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

Optimal. Leaf size=44 \[ -\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4 \]

[Out]

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

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Rubi [A]  time = 0.0350813, 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} \[ -\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0048538, size = 44, normalized size = 1. \[ -\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 39, normalized size = 0.9 \begin{align*} -{\frac{ac}{x}}+aex+{\frac{bc{x}^{2}}{2}}+{\frac{bd{x}^{3}}{3}}+{\frac{be{x}^{4}}{4}}+ad\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^2,x)

[Out]

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

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Maxima [A]  time = 0.943359, size = 51, normalized size = 1.16 \begin{align*} \frac{1}{4} \, b e x^{4} + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, b c x^{2} + a e x + a d \log \left (x\right ) - \frac{a 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)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43856, size = 113, normalized size = 2.57 \begin{align*} \frac{3 \, b e x^{5} + 4 \, b d x^{4} + 6 \, b c x^{3} + 12 \, a e x^{2} + 12 \, a d x \log \left (x\right ) - 12 \, a c}{12 \, 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)/x^2,x, algorithm="fricas")

[Out]

1/12*(3*b*e*x^5 + 4*b*d*x^4 + 6*b*c*x^3 + 12*a*e*x^2 + 12*a*d*x*log(x) - 12*a*c)/x

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Sympy [A]  time = 0.296316, size = 41, normalized size = 0.93 \begin{align*} - \frac{a c}{x} + a d \log{\left (x \right )} + a e x + \frac{b c x^{2}}{2} + \frac{b d x^{3}}{3} + \frac{b e x^{4}}{4} \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**2,x)

[Out]

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

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Giac [A]  time = 1.06735, size = 55, normalized size = 1.25 \begin{align*} \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, b c x^{2} + a x e + a d \log \left ({\left | x \right |}\right ) - \frac{a 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)/x^2,x, algorithm="giac")

[Out]

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