3.1.93 \(\int \frac {(x+x^2)^{3/2}}{1+x^2} \, dx\)

Optimal. Leaf size=130 \[ \frac {1}{4} \sqrt {x^2+x} (2 x+5)+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {-x+\sqrt {2}+1}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x^2+x}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {-x-\sqrt {2}+1}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x^2+x}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {978, 1078, 620, 206, 12, 1036, 1030, 207, 203} \begin {gather*} \frac {1}{4} \sqrt {x^2+x} (2 x+5)+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {-x+\sqrt {2}+1}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x^2+x}}\right )-\sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {-x-\sqrt {2}+1}{\sqrt {2 \left (\sqrt {2}-1\right )} \sqrt {x^2+x}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + x^2)^(3/2)/(1 + x^2),x]

[Out]

((5 + 2*x)*Sqrt[x + x^2])/4 + Sqrt[1 + Sqrt[2]]*ArcTan[(1 + Sqrt[2] - x)/(Sqrt[2*(1 + Sqrt[2])]*Sqrt[x + x^2])
] - Sqrt[-1 + Sqrt[2]]*ArcTanh[(1 - Sqrt[2] - x)/(Sqrt[2*(-1 + Sqrt[2])]*Sqrt[x + x^2])] - (5*ArcTanh[x/Sqrt[x
+ x^2]])/4

Rule 12

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

Rule 203

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

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 207

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

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 978

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

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1036

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

Rule 1078

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d
+ e*x + f*x^2]), x], x] /; FreeQ[{a, c, d, e, f, A, B, C}, x] && NeQ[e^2 - 4*d*f, 0]

Rubi steps

\begin {align*} \int \frac {\left (x+x^2\right )^{3/2}}{1+x^2} \, dx &=\frac {1}{4} (5+2 x) \sqrt {x+x^2}-\frac {1}{2} \int \frac {\frac {5}{4}+4 x+\frac {5 x^2}{4}}{\left (1+x^2\right ) \sqrt {x+x^2}} \, dx\\ &=\frac {1}{4} (5+2 x) \sqrt {x+x^2}-\frac {1}{2} \int \frac {4 x}{\left (1+x^2\right ) \sqrt {x+x^2}} \, dx-\frac {5}{8} \int \frac {1}{\sqrt {x+x^2}} \, dx\\ &=\frac {1}{4} (5+2 x) \sqrt {x+x^2}-\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right )-2 \int \frac {x}{\left (1+x^2\right ) \sqrt {x+x^2}} \, dx\\ &=\frac {1}{4} (5+2 x) \sqrt {x+x^2}-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right )+\frac {\int \frac {-1+\left (-1-\sqrt {2}\right ) x}{\left (1+x^2\right ) \sqrt {x+x^2}} \, dx}{\sqrt {2}}-\frac {\int \frac {-1+\left (-1+\sqrt {2}\right ) x}{\left (1+x^2\right ) \sqrt {x+x^2}} \, dx}{\sqrt {2}}\\ &=\frac {1}{4} (5+2 x) \sqrt {x+x^2}-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right )+\left (-2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )+x^2} \, dx,x,\frac {-1+\sqrt {2}+x}{\sqrt {x+x^2}}\right )-\left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (1+\sqrt {2}\right )+x^2} \, dx,x,\frac {-1-\sqrt {2}+x}{\sqrt {x+x^2}}\right )\\ &=\frac {1}{4} (5+2 x) \sqrt {x+x^2}+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {1+\sqrt {2}-x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+x^2}}\right )-\sqrt {-1+\sqrt {2}} \tanh ^{-1}\left (\frac {1-\sqrt {2}-x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {x+x^2}}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 120, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x} \sqrt {x+1} \left (2 \sqrt {x+1} x^{3/2}+5 \sqrt {x+1} \sqrt {x}+4 (-1+i)^{3/2} \tan ^{-1}\left (\sqrt {-1+i} \sqrt {\frac {x}{x+1}}\right )-5 \sinh ^{-1}\left (\sqrt {x}\right )+4 (1+i)^{3/2} \tanh ^{-1}\left (\sqrt {1+i} \sqrt {\frac {x}{x+1}}\right )\right )}{4 \sqrt {x (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + x^2)^(3/2)/(1 + x^2),x]

[Out]

(Sqrt[x]*Sqrt[1 + x]*(5*Sqrt[x]*Sqrt[1 + x] + 2*x^(3/2)*Sqrt[1 + x] - 5*ArcSinh[Sqrt[x]] + 4*(-1 + I)^(3/2)*Ar
cTan[Sqrt[-1 + I]*Sqrt[x/(1 + x)]] + 4*(1 + I)^(3/2)*ArcTanh[Sqrt[1 + I]*Sqrt[x/(1 + x)]]))/(4*Sqrt[x*(1 + x)]
)

