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

Optimal. Leaf size=126 \[ 3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9 \]

[Out]

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

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Rubi [A]  time = 0.0864589, antiderivative size = 126, 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} \[ 3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0090364, size = 126, normalized size = 1. \[ 3 a^2 b c x+\frac{3}{2} a^2 b d x^2+a^2 b e x^3-\frac{a^3 c}{2 x^2}-\frac{a^3 d}{x}+a^3 e \log (x)+\frac{3}{4} a b^2 c x^4+\frac{3}{5} a b^2 d x^5+\frac{1}{2} a b^2 e x^6+\frac{1}{7} b^3 c x^7+\frac{1}{8} b^3 d x^8+\frac{1}{9} b^3 e x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 111, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}c}{2\,{x}^{2}}}-{\frac{{a}^{3}d}{x}}+3\,{a}^{2}bcx+{\frac{3\,{a}^{2}bd{x}^{2}}{2}}+{a}^{2}be{x}^{3}+{\frac{3\,a{b}^{2}c{x}^{4}}{4}}+{\frac{3\,a{b}^{2}d{x}^{5}}{5}}+{\frac{a{b}^{2}e{x}^{6}}{2}}+{\frac{{b}^{3}c{x}^{7}}{7}}+{\frac{{b}^{3}d{x}^{8}}{8}}+{\frac{{b}^{3}e{x}^{9}}{9}}+{a}^{3}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)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 0.935902, size = 149, normalized size = 1.18 \begin{align*} \frac{1}{9} \, b^{3} e x^{9} + \frac{1}{8} \, b^{3} d x^{8} + \frac{1}{7} \, b^{3} c x^{7} + \frac{1}{2} \, a b^{2} e x^{6} + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{4} \, a b^{2} c x^{4} + a^{2} b e x^{3} + \frac{3}{2} \, a^{2} b d x^{2} + 3 \, a^{2} b c x + a^{3} e \log \left (x\right ) - \frac{2 \, a^{3} d x + a^{3} 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)^3/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46691, size = 308, normalized size = 2.44 \begin{align*} \frac{280 \, b^{3} e x^{11} + 315 \, b^{3} d x^{10} + 360 \, b^{3} c x^{9} + 1260 \, a b^{2} e x^{8} + 1512 \, a b^{2} d x^{7} + 1890 \, a b^{2} c x^{6} + 2520 \, a^{2} b e x^{5} + 3780 \, a^{2} b d x^{4} + 7560 \, a^{2} b c x^{3} + 2520 \, a^{3} e x^{2} \log \left (x\right ) - 2520 \, a^{3} d x - 1260 \, a^{3} c}{2520 \, 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)^3/x^3,x, algorithm="fricas")

[Out]

1/2520*(280*b^3*e*x^11 + 315*b^3*d*x^10 + 360*b^3*c*x^9 + 1260*a*b^2*e*x^8 + 1512*a*b^2*d*x^7 + 1890*a*b^2*c*x
^6 + 2520*a^2*b*e*x^5 + 3780*a^2*b*d*x^4 + 7560*a^2*b*c*x^3 + 2520*a^3*e*x^2*log(x) - 2520*a^3*d*x - 1260*a^3*
c)/x^2

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Sympy [A]  time = 0.460926, size = 129, normalized size = 1.02 \begin{align*} a^{3} e \log{\left (x \right )} + 3 a^{2} b c x + \frac{3 a^{2} b d x^{2}}{2} + a^{2} b e x^{3} + \frac{3 a b^{2} c x^{4}}{4} + \frac{3 a b^{2} d x^{5}}{5} + \frac{a b^{2} e x^{6}}{2} + \frac{b^{3} c x^{7}}{7} + \frac{b^{3} d x^{8}}{8} + \frac{b^{3} e x^{9}}{9} - \frac{a^{3} c + 2 a^{3} 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)**3/x**3,x)

[Out]

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

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Giac [A]  time = 1.0553, size = 155, normalized size = 1.23 \begin{align*} \frac{1}{9} \, b^{3} x^{9} e + \frac{1}{8} \, b^{3} d x^{8} + \frac{1}{7} \, b^{3} c x^{7} + \frac{1}{2} \, a b^{2} x^{6} e + \frac{3}{5} \, a b^{2} d x^{5} + \frac{3}{4} \, a b^{2} c x^{4} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + 3 \, a^{2} b c x + a^{3} e \log \left ({\left | x \right |}\right ) - \frac{2 \, a^{3} d x + a^{3} 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)^3/x^3,x, algorithm="giac")

[Out]

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