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

Optimal. Leaf size=88 \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{2} a b d x^4+\frac{2}{5} a b e x^5+\frac{1}{6} b^2 c x^6+\frac{1}{7} b^2 d x^7+\frac{1}{8} b^2 e x^8 \]

[Out]

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

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Rubi [A]  time = 0.0518618, antiderivative size = 88, 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} \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{2} a b d x^4+\frac{2}{5} a b e x^5+\frac{1}{6} b^2 c x^6+\frac{1}{7} b^2 d x^7+\frac{1}{8} b^2 e x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0076852, size = 88, normalized size = 1. \[ a^2 c \log (x)+a^2 d x+\frac{1}{2} a^2 e x^2+\frac{2}{3} a b c x^3+\frac{1}{2} a b d x^4+\frac{2}{5} a b e x^5+\frac{1}{6} b^2 c x^6+\frac{1}{7} b^2 d x^7+\frac{1}{8} b^2 e x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.001, size = 75, normalized size = 0.9 \begin{align*}{a}^{2}dx+{\frac{{a}^{2}e{x}^{2}}{2}}+{\frac{2\,abc{x}^{3}}{3}}+{\frac{abd{x}^{4}}{2}}+{\frac{2\,abe{x}^{5}}{5}}+{\frac{{b}^{2}c{x}^{6}}{6}}+{\frac{{b}^{2}d{x}^{7}}{7}}+{\frac{{b}^{2}e{x}^{8}}{8}}+{a}^{2}c\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,x)

[Out]

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

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Maxima [A]  time = 0.948685, size = 100, normalized size = 1.14 \begin{align*} \frac{1}{8} \, b^{2} e x^{8} + \frac{1}{7} \, b^{2} d x^{7} + \frac{1}{6} \, b^{2} c x^{6} + \frac{2}{5} \, a b e x^{5} + \frac{1}{2} \, a b d x^{4} + \frac{2}{3} \, a b c x^{3} + \frac{1}{2} \, a^{2} e x^{2} + a^{2} d x + a^{2} c \log \left (x\right ) \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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49492, size = 184, normalized size = 2.09 \begin{align*} \frac{1}{8} \, b^{2} e x^{8} + \frac{1}{7} \, b^{2} d x^{7} + \frac{1}{6} \, b^{2} c x^{6} + \frac{2}{5} \, a b e x^{5} + \frac{1}{2} \, a b d x^{4} + \frac{2}{3} \, a b c x^{3} + \frac{1}{2} \, a^{2} e x^{2} + a^{2} d x + a^{2} c \log \left (x\right ) \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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.352949, size = 88, normalized size = 1. \begin{align*} a^{2} c \log{\left (x \right )} + a^{2} d x + \frac{a^{2} e x^{2}}{2} + \frac{2 a b c x^{3}}{3} + \frac{a b d x^{4}}{2} + \frac{2 a b e x^{5}}{5} + \frac{b^{2} c x^{6}}{6} + \frac{b^{2} d x^{7}}{7} + \frac{b^{2} e x^{8}}{8} \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,x)

[Out]

a**2*c*log(x) + a**2*d*x + a**2*e*x**2/2 + 2*a*b*c*x**3/3 + a*b*d*x**4/2 + 2*a*b*e*x**5/5 + b**2*c*x**6/6 + b*
*2*d*x**7/7 + b**2*e*x**8/8

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Giac [A]  time = 1.05081, size = 105, normalized size = 1.19 \begin{align*} \frac{1}{8} \, b^{2} x^{8} e + \frac{1}{7} \, b^{2} d x^{7} + \frac{1}{6} \, b^{2} c x^{6} + \frac{2}{5} \, a b x^{5} e + \frac{1}{2} \, a b d x^{4} + \frac{2}{3} \, a b c x^{3} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x + a^{2} c \log \left ({\left | x \right |}\right ) \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,x, algorithm="giac")

[Out]

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