\(\int \frac {1}{(1+2 x)^{3/2} (2+3 x+5 x^2)^3} \, dx\) [2329]

Optimal result

Integrand size = 22, antiderivative size = 313 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{329623}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246} \]

[Out]

-81090/329623/(1+2*x)^(1/2)+1/434*(37+20*x)/(5*x^2+3*x+2)^2/(1+2*x)^(1/2)+5/94178*(2329+2080*x)/(5*x^2+3*x+2)/
(1+2*x)^(1/2)-15/143056382*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-979586
504708+168143350550*35^(1/2))^(1/2)+15/143056382*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(
1/2))^(1/2))*(-979586504708+168143350550*35^(1/2))^(1/2)-15/286112764*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*
35^(1/2))^(1/2))*(979586504708+168143350550*35^(1/2))^(1/2)+15/286112764*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+
10*35^(1/2))^(1/2))*(979586504708+168143350550*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {754, 836, 842, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (387427075 \sqrt {35}-2257111762\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{329623}+\frac {20 x+37}{434 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac {5 (2080 x+2329)}{94178 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {81090}{329623 \sqrt {2 x+1}}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{659246} \]

[In]

Int[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]

[Out]

-81090/(329623*Sqrt[1 + 2*x]) + (37 + 20*x)/(434*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + (5*(2329 + 2080*x))/(941
78*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - (15*Sqrt[(-2257111762 + 387427075*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sq
rt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/329623 + (15*Sqrt[(-2257111762 + 387427075*Sqrt[35])/4
34]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/329623 - (15*Sqrt[(22571117
62 + 387427075*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/659246 + (1
5*Sqrt[(2257111762 + 387427075*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*
x)])/659246

Rule 210

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

Rule 632

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 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), 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 754

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

Rule 836

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

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 842

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

Rule 1183

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

Rubi steps \begin{align*} \text {integral}& = \frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {1}{434} \int \frac {345+140 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx \\ & = \frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {56145+31200 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{94178} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {68655-405450 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{659246} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {542760-405450 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{329623} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {108552 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (542760+81090 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{659246 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {108552 \sqrt {10 \left (2+\sqrt {35}\right )}+\left (542760+81090 \sqrt {35}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{659246 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {\left (3 \left (94605-18092 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{4614722}-\frac {\left (3 \left (94605-18092 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{4614722}-\frac {\left (15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )}\right ) \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{659246}+\frac {\left (15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{659246} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {\left (3 \left (94605-18092 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{2307361}+\frac {\left (3 \left (94605-18092 \sqrt {35}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{2307361} \\ & = -\frac {81090}{329623 \sqrt {1+2 x}}+\frac {37+20 x}{434 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+\frac {5 (2329+2080 x)}{94178 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )}{329623}+\frac {15 \sqrt {\frac {1}{434} \left (-2257111762+387427075 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )}{329623}-\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246}+\frac {15 \sqrt {\frac {1}{434} \left (2257111762+387427075 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{659246} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.47 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {-\frac {217 \left (429487+1525635 x+4077245 x^2+4501400 x^3+4054500 x^4\right )}{2 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2}+15 \sqrt {217 \left (-2257111762+71603149 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+15 \sqrt {217 \left (-2257111762-71603149 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{71528191} \]

[In]

Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)^3),x]

[Out]

((-217*(429487 + 1525635*x + 4077245*x^2 + 4501400*x^3 + 4054500*x^4))/(2*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) +
15*Sqrt[217*(-2257111762 + (71603149*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 15*Sqrt[
217*(-2257111762 - (71603149*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/71528191

