3.391 \(\int \frac {2+5 x+x^2}{(1+4 x-7 x^2)^3 \sqrt {3+2 x+5 x^2}} \, dx\)

Optimal. Leaf size=227 \[ -\frac {7 \sqrt {5 x^2+2 x+3} (409769-1189370 x)}{62451488 \left (-7 x^2+4 x+1\right )}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}-\frac {7 \left (39370231-2538725 \sqrt {11}\right ) \tanh ^{-1}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{124902976 \sqrt {22 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (39370231+2538725 \sqrt {11}\right ) \tanh ^{-1}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{124902976 \sqrt {22 \left (125+17 \sqrt {11}\right )}} \]

[Out]

-3/11176*(40-371*x)*(5*x^2+2*x+3)^(1/2)/(-7*x^2+4*x+1)^2-7/62451488*(409769-1189370*x)*(5*x^2+2*x+3)^(1/2)/(-7
*x^2+4*x+1)-7/124902976*arctanh((23+x*(17-5*11^(1/2))-11^(1/2))/(5*x^2+2*x+3)^(1/2)/(250-34*11^(1/2))^(1/2))*(
39370231-2538725*11^(1/2))/(2750-374*11^(1/2))^(1/2)+7/124902976*arctanh((23+11^(1/2)+x*(17+5*11^(1/2)))/(5*x^
2+2*x+3)^(1/2)/(250+34*11^(1/2))^(1/2))*(39370231+2538725*11^(1/2))/(2750+374*11^(1/2))^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1060, 1032, 724, 206} \[ -\frac {7 \sqrt {5 x^2+2 x+3} (409769-1189370 x)}{62451488 \left (-7 x^2+4 x+1\right )}-\frac {3 (40-371 x) \sqrt {5 x^2+2 x+3}}{11176 \left (-7 x^2+4 x+1\right )^2}-\frac {7 \left (39370231-2538725 \sqrt {11}\right ) \tanh ^{-1}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{124902976 \sqrt {22 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (39370231+2538725 \sqrt {11}\right ) \tanh ^{-1}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{124902976 \sqrt {22 \left (125+17 \sqrt {11}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]

[Out]

(-3*(40 - 371*x)*Sqrt[3 + 2*x + 5*x^2])/(11176*(1 + 4*x - 7*x^2)^2) - (7*(409769 - 1189370*x)*Sqrt[3 + 2*x + 5
*x^2])/(62451488*(1 + 4*x - 7*x^2)) - (7*(39370231 - 2538725*Sqrt[11])*ArcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[1
1])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(124902976*Sqrt[22*(125 - 17*Sqrt[11])]) + (7*(39
370231 + 2538725*Sqrt[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqrt[3 +
2*x + 5*x^2])])/(124902976*Sqrt[22*(125 + 17*Sqrt[11])])

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 724

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

Rule 1032

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbo
l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2])
, x], x] - Dist[(2*c*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b,
c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]

