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

Optimal. Leaf size=44 \[ -\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3 \]

[Out]

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

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Rubi [A]  time = 0.0341978, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1628} \[ -\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0040648, size = 44, normalized size = 1. \[ -\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 0.9 \begin{align*} -{\frac{ac}{2\,{x}^{2}}}-{\frac{ad}{x}}+bcx+{\frac{bd{x}^{2}}{2}}+{\frac{be{x}^{3}}{3}}+ae\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)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.937409, size = 51, normalized size = 1.16 \begin{align*} \frac{1}{3} \, b e x^{3} + \frac{1}{2} \, b d x^{2} + b c x + a e \log \left (x\right ) - \frac{2 \, a d x + a 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)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50428, size = 111, normalized size = 2.52 \begin{align*} \frac{2 \, b e x^{5} + 3 \, b d x^{4} + 6 \, b c x^{3} + 6 \, a e x^{2} \log \left (x\right ) - 6 \, a d x - 3 \, a c}{6 \, 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)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.365596, size = 42, normalized size = 0.95 \begin{align*} a e \log{\left (x \right )} + b c x + \frac{b d x^{2}}{2} + \frac{b e x^{3}}{3} - \frac{a c + 2 a 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)/x**3,x)

[Out]

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

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Giac [A]  time = 1.06201, size = 55, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b x^{3} e + \frac{1}{2} \, b d x^{2} + b c x + a e \log \left ({\left | x \right |}\right ) - \frac{2 \, a d x + a 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)/x^3,x, algorithm="giac")

[Out]

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