3.90.6 \(\int \frac {-10+150 x+56 x^2-114 x^3-24 x^4+24 x^5+e^{x^2} (150 x+60 x^2-114 x^3-24 x^4+24 x^5)}{75+30 x-57 x^2-12 x^3+12 x^4} \, dx\)

Optimal. Leaf size=27 \[ -1+e^{x^2}+x^2+\frac {2 x}{3 \left (-5-x+2 x^2\right )} \]

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Rubi [B]  time = 0.54, antiderivative size = 310, normalized size of antiderivative = 11.48, number of steps used = 29, number of rules used = 12, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6742, 2209, 614, 618, 206, 638, 722, 738, 773, 632, 31, 800} \begin {gather*} \frac {4 x^3}{41}-\frac {38 (x+10) x^2}{41 \left (-2 x^2+x+5\right )}+\frac {79 x^2}{41}+\frac {56 (x+10) x}{123 \left (-2 x^2+x+5\right )}+e^{x^2}+\frac {10 (1-4 x)}{123 \left (-2 x^2+x+5\right )}+\frac {50 (x+10)}{41 \left (-2 x^2+x+5\right )}+\frac {8 (x+10) x^4}{41 \left (-2 x^2+x+5\right )}-\frac {8 (x+10) x^3}{41 \left (-2 x^2+x+5\right )}-\frac {30 x}{41}-\frac {19 \left (1681-61 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{6724}-\frac {\left (1681-661 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{1681}+\frac {\left (38663-3203 \sqrt {41}\right ) \log \left (-4 x-\sqrt {41}+1\right )}{6724}+\frac {\left (38663+3203 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{6724}-\frac {\left (1681+661 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{1681}-\frac {19 \left (1681+61 \sqrt {41}\right ) \log \left (-4 x+\sqrt {41}+1\right )}{6724}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 150*x + 56*x^2 - 114*x^3 - 24*x^4 + 24*x^5 + E^x^2*(150*x + 60*x^2 - 114*x^3 - 24*x^4 + 24*x^5))/(7
5 + 30*x - 57*x^2 - 12*x^3 + 12*x^4),x]

[Out]

E^x^2 - (30*x)/41 + (79*x^2)/41 + (4*x^3)/41 + (10*(1 - 4*x))/(123*(5 + x - 2*x^2)) + (50*(10 + x))/(41*(5 + x
- 2*x^2)) + (56*x*(10 + x))/(123*(5 + x - 2*x^2)) - (38*x^2*(10 + x))/(41*(5 + x - 2*x^2)) - (8*x^3*(10 + x))
/(41*(5 + x - 2*x^2)) + (8*x^4*(10 + x))/(41*(5 + x - 2*x^2)) + (300*ArcTanh[(1 - 4*x)/Sqrt[41]])/(41*Sqrt[41]
) + ((38663 - 3203*Sqrt[41])*Log[1 - Sqrt[41] - 4*x])/6724 - ((1681 - 661*Sqrt[41])*Log[1 - Sqrt[41] - 4*x])/1
681 - (19*(1681 - 61*Sqrt[41])*Log[1 - Sqrt[41] - 4*x])/6724 - (19*(1681 + 61*Sqrt[41])*Log[1 + Sqrt[41] - 4*x
])/6724 - ((1681 + 661*Sqrt[41])*Log[1 + Sqrt[41] - 4*x])/1681 + ((38663 + 3203*Sqrt[41])*Log[1 + Sqrt[41] - 4
*x])/6724

