3.324 \(\int \frac{(c+d x+e x^2) (a+b x^3)^2}{x^3} \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+2 a b c x+a b d x^2+\frac{2}{3} a b e x^3+\frac{1}{4} b^2 c x^4+\frac{1}{5} b^2 d x^5+\frac{1}{6} b^2 e x^6 \]

[Out]

-(a^2*c)/(2*x^2) - (a^2*d)/x + 2*a*b*c*x + a*b*d*x^2 + (2*a*b*e*x^3)/3 + (b^2*c*x^4)/4 + (b^2*d*x^5)/5 + (b^2*
e*x^6)/6 + a^2*e*Log[x]

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Rubi [A]  time = 0.0641961, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1628} \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+2 a b c x+a b d x^2+\frac{2}{3} a b e x^3+\frac{1}{4} b^2 c x^4+\frac{1}{5} b^2 d x^5+\frac{1}{6} b^2 e x^6 \]

Antiderivative was successfully verified.

[In]

Int[((c + d*x + e*x^2)*(a + b*x^3)^2)/x^3,x]

[Out]

-(a^2*c)/(2*x^2) - (a^2*d)/x + 2*a*b*c*x + a*b*d*x^2 + (2*a*b*e*x^3)/3 + (b^2*c*x^4)/4 + (b^2*d*x^5)/5 + (b^2*
e*x^6)/6 + a^2*e*Log[x]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.0084296, size = 84, normalized size = 1. \[ -\frac{a^2 c}{2 x^2}-\frac{a^2 d}{x}+a^2 e \log (x)+2 a b c x+a b d x^2+\frac{2}{3} a b e x^3+\frac{1}{4} b^2 c x^4+\frac{1}{5} b^2 d x^5+\frac{1}{6} b^2 e x^6 \]

Antiderivative was successfully verified.

[In]

Integrate[((c + d*x + e*x^2)*(a + b*x^3)^2)/x^3,x]

[Out]

-(a^2*c)/(2*x^2) - (a^2*d)/x + 2*a*b*c*x + a*b*d*x^2 + (2*a*b*e*x^3)/3 + (b^2*c*x^4)/4 + (b^2*d*x^5)/5 + (b^2*
e*x^6)/6 + a^2*e*Log[x]

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Maple [A]  time = 0.006, size = 75, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}c}{2\,{x}^{2}}}-{\frac{{a}^{2}d}{x}}+2\,abcx+abd{x}^{2}+{\frac{2\,abe{x}^{3}}{3}}+{\frac{{b}^{2}c{x}^{4}}{4}}+{\frac{{b}^{2}d{x}^{5}}{5}}+{\frac{{b}^{2}e{x}^{6}}{6}}+{a}^{2}e\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d*x+c)*(b*x^3+a)^2/x^3,x)

[Out]

-1/2*a^2*c/x^2-a^2*d/x+2*a*b*c*x+a*b*d*x^2+2/3*a*b*e*x^3+1/4*b^2*c*x^4+1/5*b^2*d*x^5+1/6*b^2*e*x^6+a^2*e*ln(x)

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Maxima [A]  time = 0.943366, size = 100, normalized size = 1.19 \begin{align*} \frac{1}{6} \, b^{2} e x^{6} + \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{4} \, b^{2} c x^{4} + \frac{2}{3} \, a b e x^{3} + a b d x^{2} + 2 \, a b c x + a^{2} e \log \left (x\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((e*x^2+d*x+c)*(b*x^3+a)^2/x^3,x, algorithm="maxima")

[Out]

1/6*b^2*e*x^6 + 1/5*b^2*d*x^5 + 1/4*b^2*c*x^4 + 2/3*a*b*e*x^3 + a*b*d*x^2 + 2*a*b*c*x + a^2*e*log(x) - 1/2*(2*
a^2*d*x + a^2*c)/x^2

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Fricas [A]  time = 1.46572, size = 198, normalized size = 2.36 \begin{align*} \frac{10 \, b^{2} e x^{8} + 12 \, b^{2} d x^{7} + 15 \, b^{2} c x^{6} + 40 \, a b e x^{5} + 60 \, a b d x^{4} + 120 \, a b c x^{3} + 60 \, a^{2} e x^{2} \log \left (x\right ) - 60 \, a^{2} d x - 30 \, a^{2} c}{60 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d*x+c)*(b*x^3+a)^2/x^3,x, algorithm="fricas")

[Out]

1/60*(10*b^2*e*x^8 + 12*b^2*d*x^7 + 15*b^2*c*x^6 + 40*a*b*e*x^5 + 60*a*b*d*x^4 + 120*a*b*c*x^3 + 60*a^2*e*x^2*
log(x) - 60*a^2*d*x - 30*a^2*c)/x^2

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Sympy [A]  time = 0.394683, size = 85, normalized size = 1.01 \begin{align*} a^{2} e \log{\left (x \right )} + 2 a b c x + a b d x^{2} + \frac{2 a b e x^{3}}{3} + \frac{b^{2} c x^{4}}{4} + \frac{b^{2} d x^{5}}{5} + \frac{b^{2} e x^{6}}{6} - \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((e*x**2+d*x+c)*(b*x**3+a)**2/x**3,x)

[Out]

a**2*e*log(x) + 2*a*b*c*x + a*b*d*x**2 + 2*a*b*e*x**3/3 + b**2*c*x**4/4 + b**2*d*x**5/5 + b**2*e*x**6/6 - (a**
2*c + 2*a**2*d*x)/(2*x**2)

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Giac [A]  time = 1.05293, size = 105, normalized size = 1.25 \begin{align*} \frac{1}{6} \, b^{2} x^{6} e + \frac{1}{5} \, b^{2} d x^{5} + \frac{1}{4} \, b^{2} c x^{4} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + 2 \, a b c 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((e*x^2+d*x+c)*(b*x^3+a)^2/x^3,x, algorithm="giac")

[Out]

1/6*b^2*x^6*e + 1/5*b^2*d*x^5 + 1/4*b^2*c*x^4 + 2/3*a*b*x^3*e + a*b*d*x^2 + 2*a*b*c*x + a^2*e*log(abs(x)) - 1/
2*(2*a^2*d*x + a^2*c)/x^2