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IntegrateAlgebraic [C]  time = 0.43, size = 103, normalized size = 0.79 \begin {gather*} \frac {1}{4} \sqrt {x^2+x} (2 x+5)+\sqrt {2+2 i} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {x^2+x}}{x}\right )+\sqrt {2-2 i} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {x^2+x}}{x}\right )-\frac {5}{4} \tanh ^{-1}\left (\frac {\sqrt {x^2+x}}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x + x^2)^(3/2)/(1 + x^2),x]

[Out]

((5 + 2*x)*Sqrt[x + x^2])/4 + Sqrt[2 + 2*I]*ArcTan[(Sqrt[-1/2 - I/2]*Sqrt[x + x^2])/x] + Sqrt[2 - 2*I]*ArcTan[
(Sqrt[-1/2 + I/2]*Sqrt[x + x^2])/x] - (5*ArcTanh[Sqrt[x + x^2]/x])/4

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fricas [B]  time = 0.46, size = 777, normalized size = 5.98

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x)^(3/2)/(x^2+1),x, algorithm="fricas")

[Out]

-1/8*8^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2)*log(8*x^2 - 8*sqrt(x^2 + x)*x + 2*(8^(1/4)*sqrt(x^2 + x)*(sqrt(
2) - 1) - 8^(1/4)*(sqrt(2)*x - x - 1))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4) + 1/8*8^(1/4)*sqrt(2*sqrt(2)
+ 4)*(sqrt(2) - 2)*log(8*x^2 - 8*sqrt(x^2 + x)*x - 2*(8^(1/4)*sqrt(x^2 + x)*(sqrt(2) - 1) - 8^(1/4)*(sqrt(2)*
x - x - 1))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4) + 1/2*8^(1/4)*sqrt(2)*sqrt(2*sqrt(2) + 4)*arctan(1/7*sq
rt(2)*(sqrt(2)*(5*x + 1) + 6*x + 4) + 1/112*sqrt(8*x^2 - 8*sqrt(x^2 + x)*x - 2*(8^(1/4)*sqrt(x^2 + x)*(sqrt(2)
- 1) - 8^(1/4)*(sqrt(2)*x - x - 1))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4)*(8*sqrt(2)*(5*sqrt(2) + 6) + (
8^(3/4)*(5*sqrt(2) + 6) + 8*8^(1/4)*(2*sqrt(2) + 1))*sqrt(2*sqrt(2) + 4) + 64*sqrt(2) + 32) - 1/7*sqrt(x^2 + x
)*(sqrt(2)*(5*sqrt(2) + 6) + 8*sqrt(2) + 4) + 1/7*sqrt(2)*(8*x + 3) + 1/56*(8^(3/4)*(sqrt(2)*(5*x + 1) + 6*x +
4) - sqrt(x^2 + x)*(8^(3/4)*(5*sqrt(2) + 6) + 8*8^(1/4)*(2*sqrt(2) + 1)) + 8*8^(1/4)*(sqrt(2)*(2*x - 1) + x +
3))*sqrt(2*sqrt(2) + 4) + 4/7*x + 5/7) + 1/2*8^(1/4)*sqrt(2)*sqrt(2*sqrt(2) + 4)*arctan(-1/7*sqrt(2)*(sqrt(2)
*(5*x + 1) + 6*x + 4) - 1/112*sqrt(8*x^2 - 8*sqrt(x^2 + x)*x + 2*(8^(1/4)*sqrt(x^2 + x)*(sqrt(2) - 1) - 8^(1/4
)*(sqrt(2)*x - x - 1))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4)*(8*sqrt(2)*(5*sqrt(2) + 6) - (8^(3/4)*(5*sqr
t(2) + 6) + 8*8^(1/4)*(2*sqrt(2) + 1))*sqrt(2*sqrt(2) + 4) + 64*sqrt(2) + 32) + 1/7*sqrt(x^2 + x)*(sqrt(2)*(5*
sqrt(2) + 6) + 8*sqrt(2) + 4) - 1/7*sqrt(2)*(8*x + 3) + 1/56*(8^(3/4)*(sqrt(2)*(5*x + 1) + 6*x + 4) - sqrt(x^2
+ x)*(8^(3/4)*(5*sqrt(2) + 6) + 8*8^(1/4)*(2*sqrt(2) + 1)) + 8*8^(1/4)*(sqrt(2)*(2*x - 1) + x + 3))*sqrt(2*sq
rt(2) + 4) - 4/7*x - 5/7) + 1/4*sqrt(x^2 + x)*(2*x + 5) + 5/8*log(-2*x + 2*sqrt(x^2 + x) - 1)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x)^(3/2)/(x^2+1),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%{poly1[293782586339316530197997184853348485491135961433723696
543014566614744867108514065050877682658325195312500,-181983623107889624631149356157406719337484171310120143823
5703732895037079303690375957661805226995422363281250,-45703756956807472602902422047635423887856021520244395471
19677612825585897607153309996504109115565063476562500,56570880284365138950603682381140450875287062211863720342
792049323742216450449018769549097192324681377441406250000]:[1,0,-31106,-99112,-65203241]%%},[6]%%%}+%%%{%%{pol