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.13

method	result	size
pseudoelliptic	\(\frac {-\frac {2378625 \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\sqrt {5}+\frac {58421 \sqrt {7}}{88802}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{20436626}+\frac {2378625 \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\sqrt {5}+\frac {58421 \sqrt {7}}{88802}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{20436626}+\frac {6784500 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {94605}{18092}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )^{2} \sqrt {1+2 x}}{2307361}-\frac {2027250 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{4}+\frac {45014}{40545} x^{3}+\frac {815449}{810900} x^{2}+\frac {33903}{90100} x +\frac {429487}{4054500}\right )}{329623}}{\sqrt {1+2 x}\, \left (5 x^{2}+3 x +2\right )^{2} \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(354\)
derivativedivides	\(-\frac {64}{343 \sqrt {1+2 x}}-\frac {1600 \left (\frac {9793 \left (1+2 x \right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (1+2 x \right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (1+2 x \right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {1+2 x}}{192200}\right )}{343 \left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}-\frac {3 \left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}+\frac {\left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}-\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(444\)
default	\(-\frac {64}{343 \sqrt {1+2 x}}-\frac {1600 \left (\frac {9793 \left (1+2 x \right )^{\frac {7}{2}}}{30752}-\frac {14343 \left (1+2 x \right )^{\frac {5}{2}}}{19220}+\frac {762223 \left (1+2 x \right )^{\frac {3}{2}}}{768800}-\frac {170877 \sqrt {1+2 x}}{192200}\right )}{343 \left (5 \left (1+2 x \right )^{2}+3-8 x \right )^{2}}-\frac {3 \left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}+\frac {\left (444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3 \left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{286112764}-\frac {15 \left (-1121704 \sqrt {5}\, \sqrt {7}-\frac {\left (-444010 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-292105 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{71528191 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(444\)
trager	\(\text {Expression too large to display}\)	\(462\)
risch	\(-\frac {4054500 x^{4}+4501400 x^{3}+4077245 x^{2}+1525635 x +429487}{659246 \left (5 x^{2}+3 x +2\right )^{2} \sqrt {1+2 x}}-\frac {95145 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626}-\frac {876315 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{286112764}-\frac {475725 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {95145 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{20436626}+\frac {876315 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{286112764}-\frac {475725 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{10218313 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {876315 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{143056382 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {542760 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{2307361 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(648\)

[In]

int(1/(1+2*x)^(3/2)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)

[Out]

6784500/2307361/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-44401/1121704*(1+2*x)^(1/2)*(x^2+3/5*x+2/5)^2*(2*5^(1/2)*7^(1/
2)+4)^(1/2)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(5^(1/2)+58421/88802*7^(1/2))*ln(5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+
4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5+10*x)+44401/1121704*(1+2*x)^(1/2)*(x^2+3/5*x+2/5)^2*(2*5^(1/2)*7^(1/2)+4)^(1/
2)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(5^(1/2)+58421/88802*7^(1/2))*ln(5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*
5^(1/2)*(1+2*x)^(1/2)+5+10*x)+(arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/
2)-20)^(1/2))-arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)))*(5
^(1/2)*7^(1/2)-94605/18092)*(x^2+3/5*x+2/5)^2*(1+2*x)^(1/2)-18921/9046*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x^4+4501
4/40545*x^3+815449/810900*x^2+33903/90100*x+429487/4054500))/(1+2*x)^(1/2)/(5*x^2+3*x+2)^2

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} {\left (58421 i \, \sqrt {31} + 560852\right )} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} \sqrt {16110708525 i \, \sqrt {31} + 507850146450} {\left (-58421 i \, \sqrt {31} - 560852\right )} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} {\left (58421 i \, \sqrt {31} - 560852\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} \log \left (\sqrt {217} {\left (-58421 i \, \sqrt {31} + 560852\right )} \sqrt {-16110708525 i \, \sqrt {31} + 507850146450} + 6305375645625 \, \sqrt {2 \, x + 1}\right ) - 217 \, {\left (4054500 \, x^{4} + 4501400 \, x^{3} + 4077245 \, x^{2} + 1525635 \, x + 429487\right )} \sqrt {2 \, x + 1}}{143056382 \, {\left (50 \, x^{5} + 85 \, x^{4} + 88 \, x^{3} + 53 \, x^{2} + 20 \, x + 4\right )}} \]

[In]

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

[Out]

1/143056382*(sqrt(217)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sqrt(16110708525*I*sqrt(31) + 5078501464
50)*log(sqrt(217)*sqrt(16110708525*I*sqrt(31) + 507850146450)*(58421*I*sqrt(31) + 560852) + 6305375645625*sqrt
(2*x + 1)) - sqrt(217)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sqrt(16110708525*I*sqrt(31) + 5078501464
50)*log(sqrt(217)*sqrt(16110708525*I*sqrt(31) + 507850146450)*(-58421*I*sqrt(31) - 560852) + 6305375645625*sqr
t(2*x + 1)) - sqrt(217)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sqrt(-16110708525*I*sqrt(31) + 50785014
6450)*log(sqrt(217)*(58421*I*sqrt(31) - 560852)*sqrt(-16110708525*I*sqrt(31) + 507850146450) + 6305375645625*s
qrt(2*x + 1)) + sqrt(217)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)*sqrt(-16110708525*I*sqrt(31) + 507850
146450)*log(sqrt(217)*(-58421*I*sqrt(31) + 560852)*sqrt(-16110708525*I*sqrt(31) + 507850146450) + 630537564562
5*sqrt(2*x + 1)) - 217*(4054500*x^4 + 4501400*x^3 + 4077245*x^2 + 1525635*x + 429487)*sqrt(2*x + 1))/(50*x^5 +
85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4)

Sympy [F]

\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \]

[In]

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

[Out]

Integral(1/((2*x + 1)**(3/2)*(5*x**2 + 3*x + 2)**3), x)

Maxima [F]

\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3} {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(1/((5*x^2 + 3*x + 2)^3*(2*x + 1)^(3/2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (222) = 444\).

