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

Optimal. Leaf size=125 \[ \frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10} \]

[Out]

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

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Rubi [A]  time = 0.0922203, antiderivative size = 125, 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} \[ \frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0083931, size = 125, normalized size = 1. \[ \frac{3}{2} a^2 b c x^2+a^2 b d x^3+\frac{3}{4} a^2 b e x^4-\frac{a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac{3}{5} a b^2 c x^5+\frac{1}{2} a b^2 d x^6+\frac{3}{7} a b^2 e x^7+\frac{1}{8} b^3 c x^8+\frac{1}{9} b^3 d x^9+\frac{1}{10} b^3 e x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}c}{x}}+{a}^{3}ex+{\frac{3\,{a}^{2}bc{x}^{2}}{2}}+{a}^{2}bd{x}^{3}+{\frac{3\,{a}^{2}be{x}^{4}}{4}}+{\frac{3\,a{b}^{2}c{x}^{5}}{5}}+{\frac{a{b}^{2}d{x}^{6}}{2}}+{\frac{3\,a{b}^{2}e{x}^{7}}{7}}+{\frac{{b}^{3}c{x}^{8}}{8}}+{\frac{{b}^{3}d{x}^{9}}{9}}+{\frac{{b}^{3}e{x}^{10}}{10}}+{a}^{3}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)^3/x^2,x)

[Out]

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

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Maxima [A]  time = 0.940401, size = 147, normalized size = 1.18 \begin{align*} \frac{1}{10} \, b^{3} e x^{10} + \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{8} \, b^{3} c x^{8} + \frac{3}{7} \, a b^{2} e x^{7} + \frac{1}{2} \, a b^{2} d x^{6} + \frac{3}{5} \, a b^{2} c x^{5} + \frac{3}{4} \, a^{2} b e x^{4} + a^{2} b d x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + a^{3} e x + a^{3} d \log \left (x\right ) - \frac{a^{3} 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)^3/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45317, size = 305, normalized size = 2.44 \begin{align*} \frac{252 \, b^{3} e x^{11} + 280 \, b^{3} d x^{10} + 315 \, b^{3} c x^{9} + 1080 \, a b^{2} e x^{8} + 1260 \, a b^{2} d x^{7} + 1512 \, a b^{2} c x^{6} + 1890 \, a^{2} b e x^{5} + 2520 \, a^{2} b d x^{4} + 3780 \, a^{2} b c x^{3} + 2520 \, a^{3} e x^{2} + 2520 \, a^{3} d x \log \left (x\right ) - 2520 \, a^{3} c}{2520 \, 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)^3/x^2,x, algorithm="fricas")

[Out]

1/2520*(252*b^3*e*x^11 + 280*b^3*d*x^10 + 315*b^3*c*x^9 + 1080*a*b^2*e*x^8 + 1260*a*b^2*d*x^7 + 1512*a*b^2*c*x
^6 + 1890*a^2*b*e*x^5 + 2520*a^2*b*d*x^4 + 3780*a^2*b*c*x^3 + 2520*a^3*e*x^2 + 2520*a^3*d*x*log(x) - 2520*a^3*
c)/x

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Sympy [A]  time = 0.39922, size = 128, normalized size = 1.02 \begin{align*} - \frac{a^{3} c}{x} + a^{3} d \log{\left (x \right )} + a^{3} e x + \frac{3 a^{2} b c x^{2}}{2} + a^{2} b d x^{3} + \frac{3 a^{2} b e x^{4}}{4} + \frac{3 a b^{2} c x^{5}}{5} + \frac{a b^{2} d x^{6}}{2} + \frac{3 a b^{2} e x^{7}}{7} + \frac{b^{3} c x^{8}}{8} + \frac{b^{3} d x^{9}}{9} + \frac{b^{3} e x^{10}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05804, size = 154, normalized size = 1.23 \begin{align*} \frac{1}{10} \, b^{3} x^{10} e + \frac{1}{9} \, b^{3} d x^{9} + \frac{1}{8} \, b^{3} c x^{8} + \frac{3}{7} \, a b^{2} x^{7} e + \frac{1}{2} \, a b^{2} d x^{6} + \frac{3}{5} \, a b^{2} c x^{5} + \frac{3}{4} \, a^{2} b x^{4} e + a^{2} b d x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + a^{3} x e + a^{3} d \log \left ({\left | x \right |}\right ) - \frac{a^{3} 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)^3/x^2,x, algorithm="giac")

[Out]

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