y1[56321484668680812198933307201849014223494499885125012071486954344416211288878769544268525719225646972656250
,-696714630564920427940328445690285054676091560133615813397538926100889157971815729832876090140026022338867187
5,-18501639301691189171876668611027555220095054439393285921280605772669698963742122921653330793619945419921875
00000,21154356016599020416908584996741185285458255144337770041070803160511584140085769728840245644554225172979
7363281250]:[1,0,-31106,-99112,-65203241]%%},[5]%%%}+%%%{%%{poly1[24975278348928816944853728681682189343260636
4903446504002320498492700674161652642483438265732405822753906250,102899415255489529312024941498712410711762889
4418263717921826093710373154539467963565591544030347656250000000,-82953311027839481960187364248758396135697444
62926182727988533904129604785618275766584587664596969303680419921875,-6213149828977078241792565041669856008835
5013579825493623515403867274663312677694266518451768241345782073974609375]:[1,0,-31106,-99112,-65203241]%%},[4
]%%%}+%%%{%%{poly1[1340616794386906006983290339583743954723469276952503292159568741091014347530565772585034852
08281677246093750,14482236354700157104111054005212626608902674674742541683251127446785370439987875476208796249
161474060058593750,-450743946228316845834503722080596923970970618377104928698371490711344578730358390494165951
8660699478698730468750,-50125908887993966523193365597130195258718946647068904700627532897584449278054518714138
3181367765715754302978515625]:[1,0,-31106,-99112,-65203241]%%},[3]%%%}+%%%{%%{poly1[-3813813867141614073487460
82633738466949446345681605193953880120440269250489670513355644056050692291259765625,-9280904840184584672813286
67589231013642654084462078716361857082560926680726081331041009281187046966552734375,12575476280638925266580991
997598805853444942419824788143547130961660366332269561014277652901709172789611816406250,1635803743690010647973
2840612571178737540638352696122276658871622820423281266441048684041183868245192871093750000]:[1,0,-31106,-9911
2,-65203241]%%},[2]%%%}+%%%{%%{poly1[-282916504651425899151886477092387517185612250847482022493155387806666315
512821307401148679503541625976562500,-110475220241836404058124193839468468131104619301709288761928266468597500
70859837003681504660562347747802734375,92622410573620422001390283280266603413021660935063007466600699398333831
52346374953014297625933910590026855468750,37994440441311536988751640112180546198638414735619593878866920608554
7964391733575642579455121057385054046630859375]:[1,0,-31106,-99112,-65203241]%%},[1]%%%}+%%%{%%{poly1[14574425
7028158078272793741263638276606104729476650851683626206635230909249444607603805733373105438232421875,451515992
4667733682618710569206479554327942753854919836504993991415145829367574203658363698710825103759765625,-49499528
22251924970018385488181468617511266061169463951678445919411830447649751834787235187730954876190185546875,-1184
64124267461539474328024609355477174113903628048164370055623880000124338618560214721618505583680160430908203125
]:[1,0,-31106,-99112,-65203241]%%},[0]%%%} / %%%{%%{poly1[-44900316662769750,278135011567527216375,69851422632
2709000750,1028664866672275019250]:[1,0,-31106,-99112,-65203241]%%},[1]%%%}+%%%{%%{[2929989585103994875,368557
731538442832750,907666859350191995250,-45807860415265619064625]:[1,0,-31106,-99112,-65203241]%%},[0]%%%} Error
: Bad Argument Value

