3.1.93 \(\int \frac {(x+x^2)^{3/2}}{1+x^2} \, dx\) [93]

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

[Out]

-5/4*arctanh(x/(x^2+x)^(1/2))+1/4*(5+2*x)*(x^2+x)^(1/2)-arctanh((1-x-2^(1/2))/(x^2+x)^(1/2)/(-2+2*2^(1/2))^(1/
2))*(2^(1/2)-1)^(1/2)+arctan((1-x+2^(1/2))/(x^2+x)^(1/2)/(2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)

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Rubi [A]
time = 0.10, 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 = {992, 1092, 634, 212, 12, 1050, 1044, 213, 209} \begin {gather*} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {-x+\sqrt {2}+1}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x^2+x}}\right )+\frac {1}{4} \sqrt {x^2+x} (2 x+5)-\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 209

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

Rule 212

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

Rule 213

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

Rule 634

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 992

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 1044

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 1050

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 1092

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} \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.12, size = 117, normalized size = 0.90 \begin {gather*} \frac {\sqrt {x} \sqrt {1+x} \left (\sqrt {x} \sqrt {1+x} (5+2 x)-5 \tanh ^{-1}\left (\sqrt {\frac {x}{1+x}}\right )+8 \text {RootSum}\left [16+32 \text {$\#$1}+16 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log \left (-2 x+2 \sqrt {x} \sqrt {1+x}+\text {$\#$1}\right ) \text {$\#$1}^2}{8+8 \text {$\#$1}+\text {$\#$1}^3}\&\right ]\right )}{4 \sqrt {x (1+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*Sqrt[1 + x]*(Sqrt[x]*Sqrt[1 + x]*(5 + 2*x) - 5*ArcTanh[Sqrt[x/(1 + x)]] + 8*RootSum[16 + 32*#1 + 16*#
1^2 + #1^4 & , (Log[-2*x + 2*Sqrt[x]*Sqrt[1 + x] + #1]*#1^2)/(8 + 8*#1 + #1^3) & ]))/(4*Sqrt[x*(1 + x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(788\) vs. \(2(98)=196\).
time = 0.51, size = 789, normalized size = 6.07 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+x)^(3/2)/(x^2+1),x,method=_RETURNVERBOSE)

[Out]

