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

Optimal. Leaf size=147 \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+\frac{1}{4} b x^4 (2 a f+b c)+a x (a f+2 b c)+\frac{1}{5} b x^5 (2 a g+b d)+\frac{1}{2} a x^2 (a g+2 b d)+\frac{2}{3} a b e x^3+\frac{h \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 e x^6+\frac{1}{7} b^2 f x^7+\frac{1}{8} b^2 g x^8 \]

[Out]

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

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Rubi [A]  time = 0.128372, antiderivative size = 147, 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} \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+\frac{1}{4} b x^4 (2 a f+b c)+a x (a f+2 b c)+\frac{1}{5} b x^5 (2 a g+b d)+\frac{1}{2} a x^2 (a g+2 b d)+\frac{2}{3} a b e x^3+\frac{h \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 e x^6+\frac{1}{7} b^2 f x^7+\frac{1}{8} b^2 g x^8 \]

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^3,x]

[Out]

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

Mathematica [A]  time = 0.0706433, size = 127, normalized size = 0.86 \[ \frac{a^2 \left (-3 c-6 d x+x^3 \left (6 f+3 g x+2 h x^2\right )\right )}{6 x^2}+a^2 e \log (x)+\frac{1}{30} a b x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+\frac{b^2 x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))}{2520} \]

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^3,x]

[Out]

(a^2*(-3*c - 6*d*x + x^3*(6*f + 3*g*x + 2*h*x^2)))/(6*x^2) + (a*b*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*
x^2 + 10*h*x^3))))/30 + (b^2*x^4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x))))))/2520 + a^2*e
*Log[x]

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Maple [A]  time = 0.006, size = 150, normalized size = 1. \begin{align*}{\frac{{b}^{2}h{x}^{9}}{9}}+{\frac{{b}^{2}g{x}^{8}}{8}}+{\frac{{b}^{2}f{x}^{7}}{7}}+{\frac{{x}^{6}abh}{3}}+{\frac{{b}^{2}e{x}^{6}}{6}}+{\frac{2\,{x}^{5}abg}{5}}+{\frac{{b}^{2}d{x}^{5}}{5}}+{\frac{{x}^{4}abf}{2}}+{\frac{{b}^{2}c{x}^{4}}{4}}+{\frac{{x}^{3}{a}^{2}h}{3}}+{\frac{2\,abe{x}^{3}}{3}}+{\frac{{x}^{2}{a}^{2}g}{2}}+abd{x}^{2}+{a}^{2}fx+2\,abcx+{a}^{2}e\ln \left ( x \right ) -{\frac{{a}^{2}c}{2\,{x}^{2}}}-{\frac{{a}^{2}d}{x}} \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^3,x)

[Out]

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

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Maxima [A]  time = 0.939303, size = 197, normalized size = 1.34 \begin{align*} \frac{1}{9} \, b^{2} h x^{9} + \frac{1}{8} \, b^{2} g x^{8} + \frac{1}{7} \, b^{2} f x^{7} + \frac{1}{6} \,{\left (b^{2} e + 2 \, a b h\right )} x^{6} + \frac{1}{5} \,{\left (b^{2} d + 2 \, a b g\right )} x^{5} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b f\right )} x^{4} + \frac{1}{3} \,{\left (2 \, a b e + a^{2} h\right )} x^{3} + a^{2} e \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, a b d + a^{2} g\right )} x^{2} +{\left (2 \, a b c + a^{2} f\right )} x - \frac{2 \, a^{2} d x + a^{2} c}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.00032, size = 383, normalized size = 2.61 \begin{align*} \frac{280 \, b^{2} h x^{11} + 315 \, b^{2} g x^{10} + 360 \, b^{2} f x^{9} + 420 \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + 504 \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + 630 \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + 840 \,{\left (2 \, a b e + a^{2} h\right )} x^{5} + 2520 \, a^{2} e x^{2} \log \left (x\right ) + 1260 \,{\left (2 \, a b d + a^{2} g\right )} x^{4} - 2520 \, a^{2} d x + 2520 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} - 1260 \, a^{2} c}{2520 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

1/2520*(280*b^2*h*x^11 + 315*b^2*g*x^10 + 360*b^2*f*x^9 + 420*(b^2*e + 2*a*b*h)*x^8 + 504*(b^2*d + 2*a*b*g)*x^
7 + 630*(b^2*c + 2*a*b*f)*x^6 + 840*(2*a*b*e + a^2*h)*x^5 + 2520*a^2*e*x^2*log(x) + 1260*(2*a*b*d + a^2*g)*x^4
- 2520*a^2*d*x + 2520*(2*a*b*c + a^2*f)*x^3 - 1260*a^2*c)/x^2

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Sympy [A]  time = 0.550674, size = 156, normalized size = 1.06 \begin{align*} a^{2} e \log{\left (x \right )} + \frac{b^{2} f x^{7}}{7} + \frac{b^{2} g x^{8}}{8} + \frac{b^{2} h x^{9}}{9} + x^{6} \left (\frac{a b h}{3} + \frac{b^{2} e}{6}\right ) + x^{5} \left (\frac{2 a b g}{5} + \frac{b^{2} d}{5}\right ) + x^{4} \left (\frac{a b f}{2} + \frac{b^{2} c}{4}\right ) + x^{3} \left (\frac{a^{2} h}{3} + \frac{2 a b e}{3}\right ) + x^{2} \left (\frac{a^{2} g}{2} + a b d\right ) + x \left (a^{2} f + 2 a b c\right ) - \frac{a^{2} c + 2 a^{2} d x}{2 x^{2}} \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**3,x)

[Out]

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

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Giac [A]  time = 1.07399, size = 207, normalized size = 1.41 \begin{align*} \frac{1}{9} \, b^{2} h x^{9} + \frac{1}{8} \, b^{2} g x^{8} + \frac{1}{7} \, b^{2} f x^{7} + \frac{1}{3} \, a b h x^{6} + \frac{1}{6} \, b^{2} x^{6} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{2}{5} \, a b g x^{5} + \frac{1}{4} \, b^{2} c x^{4} + \frac{1}{2} \, a b f x^{4} + \frac{1}{3} \, a^{2} h x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} g x^{2} + 2 \, a b c x + a^{2} f x + a^{2} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{2} d x + a^{2} c}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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