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 722

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*(2*p + 3)*(c*d
^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ
[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^{x^2} x-\frac {10}{3 \left (-5-x+2 x^2\right )^2}+\frac {50 x}{\left (-5-x+2 x^2\right )^2}+\frac {56 x^2}{3 \left (-5-x+2 x^2\right )^2}-\frac {38 x^3}{\left (-5-x+2 x^2\right )^2}-\frac {8 x^4}{\left (-5-x+2 x^2\right )^2}+\frac {8 x^5}{\left (-5-x+2 x^2\right )^2}\right ) \, dx\\ &=2 \int e^{x^2} x \, dx-\frac {10}{3} \int \frac {1}{\left (-5-x+2 x^2\right )^2} \, dx-8 \int \frac {x^4}{\left (-5-x+2 x^2\right )^2} \, dx+8 \int \frac {x^5}{\left (-5-x+2 x^2\right )^2} \, dx+\frac {56}{3} \int \frac {x^2}{\left (-5-x+2 x^2\right )^2} \, dx-38 \int \frac {x^3}{\left (-5-x+2 x^2\right )^2} \, dx+50 \int \frac {x}{\left (-5-x+2 x^2\right )^2} \, dx\\ &=e^{x^2}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8}{41} \int \frac {(-30-2 x) x^2}{-5-x+2 x^2} \, dx-\frac {8}{41} \int \frac {(-40-3 x) x^3}{-5-x+2 x^2} \, dx+\frac {40}{123} \int \frac {1}{-5-x+2 x^2} \, dx+\frac {38}{41} \int \frac {(-20-x) x}{-5-x+2 x^2} \, dx-\frac {50}{41} \int \frac {1}{-5-x+2 x^2} \, dx+\frac {560}{123} \int \frac {1}{-5-x+2 x^2} \, dx\\ &=e^{x^2}-\frac {19 x}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8}{41} \int \left (-\frac {31}{2}-x-\frac {155+41 x}{2 \left (-5-x+2 x^2\right )}\right ) \, dx-\frac {8}{41} \int \left (-\frac {113}{8}-\frac {83 x}{4}-\frac {3 x^2}{2}-\frac {565+943 x}{8 \left (-5-x+2 x^2\right )}\right ) \, dx+\frac {19}{41} \int \frac {-5-41 x}{-5-x+2 x^2} \, dx-\frac {80}{123} \operatorname {Subst}\left (\int \frac {1}{41-x^2} \, dx,x,-1+4 x\right )+\frac {100}{41} \operatorname {Subst}\left (\int \frac {1}{41-x^2} \, dx,x,-1+4 x\right )-\frac {1120}{123} \operatorname {Subst}\left (\int \frac {1}{41-x^2} \, dx,x,-1+4 x\right )\\ &=e^{x^2}-\frac {30 x}{41}+\frac {79 x^2}{41}+\frac {4 x^3}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}}+\frac {1}{41} \int \frac {565+943 x}{-5-x+2 x^2} \, dx-\frac {4}{41} \int \frac {155+41 x}{-5-x+2 x^2} \, dx-\frac {\left (19 \left (1681-61 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}-\frac {\left (19 \left (1681+61 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}\\ &=e^{x^2}-\frac {30 x}{41}+\frac {79 x^2}{41}+\frac {4 x^3}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}}-\frac {19 \left (1681-61 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{6724}-\frac {19 \left (1681+61 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{6724}+\frac {\left (38663-3203 \sqrt {41}\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}-\frac {\left (2 \left (1681-661 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {41}}{2}+2 x} \, dx}{1681}-\frac {\left (2 \left (1681+661 \sqrt {41}\right )\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {41}}{2}+2 x} \, dx}{1681}+\frac {\left (38663+3203 \sqrt {41}\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {41}}{2}+2 x} \, dx}{3362}\\ &=e^{x^2}-\frac {30 x}{41}+\frac {79 x^2}{41}+\frac {4 x^3}{41}+\frac {10 (1-4 x)}{123 \left (5+x-2 x^2\right )}+\frac {50 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {56 x (10+x)}{123 \left (5+x-2 x^2\right )}-\frac {38 x^2 (10+x)}{41 \left (5+x-2 x^2\right )}-\frac {8 x^3 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {8 x^4 (10+x)}{41 \left (5+x-2 x^2\right )}+\frac {300 \tanh ^{-1}\left (\frac {1-4 x}{\sqrt {41}}\right )}{41 \sqrt {41}}+\frac {\left (38663-3203 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{6724}-\frac {\left (1681-661 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{1681}-\frac {19 \left (1681-61 \sqrt {41}\right ) \log \left (1-\sqrt {41}-4 x\right )}{6724}-\frac {19 \left (1681+61 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{6724}-\frac {\left (1681+661 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{1681}+\frac {\left (38663+3203 \sqrt {41}\right ) \log \left (1+\sqrt {41}-4 x\right )}{6724}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 35, normalized size = 1.30 \begin {gather*} \frac {2}{3} \left (\frac {3 e^{x^2}}{2}+\frac {3 x^2}{2}+\frac {x}{-5-x+2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 150*x + 56*x^2 - 114*x^3 - 24*x^4 + 24*x^5 + E^x^2*(150*x + 60*x^2 - 114*x^3 - 24*x^4 + 24*x^
5))/(75 + 30*x - 57*x^2 - 12*x^3 + 12*x^4),x]

[Out]

(2*((3*E^x^2)/2 + (3*x^2)/2 + x/(-5 - x + 2*x^2)))/3

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fricas [B]  time = 0.54, size = 49, normalized size = 1.81 \begin {gather*} \frac {6 \, x^{4} - 3 \, x^{3} - 15 \, x^{2} + 3 \, {\left (2 \, x^{2} - x - 5\right )} e^{\left (x^{2}\right )} + 2 \, x}{3 \, {\left (2 \, x^{2} - x - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114*x^3+56*x^2+150*x-10)/(12*x^4-12*x^3
-57*x^2+30*x+75),x, algorithm="fricas")

