\(\int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} (1+2 x^2+(-2 x^2+20 x^3+4 x^4) \log (7))}{2 x^2} \, dx\) [4342]

Optimal result

Integrand size = 55, antiderivative size = 25 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=-1+\left (e^{5+x^2 \log (7)}+x\right ) \left (5-\frac {1}{2 x}+x\right ) \]

[Out]

(x+5-1/2/x)*(exp(x^2*ln(7)+5)+x)-1

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.16, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.127, Rules used = {12, 14, 6874, 2245, 2235, 2240, 2243} \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=\frac {1}{2} e^5 \sqrt {\pi \log (7)} \text {erfi}\left (x \sqrt {\log (7)}\right )-\frac {1}{2} e^5 \sqrt {\frac {\pi }{\log (7)}} \text {erfi}\left (x \sqrt {\log (7)}\right )+\frac {1}{2} e^5 (1-\log (7)) \sqrt {\frac {\pi }{\log (7)}} \text {erfi}\left (x \sqrt {\log (7)}\right )+e^5 7^{x^2} x-\frac {e^5 7^{x^2}}{2 x}+5 e^5 7^{x^2}+\frac {1}{4} (2 x+5)^2 \]

[In]

Int[(10*x^2 + 4*x^3 + E^(5 + x^2*Log[7])*(1 + 2*x^2 + (-2*x^2 + 20*x^3 + 4*x^4)*Log[7]))/(2*x^2),x]

[Out]

5*7^x^2*E^5 - (7^x^2*E^5)/(2*x) + 7^x^2*E^5*x + (5 + 2*x)^2/4 - (E^5*Erfi[x*Sqrt[Log[7]]]*Sqrt[Pi/Log[7]])/2 +
(E^5*Erfi[x*Sqrt[Log[7]]]*(1 - Log[7])*Sqrt[Pi/Log[7]])/2 + (E^5*Erfi[x*Sqrt[Log[7]]]*Sqrt[Pi*Log[7]])/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2240

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]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2245

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{x^2} \, dx \\ & = \frac {1}{2} \int \left (2 (5+2 x)+\frac {7^{x^2} e^5 \left (1+2 x^2 (1-\log (7))+20 x^3 \log (7)+4 x^4 \log (7)\right )}{x^2}\right ) \, dx \\ & = \frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \int \frac {7^{x^2} \left (1+2 x^2 (1-\log (7))+20 x^3 \log (7)+4 x^4 \log (7)\right )}{x^2} \, dx \\ & = \frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \int \left (\frac {7^{x^2}}{x^2}-2\ 7^{x^2} (-1+\log (7))+20\ 7^{x^2} x \log (7)+4\ 7^{x^2} x^2 \log (7)\right ) \, dx \\ & = \frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \int \frac {7^{x^2}}{x^2} \, dx+\left (e^5 (1-\log (7))\right ) \int 7^{x^2} \, dx+\left (2 e^5 \log (7)\right ) \int 7^{x^2} x^2 \, dx+\left (10 e^5 \log (7)\right ) \int 7^{x^2} x \, dx \\ & = 5\ 7^{x^2} e^5-\frac {7^{x^2} e^5}{2 x}+7^{x^2} e^5 x+\frac {1}{4} (5+2 x)^2+\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) (1-\log (7)) \sqrt {\frac {\pi }{\log (7)}}-e^5 \int 7^{x^2} \, dx+\left (e^5 \log (7)\right ) \int 7^{x^2} \, dx \\ & = 5\ 7^{x^2} e^5-\frac {7^{x^2} e^5}{2 x}+7^{x^2} e^5 x+\frac {1}{4} (5+2 x)^2-\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) \sqrt {\frac {\pi }{\log (7)}}+\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) (1-\log (7)) \sqrt {\frac {\pi }{\log (7)}}+\frac {1}{2} e^5 \text {erfi}\left (x \sqrt {\log (7)}\right ) \sqrt {\pi \log (7)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=5\ 7^{x^2} e^5-\frac {7^{x^2} e^5}{2 x}+5 x+7^{x^2} e^5 x+x^2 \]

[In]

Integrate[(10*x^2 + 4*x^3 + E^(5 + x^2*Log[7])*(1 + 2*x^2 + (-2*x^2 + 20*x^3 + 4*x^4)*Log[7]))/(2*x^2),x]

[Out]

5*7^x^2*E^5 - (7^x^2*E^5)/(2*x) + 5*x + 7^x^2*E^5*x + x^2

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20

method	result	size
risch	\(x^{2}+5 x +\frac {\left (2 x^{2}+10 x -1\right ) 7^{x^{2}} {\mathrm e}^{5}}{2 x}\)	\(30\)
default	\(x \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}+5 \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}-\frac {{\mathrm e}^{x^{2} \ln \left (7\right )+5}}{2 x}+x^{2}+5 x\)	\(44\)
parts	\(x \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}+5 \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}-\frac {{\mathrm e}^{x^{2} \ln \left (7\right )+5}}{2 x}+x^{2}+5 x\)	\(44\)
norman	\(\frac {x^{3}+{\mathrm e}^{x^{2} \ln \left (7\right )+5} x^{2}+5 x^{2}+5 x \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}-\frac {{\mathrm e}^{x^{2} \ln \left (7\right )+5}}{2}}{x}\)	\(50\)
parallelrisch	\(\frac {2 x^{3}+2 \,{\mathrm e}^{x^{2} \ln \left (7\right )+5} x^{2}+10 x^{2}+10 x \,{\mathrm e}^{x^{2} \ln \left (7\right )+5}-{\mathrm e}^{x^{2} \ln \left (7\right )+5}}{2 x}\)	\(54\)