Time = 0.55 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-3/981366780520*sqrt(31)*(567630*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 2703*sqrt
(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 5406*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 1135260*(7/5)^(
3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) -
35460320*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2)
+ sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 3/981366780520*sqrt(31)*(567630*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35
) + 35)*sqrt(-140*sqrt(35) + 2450) - 2703*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 5406*(7/5)^(3/4)
*(140*sqrt(35) + 2450)^(3/2) + 1135260*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 17730160*sqrt
(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 35460320*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5
)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 3/1962733561040*
sqrt(31)*(2703*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 567630*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35
) + 2450)*(2*sqrt(35) - 35) - 1135260*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 5406*(7/5)^(3
/4)*(-140*sqrt(35) + 2450)^(3/2) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 35460320*(7/5)^(1
/4)*sqrt(-140*sqrt(35) + 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) +
1) + 3/1962733561040*sqrt(31)*(2703*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 567630*sqrt(31)*(7/5)^(
3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 1135260*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) +
2450) + 5406*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 17730160*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450
) + 35460320*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2
) + 2*x + sqrt(7/5) + 1) - 64/343/sqrt(2*x + 1) - 2/47089*(34975*(2*x + 1)^(7/2) - 81960*(2*x + 1)^(5/2) + 108
889*(2*x + 1)^(3/2) - 97644*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {274928\,x}{235445}-\frac {1362758\,{\left (2\,x+1\right )}^2}{1648115}+\frac {144304\,{\left (2\,x+1\right )}^3}{329623}-\frac {81090\,{\left (2\,x+1\right )}^4}{329623}+\frac {256792}{1177225}}{\frac {49\,\sqrt {2\,x+1}}{25}-\frac {56\,{\left (2\,x+1\right )}^{3/2}}{25}+\frac {86\,{\left (2\,x+1\right )}^{5/2}}{25}-\frac {8\,{\left (2\,x+1\right )}^{7/2}}{5}+{\left (2\,x+1\right )}^{9/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}+\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (-\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762-\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,32187888{}\mathrm {i}}{1826102771022103\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}-\frac {64375776\,\sqrt {31}\,\sqrt {217}\,\sqrt {2\,x+1}\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}}{56609185901685193\,\left (\frac {1880448604848}{260871824431729}+\frac {\sqrt {31}\,582343269696{}\mathrm {i}}{260871824431729}\right )}\right )\,\sqrt {2257111762+\sqrt {31}\,71603149{}\mathrm {i}}\,15{}\mathrm {i}}{71528191} \]

[In]

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

[Out]

((274928*x)/235445 - (1362758*(2*x + 1)^2)/1648115 + (144304*(2*x + 1)^3)/329623 - (81090*(2*x + 1)^4)/329623
+ 256792/1177225)/((49*(2*x + 1)^(1/2))/25 - (56*(2*x + 1)^(3/2))/25 + (86*(2*x + 1)^(5/2))/25 - (8*(2*x + 1)^
(7/2))/5 + (2*x + 1)^(9/2)) + (217^(1/2)*atan((217^(1/2)*(2*x + 1)^(1/2)*(2257111762 - 31^(1/2)*71603149i)^(1/
2)*32187888i)/(1826102771022103*((31^(1/2)*582343269696i)/260871824431729 - 1880448604848/260871824431729)) +
(64375776*31^(1/2)*217^(1/2)*(2*x + 1)^(1/2)*(2257111762 - 31^(1/2)*71603149i)^(1/2))/(56609185901685193*((31^
(1/2)*582343269696i)/260871824431729 - 1880448604848/260871824431729)))*(2257111762 - 31^(1/2)*71603149i)^(1/2
)*15i)/71528191 - (217^(1/2)*atan((217^(1/2)*(2*x + 1)^(1/2)*(31^(1/2)*71603149i + 2257111762)^(1/2)*32187888i
)/(1826102771022103*((31^(1/2)*582343269696i)/260871824431729 + 1880448604848/260871824431729)) - (64375776*31
^(1/2)*217^(1/2)*(2*x + 1)^(1/2)*(31^(1/2)*71603149i + 2257111762)^(1/2))/(56609185901685193*((31^(1/2)*582343
269696i)/260871824431729 + 1880448604848/260871824431729)))*(31^(1/2)*71603149i + 2257111762)^(1/2)*15i)/71528
191