3.42.74 \(\int \frac {-45+x^2+2 x^3+4 e^{10+2 x^2} x^3}{x^2} \, dx\)

Optimal. Leaf size=24 \[ e^{10+2 x^2}+\frac {9 (5-x)}{x}+x+x^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 19, normalized size of antiderivative = 0.79, number of steps used = 5, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14, 2209} \begin {gather*} x^2+e^{2 x^2+10}+x+\frac {45}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-45 + x^2 + 2*x^3 + 4*E^(10 + 2*x^2)*x^3)/x^2,x]

[Out]

E^(10 + 2*x^2) + 45/x + x + x^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4 e^{10+2 x^2} x+\frac {-45+x^2+2 x^3}{x^2}\right ) \, dx\\ &=4 \int e^{10+2 x^2} x \, dx+\int \frac {-45+x^2+2 x^3}{x^2} \, dx\\ &=e^{10+2 x^2}+\int \left (1-\frac {45}{x^2}+2 x\right ) \, dx\\ &=e^{10+2 x^2}+\frac {45}{x}+x+x^2\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 19, normalized size = 0.79 \begin {gather*} e^{10+2 x^2}+\frac {45}{x}+x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-45 + x^2 + 2*x^3 + 4*E^(10 + 2*x^2)*x^3)/x^2,x]

[Out]

E^(10 + 2*x^2) + 45/x + x + x^2

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 22, normalized size = 0.92 \begin {gather*} \frac {x^{3} + x^{2} + x e^{\left (2 \, x^{2} + 10\right )} + 45}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*exp(x^2+5)^2+2*x^3+x^2-45)/x^2,x, algorithm="fricas")

[Out]

(x^3 + x^2 + x*e^(2*x^2 + 10) + 45)/x

________________________________________________________________________________________

giac [A]  time = 0.12, size = 22, normalized size = 0.92 \begin {gather*} \frac {x^{3} + x^{2} + x e^{\left (2 \, x^{2} + 10\right )} + 45}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*exp(x^2+5)^2+2*x^3+x^2-45)/x^2,x, algorithm="giac")

[Out]

(x^3 + x^2 + x*e^(2*x^2 + 10) + 45)/x

________________________________________________________________________________________

maple [A]  time = 0.03, size = 19, normalized size = 0.79

method	result	size
risch	\(x^{2}+x +\frac {45}{x}+{\mathrm e}^{2 x^{2}+10}\)	\(19\)
default	\(x^{2}+x +\frac {45}{x}+{\mathrm e}^{10} {\mathrm e}^{2 x^{2}}\)	\(22\)
norman	\(\frac {45+x^{2}+x^{3}+x \,{\mathrm e}^{2 x^{2}+10}}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*exp(x^2+5)^2+2*x^3+x^2-45)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^2+x+45/x+exp(2*x^2+10)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 18, normalized size = 0.75 \begin {gather*} x^{2} + x + \frac {45}{x} + e^{\left (2 \, x^{2} + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*exp(x^2+5)^2+2*x^3+x^2-45)/x^2,x, algorithm="maxima")

[Out]

x^2 + x + 45/x + e^(2*x^2 + 10)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 18, normalized size = 0.75 \begin {gather*} x+{\mathrm {e}}^{2\,x^2+10}+\frac {45}{x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*exp(2*x^2 + 10) + x^2 + 2*x^3 - 45)/x^2,x)

[Out]

x + exp(2*x^2 + 10) + 45/x + x^2

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 15, normalized size = 0.62 \begin {gather*} x^{2} + x + e^{2 x^{2} + 10} + \frac {45}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3*exp(x**2+5)**2+2*x**3+x**2-45)/x**2,x)

[Out]

x**2 + x + exp(2*x**2 + 10) + 45/x

________________________________________________________________________________________