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

Optimal. Leaf size=46 \[ a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4+\frac{1}{5} b e x^5 \]

[Out]

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

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Rubi [A]  time = 0.0245177, antiderivative size = 46, 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} \[ a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4+\frac{1}{5} b e x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0041212, size = 46, normalized size = 1. \[ a c \log (x)+a d x+\frac{1}{2} a e x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4+\frac{1}{5} b e x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 0.9 \begin{align*} adx+{\frac{ae{x}^{2}}{2}}+{\frac{bc{x}^{3}}{3}}+{\frac{bd{x}^{4}}{4}}+{\frac{be{x}^{5}}{5}}+ac\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,x)

[Out]

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

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Maxima [A]  time = 0.949336, size = 51, normalized size = 1.11 \begin{align*} \frac{1}{5} \, b e x^{5} + \frac{1}{4} \, b d x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{2} \, a e x^{2} + a d x + a 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)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49063, size = 103, normalized size = 2.24 \begin{align*} \frac{1}{5} \, b e x^{5} + \frac{1}{4} \, b d x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{2} \, a e x^{2} + a d x + a 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)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.272387, size = 44, normalized size = 0.96 \begin{align*} a c \log{\left (x \right )} + a d x + \frac{a e x^{2}}{2} + \frac{b c x^{3}}{3} + \frac{b d x^{4}}{4} + \frac{b e x^{5}}{5} \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,x)

[Out]

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

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Giac [A]  time = 1.08311, size = 55, normalized size = 1.2 \begin{align*} \frac{1}{5} \, b x^{5} e + \frac{1}{4} \, b d x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{2} \, a x^{2} e + a d x + a 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)/x,x, algorithm="giac")

[Out]

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