1/2*x*(x^2+x)^(1/2)+5/4*(x^2+x)^(1/2)-5/8*ln(x+1/2+(x^2+x)^(1/2))+1/2*(4*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2-3*2
^(1/2)*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+4+3*2^(1/2))^(1/2)*2^(1/2)*((-2+2*2^(1/2))^(1/2)*arctan(1/2*((3*2^(1/
2)-4)*(-(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+12*2^(1/2)+17))^(1/2)*(-2+2*2^(1/2))^(1/2)*(24*(-2^(1/2)-1+x)^2/(-2^
(1/2)+1-x)^2+17*2^(1/2)*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2-2^(1/2))*(-2^(1/2)-1+x)/(-2^(1/2)+1-x)*(3*2^(1/2)-4)
/((-2^(1/2)-1+x)^4/(-2^(1/2)+1-x)^4-34*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+1))*(1+2^(1/2))^(1/2)*2^(1/2)-2*(-2+2
*2^(1/2))^(1/2)*arctan(1/2*((3*2^(1/2)-4)*(-(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+12*2^(1/2)+17))^(1/2)*(-2+2*2^(1
/2))^(1/2)*(24*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+17*2^(1/2)*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2-2^(1/2))*(-2^(1/
2)-1+x)/(-2^(1/2)+1-x)*(3*2^(1/2)-4)/((-2^(1/2)-1+x)^4/(-2^(1/2)+1-x)^4-34*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+1
))*(1+2^(1/2))^(1/2)-4*arctanh(1/2*(4*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2-3*2^(1/2)*(-2^(1/2)-1+x)^2/(-2^(1/2)+1
-x)^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/(-2^(1/2)+1-x)^2-3*2^(1/
2)*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2+4+3*2^(1/2))^(1/2)/(1+2^(1/2))^(1/2)))/(-(3*2^(1/2)*(-2^(1/2)-1+x)^2/(-2^
(1/2)+1-x)^2-4*(-2^(1/2)-1+x)^2/(-2^(1/2)+1-x)^2-3*2^(1/2)-4)/(1+(-2^(1/2)-1+x)/(-2^(1/2)+1-x))^2)^(1/2)/(1+(-
2^(1/2)-1+x)/(-2^(1/2)+1-x))/(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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (96) = 192\).
time = 0.37, size = 777, normalized size = 5.98 \begin {gather*} -\frac {1}{8} \cdot 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} \log \left (8 \, x^{2} - 8 \, \sqrt {x^{2} + x} x + 2 \, {\left (8^{\frac {1}{4}} \sqrt {x^{2} + x} {\left (\sqrt {2} - 1\right )} - 8^{\frac {1}{4}} {\left (\sqrt {2} x - x - 1\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) + \frac {1}{8} \cdot 8^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} \log \left (8 \, x^{2} - 8 \, \sqrt {x^{2} + x} x - 2 \, {\left (8^{\frac {1}{4}} \sqrt {x^{2} + x} {\left (\sqrt {2} - 1\right )} - 8^{\frac {1}{4}} {\left (\sqrt {2} x - x - 1\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) + \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{7} \, \sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 1\right )} + 6 \, x + 4\right )} + \frac {1}{112} \, \sqrt {8 \, x^{2} - 8 \, \sqrt {x^{2} + x} x - 2 \, {\left (8^{\frac {1}{4}} \sqrt {x^{2} + x} {\left (\sqrt {2} - 1\right )} - 8^{\frac {1}{4}} {\left (\sqrt {2} x - x - 1\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} {\left (8 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + {\left (8^{\frac {3}{4}} {\left (5 \, \sqrt {2} + 6\right )} + 8 \cdot 8^{\frac {1}{4}} {\left (2 \, \sqrt {2} + 1\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 64 \, \sqrt {2} + 32\right )} - \frac {1}{7} \, \sqrt {x^{2} + x} {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} + \frac {1}{7} \, \sqrt {2} {\left (8 \, x + 3\right )} + \frac {1}{56} \, {\left (8^{\frac {3}{4}} {\left (\sqrt {2} {\left (5 \, x + 1\right )} + 6 \, x + 4\right )} - \sqrt {x^{2} + x} {\left (8^{\frac {3}{4}} {\left (5 \, \sqrt {2} + 6\right )} + 8 \cdot 8^{\frac {1}{4}} {\left (2 \, \sqrt {2} + 1\right )}\right )} + 8 \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} {\left (2 \, x - 1\right )} + x + 3\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + \frac {4}{7} \, x + \frac {5}{7}\right ) + \frac {1}{2} \cdot 8^{\frac {1}{4}} \sqrt {2} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{7} \, \sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 1\right )} + 6 \, x + 4\right )} - \frac {1}{112} \, \sqrt {8 \, x^{2} - 8 \, \sqrt {x^{2} + x} x + 2 \, {\left (8^{\frac {1}{4}} \sqrt {x^{2} + x} {\left (\sqrt {2} - 1\right )} - 8^{\frac {1}{4}} {\left (\sqrt {2} x - x - 1\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} {\left (8 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} - {\left (8^{\frac {3}{4}} {\left (5 \, \sqrt {2} + 6\right )} + 8 \cdot 8^{\frac {1}{4}} {\left (2 \, \sqrt {2} + 1\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 64 \, \sqrt {2} + 32\right )} + \frac {1}{7} \, \sqrt {x^{2} + x} {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} - \frac {1}{7} \, \sqrt {2} {\left (8 \, x + 3\right )} + \frac {1}{56} \, {\left (8^{\frac {3}{4}} {\left (\sqrt {2} {\left (5 \, x + 1\right )} + 6 \, x + 4\right )} - \sqrt {x^{2} + x} {\left (8^{\frac {3}{4}} {\left (5 \, \sqrt {2} + 6\right )} + 8 \cdot 8^{\frac {1}{4}} {\left (2 \, \sqrt {2} + 1\right )}\right )} + 8 \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} {\left (2 \, x - 1\right )} + x + 3\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} - \frac {4}{7} \, x - \frac {5}{7}\right ) + \frac {1}{4} \, \sqrt {x^{2} + x} {\left (2 \, x + 5\right )} + \frac {5}{8} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \end {gather*}

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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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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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,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%{poly1[29378258633931653019799718485334848549113596143
37236965430

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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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