Rule 1060

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x_
)^2)^(q_), x_Symbol] :> Simp[((a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q + 1)*((A*c - a*C)*(2*a*c*e - b*(c
*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b
*c*d - 2*a*c*e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)
*(c*e - b*f))*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)), Int[(a
+ b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f)
)*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C
*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f -
c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c
*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(
c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4)))*x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*
e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b,
c, d, e, f, A, B, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 -
(b*d - a*e)*(c*e - b*f), 0] &&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right )^3 \sqrt {3+2 x+5 x^2}} \, dx &=-\frac {3 (40-371 x) \sqrt {3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac {\int \frac {-130024-81000 x-89040 x^2}{\left (1+4 x-7 x^2\right )^2 \sqrt {3+2 x+5 x^2}} \, dx}{89408}\\ &=-\frac {3 (40-371 x) \sqrt {3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac {7 (409769-1189370 x) \sqrt {3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}+\frac {\int \frac {2194737984+1137348800 x}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx}{3996895232}\\ &=-\frac {3 (40-371 x) \sqrt {3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac {7 (409769-1189370 x) \sqrt {3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}+\frac {\left (7 \left (27925975-39370231 \sqrt {11}\right )\right ) \int \frac {1}{\left (4-2 \sqrt {11}-14 x\right ) \sqrt {3+2 x+5 x^2}} \, dx}{686966368}+\frac {\left (7 \left (27925975+39370231 \sqrt {11}\right )\right ) \int \frac {1}{\left (4+2 \sqrt {11}-14 x\right ) \sqrt {3+2 x+5 x^2}} \, dx}{686966368}\\ &=-\frac {3 (40-371 x) \sqrt {3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac {7 (409769-1189370 x) \sqrt {3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}-\frac {\left (7 \left (27925975-39370231 \sqrt {11}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2352+112 \left (4-2 \sqrt {11}\right )+20 \left (4-2 \sqrt {11}\right )^2-x^2} \, dx,x,\frac {-84-2 \left (4-2 \sqrt {11}\right )-\left (28+10 \left (4-2 \sqrt {11}\right )\right ) x}{\sqrt {3+2 x+5 x^2}}\right )}{343483184}-\frac {\left (7 \left (27925975+39370231 \sqrt {11}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2352+112 \left (4+2 \sqrt {11}\right )+20 \left (4+2 \sqrt {11}\right )^2-x^2} \, dx,x,\frac {-84-2 \left (4+2 \sqrt {11}\right )-\left (28+10 \left (4+2 \sqrt {11}\right )\right ) x}{\sqrt {3+2 x+5 x^2}}\right )}{343483184}\\ &=-\frac {3 (40-371 x) \sqrt {3+2 x+5 x^2}}{11176 \left (1+4 x-7 x^2\right )^2}-\frac {7 (409769-1189370 x) \sqrt {3+2 x+5 x^2}}{62451488 \left (1+4 x-7 x^2\right )}-\frac {7 \left (39370231-2538725 \sqrt {11}\right ) \tanh ^{-1}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{124902976 \sqrt {22 \left (125-17 \sqrt {11}\right )}}+\frac {7 \left (39370231+2538725 \sqrt {11}\right ) \tanh ^{-1}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{124902976 \sqrt {22 \left (125+17 \sqrt {11}\right )}}\\ \end {align*}

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Mathematica [A]  time = 1.17, size = 371, normalized size = 1.63 \[ \frac {\frac {732651920 \sqrt {5 x^2+2 x+3} x}{-7 x^2+4 x+1}+\frac {547311072 \sqrt {5 x^2+2 x+3} x}{\left (-7 x^2+4 x+1\right )^2}-\frac {59009280 \sqrt {5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2}+\frac {252417704 \sqrt {5 x^2+2 x+3}}{7 x^2-4 x-1}[phone] \sqrt {\frac {22}{125+17 \sqrt {11}}} \log \left (\sqrt {2750+374 \sqrt {11}} \sqrt {5 x^2+2 x+3}+\left (55+17 \sqrt {11}\right ) x+23 \sqrt {11}+11\right )[phone] \sqrt {\frac {2}{125+17 \sqrt {11}}} \log \left (\sqrt {2750+374 \sqrt {11}} \sqrt {5 x^2+2 x+3}+\left (55+17 \sqrt {11}\right ) x+23 \sqrt {11}+11\right )+14 \sqrt {\frac {2}{125-17 \sqrt {11}}} \left (39370231 \sqrt {11}-27925975\right ) \tanh ^{-1}\left (\frac {\sqrt {250-34 \sqrt {11}} \sqrt {5 x^2+2 x+3}}{\left (5 \sqrt {11}-17\right ) x+\sqrt {11}-23}\right )-14 \sqrt {\frac {2}{125+17 \sqrt {11}}} \left (27925975+39370231 \sqrt {11}\right ) \log \left (-7 x+\sqrt {11}+2\right )}{5495730944} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x + x^2)/((1 + 4*x - 7*x^2)^3*Sqrt[3 + 2*x + 5*x^2]),x]

[Out]

