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

Optimal. Leaf size=25 \[ -1+\left (e^{5+x^2 \log (7)}+x\right ) \left (5-\frac {1}{2 x}+x\right ) \]

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Rubi [C]  time = 0.31, antiderivative size = 129, normalized size of antiderivative = 5.16, number of steps used = 11, number of rules used = 7, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.127, Rules used = {12, 14, 6742, 2214, 2204, 2209, 2212} \begin {gather*} \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 \end {gather*}

Antiderivative was successfully verified.

[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 2204

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 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]

Rule 2212

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 2214

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 42, normalized size = 1.68 \begin {gather*} 5\ 7^{x^2} e^5-\frac {7^{x^2} e^5}{2 x}+5 x+7^{x^2} e^5 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.84, size = 36, normalized size = 1.44 \begin {gather*} \frac {2 \, x^{3} + 10 \, x^{2} + {\left (2 \, x^{2} + 10 \, x - 1\right )} e^{\left (x^{2} \log \relax (7) + 5\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.23, size = 47, normalized size = 1.88 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.06, size = 30, normalized size = 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	\(x \,{\mathrm e}^{x^{2} \ln \relax (7)+5}+5 \,{\mathrm e}^{x^{2} \ln \relax (7)+5}-\frac {{\mathrm e}^{x^{2} \ln \relax (7)+5}}{2 x}+x^{2}+5 x\)	\(44\)
norman	\(\frac {x^{3}+{\mathrm e}^{x^{2} \ln \relax (7)+5} x^{2}+5 x^{2}+5 x \,{\mathrm e}^{x^{2} \ln \relax (7)+5}-\frac {{\mathrm e}^{x^{2} \ln \relax (7)+5}}{2}}{x}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [C]  time = 0.48, size = 135, normalized size = 5.40 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \relax (7)}\right ) e^{5} \log \relax (7)}{2 \, \sqrt {-\log \relax (7)}} + x^{2} + \frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \relax (7)}\right ) e^{5}}{2 \, \sqrt {-\log \relax (7)}} - \frac {1}{2} \, {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {-\log \relax (7)}\right ) e^{5}}{\sqrt {-\log \relax (7)} \log \relax (7)} - \frac {2 \, x e^{\left (x^{2} \log \relax (7) + 5\right )}}{\log \relax (7)}\right )} \log \relax (7) - \frac {\sqrt {-x^{2} \log \relax (7)} e^{5} \Gamma \left (-\frac {1}{2}, -x^{2} \log \relax (7)\right )}{4 \, x} + 5 \, x + 5 \, e^{\left (x^{2} \log \relax (7) + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 3.08, size = 37, normalized size = 1.48 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.15, size = 29, normalized size = 1.16 \begin {gather*} x^{2} + 5 x + \frac {\left (2 x^{2} + 10 x - 1\right ) e^{x^{2} \log {\relax (7 )} + 5}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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