3.2329 \(\int \frac{1}{(1+2 x)^{3/2} (2+3 x+5 x^2)^3} \, dx\)

Optimal. Leaf size=313 \[ \frac{20 x+37}{434 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac{5 (2080 x+2329)}{94178 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{81090}{329623 \sqrt{2 x+1}}-\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}+\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}-\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623}+\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.423517, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {740, 822, 828, 826, 1169, 634, 618, 204, 628} \[ \frac{20 x+37}{434 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )^2}+\frac{5 (2080 x+2329)}{94178 \sqrt{2 x+1} \left (5 x^2+3 x+2\right )}-\frac{81090}{329623 \sqrt{2 x+1}}-\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}+\frac{15 \sqrt{\frac{1}{434} \left (2257111762+387427075 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{659246}-\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623}+\frac{15 \sqrt{\frac{1}{434} \left (387427075 \sqrt{35}-2257111762\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{329623} \]

Antiderivative was successfully verified.

[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 740

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 822

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 828

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 826

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 1169

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]

Rule 634

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 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 204

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx &=\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{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.870828, size = 156, normalized size = 0.5 \[ \frac{-\frac{217 \left (4054500 x^4+4501400 x^3+4077245 x^2+1525635 x+429487\right )}{\sqrt{2 x+1} \left (5 x^2+3 x+2\right )^2}+6 \sqrt{10-5 i \sqrt{31}} \left (560852+58421 i \sqrt{31}\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )+6 \sqrt{10+5 i \sqrt{31}} \left (560852-58421 i \sqrt{31}\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )}{143056382} \]

Antiderivative was successfully verified.

[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))/(Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)^2) + 6
*Sqrt[10 - (5*I)*Sqrt[31]]*(560852 + (58421*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 - I*Sqrt[31]]] + 6*Sqrt
[10 + (5*I)*Sqrt[31]]*(560852 - (58421*I)*Sqrt[31])*ArcTanh[Sqrt[5 + 10*x]/Sqrt[2 + I*Sqrt[31]]])/143056382

________________________________________________________________________________________

Maple [B]  time = 0.082, size = 671, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1600/343*(9793/30752*(1+2*x)^(7/2)-14343/19220*(1+2*x)^(5/2)+762223/768800*(1+2*x)^(3/2)-170877/192200*(1+2*x
)^(1/2))/(5*(1+2*x)^2-8*x+3)^2+95145/20436626*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1
+2*x)^(1/2))*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+876315/286112764*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)
+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-475725/10218313/(10*5^(1/2)*7^(1/2)-20)^(
1/2)*arctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7
^(1/2)+4)-876315/143056382/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4
)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)+542760/2307361/(10*5^(1/2)*7^(1/
2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1
/2)*7^(1/2)-95145/20436626*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*5^(1/
2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-876315/286112764*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*
7^(1/2)+10*x+5)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-475725/10218313/(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))*(2*5^(1/2)*7^(1/2)+4)-876315
/143056382/(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)*(2*5^(1/2)*7^(1/2)+4)*7^(1/2)+542760/2307361/(10*5^(1/2)*7^(1/2)-20)^(1/2)*ar
ctan((-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)-64
/343/(1+2*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}{\left (2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.95809, size = 3787, normalized size = 12.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49209733431234418124460140*(26636371428*3500868535115^(1/4)*sqrt(217)*sqrt(35)*(50*x^5 + 85*x^4 + 88*x^3 +
53*x^2 + 20*x + 4)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)*arctan(1/15090017683016917385677
87671747064606625*3500868535115^(3/4)*sqrt(2309779)*sqrt(316267)*sqrt(217)*sqrt(3500868535115^(1/4)*sqrt(217)*
(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x + 1)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893
750) + 1719941911827268850*x + 171994191182726885*sqrt(35) + 859970955913634425)*(18092*sqrt(35) + 94605)*sqrt
(-1748932415799512300*sqrt(35) + 10506981691013893750) - 1/1903863040197561930840325*3500868535115^(3/4)*sqrt(
217)*sqrt(2*x + 1)*(18092*sqrt(35) + 94605)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750) - 1/31*
sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 26636371428*3500868535115^(1/4)*sqrt(217)*sqrt(35)*(50*x^5 + 85*x^4 + 88*
x^3 + 53*x^2 + 20*x + 4)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)*arctan(1/27727907492543585
696183098468352312146734375*3500868535115^(3/4)*sqrt(2309779)*sqrt(217)*sqrt(-106784587546875*3500868535115^(1
/4)*sqrt(217)*(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x + 1)*sqrt(-1748932415799512300*sqrt(35) + 105
06981691013893750) + 183663287659058552515752602343750*x + 18366328765905855251575260234375*sqrt(35) + 9183164
3829529276257876301171875)*(18092*sqrt(35) + 94605)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)
- 1/1903863040197561930840325*3500868535115^(3/4)*sqrt(217)*sqrt(2*x + 1)*(18092*sqrt(35) + 94605)*sqrt(-1748
932415799512300*sqrt(35) + 10506981691013893750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) - 3*3500868535115^(
1/4)*sqrt(217)*(2257111762*sqrt(35)*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4) + 13559947625*sqrt
(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4))*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750
)*log(106784587546875/2309779*3500868535115^(1/4)*sqrt(217)*(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x
+ 1)*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750) + 79515524064881771163281250*x + 795155240648
8177116328125*sqrt(35) + 39757762032440885581640625) + 3*3500868535115^(1/4)*sqrt(217)*(2257111762*sqrt(35)*sq
rt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^2 + 20*x + 4) + 13559947625*sqrt(31)*(50*x^5 + 85*x^4 + 88*x^3 + 53*x^
2 + 20*x + 4))*sqrt(-1748932415799512300*sqrt(35) + 10506981691013893750)*log(-106784587546875/2309779*3500868
535115^(1/4)*sqrt(217)*(2703*sqrt(35)*sqrt(31) + 18092*sqrt(31))*sqrt(2*x + 1)*sqrt(-1748932415799512300*sqrt(
35) + 10506981691013893750) + 79515524064881771163281250*x + 7951552406488177116328125*sqrt(35) + 397577620324
40885581640625) + 74645478973303468090*(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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (2 x + 1\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}{\left (2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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