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

Optimal. Leaf size=83 \[ -\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+a b c x^2+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{1}{5} b^2 c x^5+\frac{1}{6} b^2 d x^6+\frac{1}{7} b^2 e x^7 \]

[Out]

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

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Rubi [A]  time = 0.0629493, antiderivative size = 83, 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} \[ -\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+a b c x^2+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{1}{5} b^2 c x^5+\frac{1}{6} b^2 d x^6+\frac{1}{7} b^2 e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0077615, size = 83, normalized size = 1. \[ -\frac{a^2 c}{x}+a^2 d \log (x)+a^2 e x+a b c x^2+\frac{2}{3} a b d x^3+\frac{1}{2} a b e x^4+\frac{1}{5} b^2 c x^5+\frac{1}{6} b^2 d x^6+\frac{1}{7} b^2 e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 74, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}c}{x}}+{a}^{2}ex+abc{x}^{2}+{\frac{2\,abd{x}^{3}}{3}}+{\frac{abe{x}^{4}}{2}}+{\frac{{b}^{2}c{x}^{5}}{5}}+{\frac{{b}^{2}d{x}^{6}}{6}}+{\frac{{b}^{2}e{x}^{7}}{7}}+{a}^{2}d\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^2,x)

[Out]

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

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Maxima [A]  time = 0.948298, size = 99, normalized size = 1.19 \begin{align*} \frac{1}{7} \, b^{2} e x^{7} + \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{5} \, b^{2} c x^{5} + \frac{1}{2} \, a b e x^{4} + \frac{2}{3} \, a b d x^{3} + a b c x^{2} + a^{2} e x + a^{2} d \log \left (x\right ) - \frac{a^{2} c}{x} \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^2,x, algorithm="maxima")

[Out]

1/7*b^2*e*x^7 + 1/6*b^2*d*x^6 + 1/5*b^2*c*x^5 + 1/2*a*b*e*x^4 + 2/3*a*b*d*x^3 + a*b*c*x^2 + a^2*e*x + a^2*d*lo
g(x) - a^2*c/x

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Fricas [A]  time = 1.53432, size = 204, normalized size = 2.46 \begin{align*} \frac{30 \, b^{2} e x^{8} + 35 \, b^{2} d x^{7} + 42 \, b^{2} c x^{6} + 105 \, a b e x^{5} + 140 \, a b d x^{4} + 210 \, a b c x^{3} + 210 \, a^{2} e x^{2} + 210 \, a^{2} d x \log \left (x\right ) - 210 \, a^{2} c}{210 \, x} \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^2,x, algorithm="fricas")

[Out]

1/210*(30*b^2*e*x^8 + 35*b^2*d*x^7 + 42*b^2*c*x^6 + 105*a*b*e*x^5 + 140*a*b*d*x^4 + 210*a*b*c*x^3 + 210*a^2*e*
x^2 + 210*a^2*d*x*log(x) - 210*a^2*c)/x

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Sympy [A]  time = 0.339739, size = 82, normalized size = 0.99 \begin{align*} - \frac{a^{2} c}{x} + a^{2} d \log{\left (x \right )} + a^{2} e x + a b c x^{2} + \frac{2 a b d x^{3}}{3} + \frac{a b e x^{4}}{2} + \frac{b^{2} c x^{5}}{5} + \frac{b^{2} d x^{6}}{6} + \frac{b^{2} e x^{7}}{7} \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**2,x)

[Out]

-a**2*c/x + a**2*d*log(x) + a**2*e*x + a*b*c*x**2 + 2*a*b*d*x**3/3 + a*b*e*x**4/2 + b**2*c*x**5/5 + b**2*d*x**
6/6 + b**2*e*x**7/7

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Giac [A]  time = 1.05151, size = 104, normalized size = 1.25 \begin{align*} \frac{1}{7} \, b^{2} x^{7} e + \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{5} \, b^{2} c x^{5} + \frac{1}{2} \, a b x^{4} e + \frac{2}{3} \, a b d x^{3} + a b c 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((e*x^2+d*x+c)*(b*x^3+a)^2/x^2,x, algorithm="giac")

[Out]

1/7*b^2*x^7*e + 1/6*b^2*d*x^6 + 1/5*b^2*c*x^5 + 1/2*a*b*x^4*e + 2/3*a*b*d*x^3 + a*b*c*x^2 + a^2*x*e + a^2*d*lo
g(abs(x)) - a^2*c/x