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+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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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 46, normalized size = 1.00 \[ a c \log (x)+a d x+\frac {1}{2} a e x^2+\frac {1}{3} b c x^3+\frac {1}{4} b d x^4+\frac {1}{5} b e x^5 \]

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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fricas [A]  time = 0.72, size = 38, normalized size = 0.83 \[ \frac {1}{5} \, b e x^{5} + \frac {1}{4} \, b d x^{4} + \frac {1}{3} \, b c x^{3} + \frac {1}{2} \, a e x^{2} + a d x + a c \log \relax (x) \]

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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giac [A]  time = 0.15, size = 41, normalized size = 0.89 \[ \frac {1}{5} \, b x^{5} e + \frac {1}{4} \, b d x^{4} + \frac {1}{3} \, b c x^{3} + \frac {1}{2} \, a x^{2} e + a d x + a c \log \left ({\left | x \right |}\right ) \]

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))

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maple [A]  time = 0.05, size = 39, normalized size = 0.85 \[ \frac {b e \,x^{5}}{5}+\frac {b d \,x^{4}}{4}+\frac {b c \,x^{3}}{3}+\frac {a e \,x^{2}}{2}+a c \ln \relax (x )+a d x \]

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 = 1.33, size = 38, normalized size = 0.83 \[ \frac {1}{5} \, b e x^{5} + \frac {1}{4} \, b d x^{4} + \frac {1}{3} \, b c x^{3} + \frac {1}{2} \, a e x^{2} + a d x + a c \log \relax (x) \]

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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mupad [B]  time = 0.03, size = 38, normalized size = 0.83 \[ a\,c\,\ln \relax (x)+a\,d\,x+\frac {b\,c\,x^3}{3}+\frac {a\,e\,x^2}{2}+\frac {b\,d\,x^4}{4}+\frac {b\,e\,x^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.14, size = 44, normalized size = 0.96 \[ a c \log {\relax (x )} + a d x + \frac {a e x^{2}}{2} + \frac {b c x^{3}}{3} + \frac {b d x^{4}}{4} + \frac {b e x^{5}}{5} \]

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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