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

Optimal. Leaf size=147 \[ -\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+\frac{1}{5} b x^5 (2 a f+b c)+\frac{1}{2} a x^2 (a f+2 b c)+\frac{2}{3} a b d x^3+\frac{1}{7} b x^7 (2 a h+b e)+\frac{1}{4} a x^4 (a h+2 b e)+\frac{g \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 d x^6+\frac{1}{8} b^2 f x^8+\frac{1}{10} b^2 h x^{10} \]

[Out]

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

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Rubi [A]  time = 0.127544, 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} \[ -\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+\frac{1}{5} b x^5 (2 a f+b c)+\frac{1}{2} a x^2 (a f+2 b c)+\frac{2}{3} a b d x^3+\frac{1}{7} b x^7 (2 a h+b e)+\frac{1}{4} a x^4 (a h+2 b e)+\frac{g \left (a+b x^3\right )^3}{9 b}+\frac{1}{6} b^2 d x^6+\frac{1}{8} b^2 f x^8+\frac{1}{10} b^2 h x^{10} \]

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^2,x]

[Out]

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

Mathematica [A]  time = 0.0605502, size = 152, normalized size = 1.03 \[ -\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+\frac{1}{5} b x^5 (2 a f+b c)+\frac{1}{2} a x^2 (a f+2 b c)+\frac{1}{6} b x^6 (2 a g+b d)+\frac{1}{3} a x^3 (a g+2 b d)+\frac{1}{7} b x^7 (2 a h+b e)+\frac{1}{4} a x^4 (a h+2 b e)+\frac{1}{8} b^2 f x^8+\frac{1}{9} b^2 g x^9+\frac{1}{10} b^2 h x^{10} \]

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^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 152, normalized size = 1. \begin{align*}{\frac{{b}^{2}h{x}^{10}}{10}}+{\frac{{b}^{2}g{x}^{9}}{9}}+{\frac{{b}^{2}f{x}^{8}}{8}}+{\frac{2\,{x}^{7}abh}{7}}+{\frac{{b}^{2}e{x}^{7}}{7}}+{\frac{{x}^{6}abg}{3}}+{\frac{{b}^{2}d{x}^{6}}{6}}+{\frac{2\,{x}^{5}abf}{5}}+{\frac{{b}^{2}c{x}^{5}}{5}}+{\frac{{x}^{4}{a}^{2}h}{4}}+{\frac{abe{x}^{4}}{2}}+{\frac{{x}^{3}{a}^{2}g}{3}}+{\frac{2\,abd{x}^{3}}{3}}+{\frac{{a}^{2}f{x}^{2}}{2}}+abc{x}^{2}+{a}^{2}ex+{a}^{2}d\ln \left ( x \right ) -{\frac{{a}^{2}c}{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^2,x)

[Out]

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

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Maxima [A]  time = 0.958382, size = 197, normalized size = 1.34 \begin{align*} \frac{1}{10} \, b^{2} h x^{10} + \frac{1}{9} \, b^{2} g x^{9} + \frac{1}{8} \, b^{2} f x^{8} + \frac{1}{7} \,{\left (b^{2} e + 2 \, a b h\right )} x^{7} + \frac{1}{6} \,{\left (b^{2} d + 2 \, a b g\right )} x^{6} + \frac{1}{5} \,{\left (b^{2} c + 2 \, a b f\right )} x^{5} + \frac{1}{4} \,{\left (2 \, a b e + a^{2} h\right )} x^{4} + a^{2} e x + \frac{1}{3} \,{\left (2 \, a b d + a^{2} g\right )} x^{3} + a^{2} d \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} f\right )} x^{2} - \frac{a^{2} c}{x} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.992344, size = 379, normalized size = 2.58 \begin{align*} \frac{252 \, b^{2} h x^{11} + 280 \, b^{2} g x^{10} + 315 \, b^{2} f x^{9} + 360 \,{\left (b^{2} e + 2 \, a b h\right )} x^{8} + 420 \,{\left (b^{2} d + 2 \, a b g\right )} x^{7} + 504 \,{\left (b^{2} c + 2 \, a b f\right )} x^{6} + 630 \,{\left (2 \, a b e + a^{2} h\right )} x^{5} + 2520 \, a^{2} e x^{2} + 840 \,{\left (2 \, a b d + a^{2} g\right )} x^{4} + 2520 \, a^{2} d x \log \left (x\right ) + 1260 \,{\left (2 \, a b c + a^{2} f\right )} x^{3} - 2520 \, a^{2} c}{2520 \, x} \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^2,x, algorithm="fricas")

[Out]

1/2520*(252*b^2*h*x^11 + 280*b^2*g*x^10 + 315*b^2*f*x^9 + 360*(b^2*e + 2*a*b*h)*x^8 + 420*(b^2*d + 2*a*b*g)*x^
7 + 504*(b^2*c + 2*a*b*f)*x^6 + 630*(2*a*b*e + a^2*h)*x^5 + 2520*a^2*e*x^2 + 840*(2*a*b*d + a^2*g)*x^4 + 2520*
a^2*d*x*log(x) + 1260*(2*a*b*c + a^2*f)*x^3 - 2520*a^2*c)/x

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Sympy [A]  time = 0.499578, size = 156, normalized size = 1.06 \begin{align*} - \frac{a^{2} c}{x} + a^{2} d \log{\left (x \right )} + a^{2} e x + \frac{b^{2} f x^{8}}{8} + \frac{b^{2} g x^{9}}{9} + \frac{b^{2} h x^{10}}{10} + x^{7} \left (\frac{2 a b h}{7} + \frac{b^{2} e}{7}\right ) + x^{6} \left (\frac{a b g}{3} + \frac{b^{2} d}{6}\right ) + x^{5} \left (\frac{2 a b f}{5} + \frac{b^{2} c}{5}\right ) + x^{4} \left (\frac{a^{2} h}{4} + \frac{a b e}{2}\right ) + x^{3} \left (\frac{a^{2} g}{3} + \frac{2 a b d}{3}\right ) + x^{2} \left (\frac{a^{2} f}{2} + a b c\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**2,x)

[Out]

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

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Giac [A]  time = 1.05546, size = 209, normalized size = 1.42 \begin{align*} \frac{1}{10} \, b^{2} h x^{10} + \frac{1}{9} \, b^{2} g x^{9} + \frac{1}{8} \, b^{2} f x^{8} + \frac{2}{7} \, a b h x^{7} + \frac{1}{7} \, b^{2} x^{7} e + \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{3} \, a b g x^{6} + \frac{1}{5} \, b^{2} c x^{5} + \frac{2}{5} \, a b f x^{5} + \frac{1}{4} \, a^{2} h x^{4} + \frac{1}{2} \, a b x^{4} e + \frac{2}{3} \, a b d x^{3} + \frac{1}{3} \, a^{2} g x^{3} + a b c x^{2} + \frac{1}{2} \, a^{2} f x^{2} + a^{2} x e + a^{2} d \log \left ({\left | x \right |}\right ) - \frac{a^{2} c}{x} \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^2,x, algorithm="giac")

[Out]

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