((-59009280*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)^2 + (547311072*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2)
^2 + (732651920*x*Sqrt[3 + 2*x + 5*x^2])/(1 + 4*x - 7*x^2) + (252417704*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x
^2) + 14*Sqrt[2/(125 - 17*Sqrt[11])]*(-27925975 + 39370231*Sqrt[11])*ArcTanh[(Sqrt[250 - 34*Sqrt[11]]*Sqrt[3 +
2*x + 5*x^2])/(-23 + Sqrt[11] + (-17 + 5*Sqrt[11])*x)] - 14*Sqrt[2/(125 + 17*Sqrt[11])]*(27925975 + 39370231*
Sqrt[11])*Log[2 + Sqrt[11] - 7*x] + 390963650*Sqrt[2/(125 + 17*Sqrt[11])]*Log[11 + 23*Sqrt[11] + (55 + 17*Sqrt
[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]] + 551183234*Sqrt[22/(125 + 17*Sqrt[11])]*Log[11 + 2
3*Sqrt[11] + (55 + 17*Sqrt[11])*x + Sqrt[2750 + 374*Sqrt[11]]*Sqrt[3 + 2*x + 5*x^2]])/5495730944

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fricas [B]  time = 1.10, size = 390, normalized size = 1.72 \[ -\frac {\sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {1283973697005131 \, \sqrt {11} + 82616280769148425} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {1283973697005131 \, \sqrt {11} + 82616280769148425} {\left (358684877 \, \sqrt {11} + 2940638404\right )} + 7232150972206110797 \, \sqrt {11} {\left (x + 3\right )} - 21696452916618332391 \, x + 36160754861030553985}{x}\right ) - \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {1283973697005131 \, \sqrt {11} + 82616280769148425} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {1283973697005131 \, \sqrt {11} + 82616280769148425} {\left (358684877 \, \sqrt {11} + 2940638404\right )} - 7232150972206110797 \, \sqrt {11} {\left (x + 3\right )} + 21696452916618332391 \, x - 36160754861030553985}{x}\right ) + \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-1283973697005131 \, \sqrt {11} + 82616280769148425} \log \left (\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (358684877 \, \sqrt {11} - 2940638404\right )} \sqrt {-1283973697005131 \, \sqrt {11} + 82616280769148425} + 7232150972206110797 \, \sqrt {11} {\left (x + 3\right )} + 21696452916618332391 \, x - 36160754861030553985}{x}\right ) - \sqrt {2794} {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )} \sqrt {-1283973697005131 \, \sqrt {11} + 82616280769148425} \log \left (-\frac {\sqrt {2794} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (358684877 \, \sqrt {11} - 2940638404\right )} \sqrt {-1283973697005131 \, \sqrt {11} + 82616280769148425} - 7232150972206110797 \, \sqrt {11} {\left (x + 3\right )} - 21696452916618332391 \, x + 36160754861030553985}{x}\right ) + 11176 \, {\left (58279130 \, x^{3} - 53381041 \, x^{2} - 3071502 \, x + 3538943\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{697957829888 \, {\left (49 \, x^{4} - 56 \, x^{3} + 2 \, x^{2} + 8 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/697957829888*(sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(1283973697005131*sqrt(11) + 8261628076914
8425)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1283973697005131*sqrt(11) + 82616280769148425)*(358684877*sq
rt(11) + 2940638404) + 7232150972206110797*sqrt(11)*(x + 3) - 21696452916618332391*x + 36160754861030553985)/x
) - sqrt(2794)*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(1283973697005131*sqrt(11) + 82616280769148425)*log((sq
rt(2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1283973697005131*sqrt(11) + 82616280769148425)*(358684877*sqrt(11) + 29406
38404) - 7232150972206110797*sqrt(11)*(x + 3) + 21696452916618332391*x - 36160754861030553985)/x) + sqrt(2794)
*(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425)*log((sqrt(2794)*sqrt
(5*x^2 + 2*x + 3)*(358684877*sqrt(11) - 2940638404)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425) + 723
2150972206110797*sqrt(11)*(x + 3) + 21696452916618332391*x - 36160754861030553985)/x) - sqrt(2794)*(49*x^4 - 5
6*x^3 + 2*x^2 + 8*x + 1)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425)*log(-(sqrt(2794)*sqrt(5*x^2 + 2*
x + 3)*(358684877*sqrt(11) - 2940638404)*sqrt(-1283973697005131*sqrt(11) + 82616280769148425) - 72321509722061
10797*sqrt(11)*(x + 3) - 21696452916618332391*x + 36160754861030553985)/x) + 11176*(58279130*x^3 - 53381041*x^
2 - 3071502*x + 3538943)*sqrt(5*x^2 + 2*x + 3))/(49*x^4 - 56*x^3 + 2*x^2 + 8*x + 1)