[In]

int(1/2*(((4*x^4+20*x^3-2*x^2)*ln(7)+2*x^2+1)*exp(x^2*ln(7)+5)+4*x^3+10*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^2+5*x+1/2*(2*x^2+10*x-1)/x*7^(x^2)*exp(5)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=\frac {2 \, x^{3} + 10 \, x^{2} + {\left (2 \, x^{2} + 10 \, x - 1\right )} e^{\left (x^{2} \log \left (7\right ) + 5\right )}}{2 \, x} \]

[In]

integrate(1/2*(((4*x^4+20*x^3-2*x^2)*log(7)+2*x^2+1)*exp(x^2*log(7)+5)+4*x^3+10*x^2)/x^2,x, algorithm="fricas"
)

[Out]

1/2*(2*x^3 + 10*x^2 + (2*x^2 + 10*x - 1)*e^(x^2*log(7) + 5))/x

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=x^{2} + 5 x + \frac {\left (2 x^{2} + 10 x - 1\right ) e^{x^{2} \log {\left (7 \right )} + 5}}{2 x} \]

[In]

integrate(1/2*(((4*x**4+20*x**3-2*x**2)*ln(7)+2*x**2+1)*exp(x**2*ln(7)+5)+4*x**3+10*x**2)/x**2,x)

[Out]

x**2 + 5*x + (2*x**2 + 10*x - 1)*exp(x**2*log(7) + 5)/(2*x)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.40 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \left (7\right )}\right ) e^{5} \log \left (7\right )}{2 \, \sqrt {-\log \left (7\right )}} + x^{2} + \frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \left (7\right )}\right ) e^{5}}{2 \, \sqrt {-\log \left (7\right )}} - \frac {1}{2} \, {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \left (7\right )}\right ) e^{5}}{\sqrt {-\log \left (7\right )} \log \left (7\right )} - \frac {2 \, x e^{\left (x^{2} \log \left (7\right ) + 5\right )}}{\log \left (7\right )}\right )} \log \left (7\right ) - \frac {\sqrt {-x^{2} \log \left (7\right )} e^{5} \Gamma \left (-\frac {1}{2}, -x^{2} \log \left (7\right )\right )}{4 \, x} + 5 \, x + 5 \, e^{\left (x^{2} \log \left (7\right ) + 5\right )} \]

[In]

integrate(1/2*(((4*x^4+20*x^3-2*x^2)*log(7)+2*x^2+1)*exp(x^2*log(7)+5)+4*x^3+10*x^2)/x^2,x, algorithm="maxima"
)

[Out]

-1/2*sqrt(pi)*erf(x*sqrt(-log(7)))*e^5*log(7)/sqrt(-log(7)) + x^2 + 1/2*sqrt(pi)*erf(x*sqrt(-log(7)))*e^5/sqrt
(-log(7)) - 1/2*(sqrt(pi)*erf(x*sqrt(-log(7)))*e^5/(sqrt(-log(7))*log(7)) - 2*x*e^(x^2*log(7) + 5)/log(7))*log
(7) - 1/4*sqrt(-x^2*log(7))*e^5*gamma(-1/2, -x^2*log(7))/x + 5*x + 5*e^(x^2*log(7) + 5)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=\frac {2 \cdot 7^{\left (x^{2}\right )} x^{2} e^{5} + 2 \, x^{3} + 10 \cdot 7^{\left (x^{2}\right )} x e^{5} + 10 \, x^{2} - 7^{\left (x^{2}\right )} e^{5}}{2 \, x} \]

[In]

integrate(1/2*(((4*x^4+20*x^3-2*x^2)*log(7)+2*x^2+1)*exp(x^2*log(7)+5)+4*x^3+10*x^2)/x^2,x, algorithm="giac")

[Out]

1/2*(2*7^(x^2)*x^2*e^5 + 2*x^3 + 10*7^(x^2)*x*e^5 + 10*x^2 - 7^(x^2)*e^5)/x

Mupad [B] (verification not implemented)

Time = 10.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {10 x^2+4 x^3+e^{5+x^2 \log (7)} \left (1+2 x^2+\left (-2 x^2+20 x^3+4 x^4\right ) \log (7)\right )}{2 x^2} \, dx=x\,\left (7^{x^2}\,{\mathrm {e}}^5+5\right )+x^2+5\,7^{x^2}\,{\mathrm {e}}^5-\frac {7^{x^2}\,{\mathrm {e}}^5}{2\,x} \]

[In]

int(((exp(x^2*log(7) + 5)*(log(7)*(20*x^3 - 2*x^2 + 4*x^4) + 2*x^2 + 1))/2 + 5*x^2 + 2*x^3)/x^2,x)

[Out]

x*(7^(x^2)*exp(5) + 5) + x^2 + 5*7^(x^2)*exp(5) - (7^(x^2)*exp(5))/(2*x)