} e + a^{2} b h\right )} x^{8} + 20790 \,{\left (a b^{2} d + a^{2} b g\right )} x^{7} + 27720 \,{\left (a b^{2} c + a^{2} b f\right )} x^{6} - 27720 \, a^{3} e x^{2} + 13860 \,{\left (3 \, a^{2} b e + a^{3} h\right )} x^{5} - 13860 \, a^{3} d x + 27720 \,{\left (3 \, a^{2} b d + a^{3} g\right )} x^{4} + 27720 \,{\left (3 \, a^{2} b c + a^{3} f\right )} x^{3} \log \left (x\right ) - 9240 \, a^{3} c}{27720 \, 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^4,x, algorithm="fricas")

[Out]

1/27720*(2520*b^3*h*x^14 + 2772*b^3*g*x^13 + 3080*b^3*f*x^12 + 3465*(b^3*e + 3*a*b^2*h)*x^11 + 3960*(b^3*d + 3
*a*b^2*g)*x^10 + 4620*(b^3*c + 3*a*b^2*f)*x^9 + 16632*(a*b^2*e + a^2*b*h)*x^8 + 20790*(a*b^2*d + a^2*b*g)*x^7
+ 27720*(a*b^2*c + a^2*b*f)*x^6 - 27720*a^3*e*x^2 + 13860*(3*a^2*b*e + a^3*h)*x^5 - 13860*a^3*d*x + 27720*(3*a
^2*b*d + a^3*g)*x^4 + 27720*(3*a^2*b*c + a^3*f)*x^3*log(x) - 9240*a^3*c)/x^3

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Sympy [A]  time = 1.06122, size = 235, normalized size = 1.12 \begin{align*} a^{2} \left (a f + 3 b c\right ) \log{\left (x \right )} + \frac{b^{3} f x^{9}}{9} + \frac{b^{3} g x^{10}}{10} + \frac{b^{3} h x^{11}}{11} + x^{8} \left (\frac{3 a b^{2} h}{8} + \frac{b^{3} e}{8}\right ) + x^{7} \left (\frac{3 a b^{2} g}{7} + \frac{b^{3} d}{7}\right ) + x^{6} \left (\frac{a b^{2} f}{2} + \frac{b^{3} c}{6}\right ) + x^{5} \left (\frac{3 a^{2} b h}{5} + \frac{3 a b^{2} e}{5}\right ) + x^{4} \left (\frac{3 a^{2} b g}{4} + \frac{3 a b^{2} d}{4}\right ) + x^{3} \left (a^{2} b f + a b^{2} c\right ) + x^{2} \left (\frac{a^{3} h}{2} + \frac{3 a^{2} b e}{2}\right ) + x \left (a^{3} g + 3 a^{2} b d\right ) - \frac{2 a^{3} c + 3 a^{3} d x + 6 a^{3} e x^{2}}{6 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**4,x)

[Out]

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

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Giac [A]  time = 1.06246, size = 304, normalized size = 1.45 \begin{align*} \frac{1}{11} \, b^{3} h x^{11} + \frac{1}{10} \, b^{3} g x^{10} + \frac{1}{9} \, b^{3} f x^{9} + \frac{3}{8} \, a b^{2} h x^{8} + \frac{1}{8} \, b^{3} x^{8} e + \frac{1}{7} \, b^{3} d x^{7} + \frac{3}{7} \, a b^{2} g x^{7} + \frac{1}{6} \, b^{3} c x^{6} + \frac{1}{2} \, a b^{2} f x^{6} + \frac{3}{5} \, a^{2} b h x^{5} + \frac{3}{5} \, a b^{2} x^{5} e + \frac{3}{4} \, a b^{2} d x^{4} + \frac{3}{4} \, a^{2} b g x^{4} + a b^{2} c x^{3} + a^{2} b f x^{3} + \frac{1}{2} \, a^{3} h x^{2} + \frac{3}{2} \, a^{2} b x^{2} e + 3 \, a^{2} b d x + a^{3} g x +{\left (3 \, a^{2} b c + a^{3} f\right )} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{3} x^{2} e + 3 \, a^{3} d x + 2 \, a^{3} c}{6 \, 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^4,x, algorithm="giac")

[Out]

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