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maple [B]  time = 0.14, size = 789, normalized size = 6.07 \begin {gather*} \frac {\sqrt {x^{2}+x}\, x}{2}-\frac {5 \ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{8}+\frac {5 \sqrt {x^{2}+x}}{4}+\frac {\sqrt {\frac {4 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-\frac {3 \sqrt {2}\, \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+4+3 \sqrt {2}}\, \sqrt {2}\, \left (-4 \sqrt {2}\, \arctanh \left (\frac {\sqrt {\frac {4 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-\frac {3 \sqrt {2}\, \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+4+3 \sqrt {2}}}{2 \sqrt {1+\sqrt {2}}}\right )+6 \arctanh \left (\frac {\sqrt {\frac {4 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-\frac {3 \sqrt {2}\, \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+4+3 \sqrt {2}}}{2 \sqrt {1+\sqrt {2}}}\right )+\sqrt {-2+2 \sqrt {2}}\, \sqrt {1+\sqrt {2}}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}\, \sqrt {\left (3 \sqrt {2}-4\right ) \left (-\frac {\left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+12 \sqrt {2}+17\right )}\, \left (\frac {24 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+\frac {17 \sqrt {2}\, \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-\sqrt {2}\right ) \left (x -\sqrt {2}-1\right ) \left (3 \sqrt {2}-4\right )}{2 \left (-x +1-\sqrt {2}\right ) \left (\frac {\left (x -\sqrt {2}-1\right )^{4}}{\left (-x +1-\sqrt {2}\right )^{4}}-\frac {34 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+1\right )}\right )-2 \sqrt {-2+2 \sqrt {2}}\, \sqrt {1+\sqrt {2}}\, \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}\, \sqrt {\left (3 \sqrt {2}-4\right ) \left (-\frac {\left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+12 \sqrt {2}+17\right )}\, \left (\frac {24 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+\frac {17 \sqrt {2}\, \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-\sqrt {2}\right ) \left (x -\sqrt {2}-1\right ) \left (3 \sqrt {2}-4\right )}{2 \left (-x +1-\sqrt {2}\right ) \left (\frac {\left (x -\sqrt {2}-1\right )^{4}}{\left (-x +1-\sqrt {2}\right )^{4}}-\frac {34 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}+1\right )}\right )\right )}{2 \sqrt {-\frac {\frac {3 \sqrt {2}\, \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-\frac {4 \left (x -\sqrt {2}-1\right )^{2}}{\left (-x +1-\sqrt {2}\right )^{2}}-3 \sqrt {2}-4}{\left (\frac {x -\sqrt {2}-1}{-x +1-\sqrt {2}}+1\right )^{2}}}\, \left (\frac {x -\sqrt {2}-1}{-x +1-\sqrt {2}}+1\right ) \left (3 \sqrt {2}-4\right ) \sqrt {1+\sqrt {2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+x)^(3/2)/(x^2+1),x)

[Out]

1/2*x*(x^2+x)^(1/2)+5/4*(x^2+x)^(1/2)-5/8*ln(1/2+x+(x^2+x)^(1/2))+1/2*(4*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^
(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+4+3*2^(1/2))^(1/2)*2^(1/2)*((-2+2*2^(1/2))^(1/2)*arctan(1/2*(-2+2*2^(1/
2))^(1/2)*((3*2^(1/2)-4)*(-(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+12*2^(1/2)+17))^(1/2)*(24*(-2^(1/2)-1+x)^2/(1-x-2^
(1/2))^2+17*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-2^(1/2))*(-2^(1/2)-1+x)/(1-x-2^(1/2))*(3*2^(1/2)-4)/((-2^
(1/2)-1+x)^4/(1-x-2^(1/2))^4-34*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+1))*(1+2^(1/2))^(1/2)*2^(1/2)-2*(-2+2*2^(1/2)
)^(1/2)*arctan(1/2*(-2+2*2^(1/2))^(1/2)*((3*2^(1/2)-4)*(-(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+12*2^(1/2)+17))^(1/2
)*(24*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+17*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-2^(1/2))*(-2^(1/2)-1+x)/(1-
x-2^(1/2))*(3*2^(1/2)-4)/((-2^(1/2)-1+x)^4/(1-x-2^(1/2))^4-34*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+1))*(1+2^(1/2))
^(1/2)-4*arctanh(1/2*(4*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+4+3*2^(1/2
))^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)+6*arctanh(1/2*(4*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)*(-2^(1/2)-1+x)
^2/(1-x-2^(1/2))^2+4+3*2^(1/2))^(1/2)/(1+2^(1/2))^(1/2)))/(-(3*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-4*(-2^
(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)-4)/((-2^(1/2)-1+x)/(1-x-2^(1/2))+1)^2)^(1/2)/((-2^(1/2)-1+x)/(1-x-2^(1/
2))+1)/(3*2^(1/2)-4)/(1+2^(1/2))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} + x\right )}^{\frac {3}{2}}}{x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x)^(3/2)/(x^2+1),x, algorithm="maxima")

[Out]

integrate((x^2 + x)^(3/2)/(x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^2+x\right )}^{3/2}}{x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^2)^(3/2)/(x^2 + 1),x)

[Out]

int((x + x^2)^(3/2)/(x^2 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (x + 1\right )\right )^{\frac {3}{2}}}{x^{2} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+x)**(3/2)/(x**2+1),x)

[Out]

Integral((x*(x + 1))**(3/2)/(x**2 + 1), x)

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