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giac [B]  time = 0.32, size = 378, normalized size = 1.67 \[ \frac {124397525 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{7} + 26796567 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{6} - 3595807617 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{5} - 1719888775 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} + 17096132999 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} + 8328401413 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} - 16383202915 \, \sqrt {5} x - 7800623485 \, \sqrt {5} + 16383202915 \, \sqrt {5 \, x^{2} + 2 \, x + 3}}{31225744 \, {\left (7 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{4} - 8 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{3} - 70 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} + 83\right )}^{2}} + 0.0423989586659649 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.0446437606656958 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.0423989586659649 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.0446437606656958 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/31225744*(124397525*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^7 + 26796567*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x +
3))^6 - 3595807617*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^5 - 1719888775*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x +
3))^4 + 17096132999*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 + 8328401413*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x
+ 3))^2 - 16383202915*sqrt(5)*x - 7800623485*sqrt(5) + 16383202915*sqrt(5*x^2 + 2*x + 3))/(7*(sqrt(5)*x - sqrt
(5*x^2 + 2*x + 3))^4 - 8*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3))^3 - 70*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)
)^2 + 16*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) + 83)^2 + 0.0423989586659649*log(-sqrt(5)*x + sqrt(5*x^2
+ 2*x + 3) + 4.41924736459000) - 0.0446437606656958*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.25295163054000)
- 0.0423989586659649*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 1.02258038113000) + 0.0446437606656958*log(-sqr
t(5)*x + sqrt(5*x^2 + 2*x + 3) - 2.09411235400000)