[Out]

1/3*(6*x^4 - 3*x^3 - 15*x^2 + 3*(2*x^2 - x - 5)*e^(x^2) + 2*x)/(2*x^2 - x - 5)

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giac [B]  time = 0.18, size = 55, normalized size = 2.04 \begin {gather*} \frac {6 \, x^{4} - 3 \, x^{3} + 6 \, x^{2} e^{\left (x^{2}\right )} - 15 \, x^{2} - 3 \, x e^{\left (x^{2}\right )} + 2 \, x - 15 \, e^{\left (x^{2}\right )}}{3 \, {\left (2 \, x^{2} - x - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114*x^3+56*x^2+150*x-10)/(12*x^4-12*x^3
-57*x^2+30*x+75),x, algorithm="giac")

[Out]

1/3*(6*x^4 - 3*x^3 + 6*x^2*e^(x^2) - 15*x^2 - 3*x*e^(x^2) + 2*x - 15*e^(x^2))/(2*x^2 - x - 5)

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maple [A]  time = 0.12, size = 22, normalized size = 0.81

method	result	size
risch	\(x^{2}+\frac {x}{3 x^{2}-\frac {3}{2} x -\frac {15}{2}}+{\mathrm e}^{x^{2}}\)	\(22\)
norman	\(\frac {-\frac {11 x^{2}}{3}-x^{3}+2 x^{4}+2 x^{2} {\mathrm e}^{x^{2}}-{\mathrm e}^{x^{2}} x -5 \,{\mathrm e}^{x^{2}}-\frac {10}{3}}{2 x^{2}-x -5}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114*x^3+56*x^2+150*x-10)/(12*x^4-12*x^3-57*x^
2+30*x+75),x,method=_RETURNVERBOSE)

[Out]

x^2+1/3*x/(x^2-1/2*x-5/2)+exp(x^2)

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maxima [B]  time = 0.51, size = 120, normalized size = 4.44 \begin {gather*} x^{2} - \frac {551 \, x + 1205}{82 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {241 \, x + 155}{41 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {19 \, {\left (31 \, x + 105\right )}}{82 \, {\left (2 \, x^{2} - x - 5\right )}} - \frac {28 \, {\left (21 \, x + 5\right )}}{123 \, {\left (2 \, x^{2} - x - 5\right )}} + \frac {10 \, {\left (4 \, x - 1\right )}}{123 \, {\left (2 \, x^{2} - x - 5\right )}} - \frac {50 \, {\left (x + 10\right )}}{41 \, {\left (2 \, x^{2} - x - 5\right )}} + e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^5-24*x^4-114*x^3+60*x^2+150*x)*exp(x^2)+24*x^5-24*x^4-114*x^3+56*x^2+150*x-10)/(12*x^4-12*x^3
-57*x^2+30*x+75),x, algorithm="maxima")

[Out]

x^2 - 1/82*(551*x + 1205)/(2*x^2 - x - 5) + 1/41*(241*x + 155)/(2*x^2 - x - 5) + 19/82*(31*x + 105)/(2*x^2 - x
- 5) - 28/123*(21*x + 5)/(2*x^2 - x - 5) + 10/123*(4*x - 1)/(2*x^2 - x - 5) - 50/41*(x + 10)/(2*x^2 - x - 5)
+ e^(x^2)

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mupad [B]  time = 5.50, size = 23, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^{x^2}-\frac {2\,x}{3\,\left (-2\,x^2+x+5\right )}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((150*x + exp(x^2)*(150*x + 60*x^2 - 114*x^3 - 24*x^4 + 24*x^5) + 56*x^2 - 114*x^3 - 24*x^4 + 24*x^5 - 10)/
(30*x - 57*x^2 - 12*x^3 + 12*x^4 + 75),x)

[Out]

exp(x^2) - (2*x)/(3*(x - 2*x^2 + 5)) + x^2

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sympy [A]  time = 0.17, size = 20, normalized size = 0.74 \begin {gather*} x^{2} + \frac {2 x}{6 x^{2} - 3 x - 15} + e^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**5-24*x**4-114*x**3+60*x**2+150*x)*exp(x**2)+24*x**5-24*x**4-114*x**3+56*x**2+150*x-10)/(12*x
**4-12*x**3-57*x**2+30*x+75),x)

[Out]

x**2 + 2*x/(6*x**2 - 3*x - 15) + exp(x**2)

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