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maple [B]  time = 0.02, size = 1194, normalized size = 5.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3535/21296*11^(1/2)/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(34/7-10/7*11^(1/2))*(x-2/7+1
/7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))
+250-34*11^(1/2))^(1/2))-21/968*(-61+13*11^(1/2))*11^(1/2)*(-1/686/(250/49-34/49*11^(1/2))/(x-2/7+1/7*11^(1/2)
)^2*(5*(x-2/7+1/7*11^(1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250/49-34/49*11^(1/2))^(1/2)-3/1372*(3
4/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))*(-1/(250/49-34/49*11^(1/2))/(x-2/7+1/7*11^(1/2))*(5*(x-2/7+1/7*11^(
1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250/49-34/49*11^(1/2))^(1/2)+7/2*(34/7-10/7*11^(1/2))/(250/4
9-34/49*11^(1/2))/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(34/7-10/7*11^(1/2))*(x-2/7+1/7*
11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+25
0-34*11^(1/2))^(1/2)))+5/98/(250/49-34/49*11^(1/2))/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2
)+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7-10/7
*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2)))-(-3535/1936+273/1936*11^(1/2))*(-1/49/(250/49-34/49*1
1^(1/2))/(x-2/7+1/7*11^(1/2))*(5*(x-2/7+1/7*11^(1/2))^2+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250/49-34/49
*11^(1/2))^(1/2)+1/14*(34/7-10/7*11^(1/2))/(250/49-34/49*11^(1/2))/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/4
9-68/49*11^(1/2)+(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^
2+49*(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2)))+3535/21296*11^(1/2)/(250+34*11^(1/2))^
(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2)))/(250+34*11^(1/2))^(1/2)/(
245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250+34*11^(1/2))^(1/2))-(-3535/1936-27
3/1936*11^(1/2))*(-1/49/(250/49+34/49*11^(1/2))/(x-2/7-1/7*11^(1/2))*(5*(x-2/7-1/7*11^(1/2))^2+(34/7+10/7*11^(
1/2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(1/2)+1/14*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))/(250
+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2)))/(250+34*11^
(1/2))^(1/2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250+34*11^(1/2))^(1/2)))
-21/968*(61+13*11^(1/2))*11^(1/2)*(-1/686/(250/49+34/49*11^(1/2))/(x-2/7-1/7*11^(1/2))^2*(5*(x-2/7-1/7*11^(1/2
))^2+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(1/2)-3/1372*(34/7+10/7*11^(1/2))/(250/4
9+34/49*11^(1/2))*(-1/(250/49+34/49*11^(1/2))/(x-2/7-1/7*11^(1/2))*(5*(x-2/7-1/7*11^(1/2))^2+(34/7+10/7*11^(1/
2))*(x-2/7-1/7*11^(1/2))+250/49+34/49*11^(1/2))^(1/2)+7/2*(34/7+10/7*11^(1/2))/(250/49+34/49*11^(1/2))/(250+34
*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2)))/(250+34*11^(1/
2))^(1/2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2))+250+34*11^(1/2))^(1/2)))+5/
98/(250/49+34/49*11^(1/2))/(250+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x
-2/7-1/7*11^(1/2)))/(250+34*11^(1/2))^(1/2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^
(1/2))+250+34*11^(1/2))^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{2} + 5 \, x + 2}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{3} \sqrt {5 \, x^{2} + 2 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((x^2 + 5*x + 2)/((7*x^2 - 4*x - 1)^3*sqrt(5*x^2 + 2*x + 3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2+5\,x+2}{\sqrt {5\,x^2+2\,x+3}\,{\left (-7\,x^2+4\,x+1\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {5 x}{343 x^{6} \sqrt {5 x^{2} + 2 x + 3} - 588 x^{5} \sqrt {5 x^{2} + 2 x + 3} + 189 x^{4} \sqrt {5 x^{2} + 2 x + 3} + 104 x^{3} \sqrt {5 x^{2} + 2 x + 3} - 27 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 12 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {x^{2}}{343 x^{6} \sqrt {5 x^{2} + 2 x + 3} - 588 x^{5} \sqrt {5 x^{2} + 2 x + 3} + 189 x^{4} \sqrt {5 x^{2} + 2 x + 3} + 104 x^{3} \sqrt {5 x^{2} + 2 x + 3} - 27 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 12 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {2}{343 x^{6} \sqrt {5 x^{2} + 2 x + 3} - 588 x^{5} \sqrt {5 x^{2} + 2 x + 3} + 189 x^{4} \sqrt {5 x^{2} + 2 x + 3} + 104 x^{3} \sqrt {5 x^{2} + 2 x + 3} - 27 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 12 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+5*x+2)/(-7*x**2+4*x+1)**3/(5*x**2+2*x+3)**(1/2),x)

[Out]

-Integral(5*x/(343*x**6*sqrt(5*x**2 + 2*x + 3) - 588*x**5*sqrt(5*x**2 + 2*x + 3) + 189*x**4*sqrt(5*x**2 + 2*x
+ 3) + 104*x**3*sqrt(5*x**2 + 2*x + 3) - 27*x**2*sqrt(5*x**2 + 2*x + 3) - 12*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5
*x**2 + 2*x + 3)), x) - Integral(x**2/(343*x**6*sqrt(5*x**2 + 2*x + 3) - 588*x**5*sqrt(5*x**2 + 2*x + 3) + 189
*x**4*sqrt(5*x**2 + 2*x + 3) + 104*x**3*sqrt(5*x**2 + 2*x + 3) - 27*x**2*sqrt(5*x**2 + 2*x + 3) - 12*x*sqrt(5*
x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x) - Integral(2/(343*x**6*sqrt(5*x**2 + 2*x + 3) - 588*x**5*sqrt(5*
x**2 + 2*x + 3) + 189*x**4*sqrt(5*x**2 + 2*x + 3) + 104*x**3*sqrt(5*x**2 + 2*x + 3) - 27*x**2*sqrt(5*x**2 + 2*
x + 3) - 12*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x)

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