3.24.22 \(\int \frac {1}{(1+2 x)^{5/2} (2+3 x+5 x^2)^2} \, dx\) [2322]

Optimal. Leaf size=296 \[ -\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {1}{434} \left (-12504542+2632525 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {5 \sqrt {\frac {1}{434} \left (-12504542+2632525 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633} \]

[Out]

-820/4557/(1+2*x)^(3/2)+1/217*(37+20*x)/(1+2*x)^(3/2)/(5*x^2+3*x+2)-4680/10633/(1+2*x)^(1/2)-5/4614722*ln(5+10
*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-5426971228+1142515850*35^(1/2))^(1/2)+5/4614722*ln(5+10*x+
35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-5426971228+1142515850*35^(1/2))^(1/2)+5/2307361*arctan((-10*(
1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(5426971228+1142515850*35^(1/2))^(1/2)-5/2307361
*arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(5426971228+1142515850*35^(1/2))^(1
/2)

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Rubi [A]
time = 0.32, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {754, 842, 840, 1183, 648, 632, 210, 642} \begin {gather*} \frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \text {ArcTan}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{10633}-\frac {5 \sqrt {\frac {2}{217} \left (12504542+2632525 \sqrt {35}\right )} \text {ArcTan}\left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{10633}+\frac {20 x+37}{217 (2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}-\frac {4680}{10633 \sqrt {2 x+1}}-\frac {820}{4557 (2 x+1)^{3/2}}-\frac {5 \sqrt {\frac {1}{434} \left (2632525 \sqrt {35}-12504542\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{10633}+\frac {5 \sqrt {\frac {1}{434} \left (2632525 \sqrt {35}-12504542\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{10633} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-820/(4557*(1 + 2*x)^(3/2)) - 4680/(10633*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*(1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2))
+ (5*Sqrt[(2*(12504542 + 2632525*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*
(-2 + Sqrt[35])]])/10633 - (5*Sqrt[(2*(12504542 + 2632525*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10
*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/10633 - (5*Sqrt[(-12504542 + 2632525*Sqrt[35])/434]*Log[Sqrt[35] -
Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10633 + (5*Sqrt[(-12504542 + 2632525*Sqrt[35])/434]*Log[
Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10633

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 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*} \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx &=\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {1}{217} \int \frac {255+100 x}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {145-2050 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{1519}\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {-8345-11700 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{10633}\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {2 \text {Subst}\left (\int \frac {-4990-11700 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{10633}\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {-998 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (-4990+2340 \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 )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-998 \sqrt {10 \left (2+\sqrt {35}\right )}+\left (-4990+2340 \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 )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\left (5 \left (499-234 \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 )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\left (5 \left (499-234 \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 )}{10633 \sqrt {14 \left (2+\sqrt {35}\right )}}-\frac {\left (8190+499 \sqrt {35}\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 )}{74431}-\frac {\left (8190+499 \sqrt {35}\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 )}{74431}\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}-\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {\left (2 \left (8190+499 \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 )}{74431}+\frac {\left (2 \left (8190+499 \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 )}{74431}\\ &=-\frac {820}{4557 (1+2 x)^{3/2}}-\frac {4680}{10633 \sqrt {1+2 x}}+\frac {37+20 x}{217 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}+\frac {5 \sqrt {\frac {2}{7 \left (-2+\sqrt {35}\right )}} \left (499+234 \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 )}{10633}-\frac {5 \sqrt {\frac {2}{7 \left (-2+\sqrt {35}\right )}} \left (499+234 \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 )}{10633}-\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}+\frac {5 \sqrt {-\frac {6252271}{217}+\frac {376075 \sqrt {35}}{62}} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{10633}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.25, size = 143, normalized size = 0.48 \begin {gather*} \frac {2 \left (-\frac {217 \left (34121+112560 x+183140 x^2+140400 x^3\right )}{2 (1+2 x)^{3/2} \left (2+3 x+5 x^2\right )}-15 \sqrt {217 \left (12504542-1667459 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-15 \sqrt {217 \left (12504542+1667459 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{6922083} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-217*(34121 + 112560*x + 183140*x^2 + 140400*x^3))/(2*(1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)) - 15*Sqrt[217*(1
2504542 - (1667459*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - 15*Sqrt[217*(12504542 + (16
67459*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/6922083

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(432\) vs. \(2(202)=404\).
time = 1.80, size = 433, normalized size = 1.46 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/343*(89/62*(2*x+1)^(3/2)+233/620*(2*x+1)^(1/2))/((2*x+1)^2-8/5*x+3/5)+1/4614722*(33845*(2*5^(1/2)*7^(1/2)+
4)^(1/2)*5^(1/2)-45940*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1
/2)+5^(1/2)*7^(1/2)+10*x+5)+10/2307361*(-30938*5^(1/2)*7^(1/2)+1/10*(33845*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)
-45940*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)
*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*(2*x+1)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))-1/4614722*(3384
5*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-45940*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*ln(5^(1/2)*7^(1/2)+10*x+5+(2*
5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(2*x+1)^(1/2))-10/2307361*(30938*5^(1/2)*7^(1/2)-1/10*(33845*(2*5^(1/2)*7^(1/
2)+4)^(1/2)*5^(1/2)-45940*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2))*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2
)*7^(1/2)-20)^(1/2)*arctan((10*(2*x+1)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2
))-16/147/(2*x+1)^(3/2)-128/343/(2*x+1)^(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(1/(1+2*x)^(5/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (205) = 410\).
time = 2.47, size = 653, normalized size = 2.21 \begin {gather*} -\frac {620294748 \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} \sqrt {35} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {12504542 \, \sqrt {35} + 92138375} \arctan \left (\frac {1}{55152316249116723757744225} \cdot 161637035^{\frac {3}{4}} \sqrt {4298} \sqrt {1535} \sqrt {217} \sqrt {149} \sqrt {161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} {\left (234 \, \sqrt {35} \sqrt {31} - 499 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {12504542 \, \sqrt {35} + 92138375} + 7775911578470 \, x + 777591157847 \, \sqrt {35} + 3887955789235} \sqrt {12504542 \, \sqrt {35} + 92138375} {\left (499 \, \sqrt {35} - 8190\right )} - \frac {1}{58486518937462105} \cdot 161637035^{\frac {3}{4}} \sqrt {4298} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {12504542 \, \sqrt {35} + 92138375} {\left (499 \, \sqrt {35} - 8190\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 620294748 \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} \sqrt {35} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} \sqrt {12504542 \, \sqrt {35} + 92138375} \arctan \left (\frac {1}{4729311118361759062226567293750} \cdot 161637035^{\frac {3}{4}} \sqrt {4298} \sqrt {217} \sqrt {149} \sqrt {-11286950937500 \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} {\left (234 \, \sqrt {35} \sqrt {31} - 499 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {12504542 \, \sqrt {35} + 92138375} + 87766332480529071315625000 \, x + 8776633248052907131562500 \, \sqrt {35} + 43883166240264535657812500} \sqrt {12504542 \, \sqrt {35} + 92138375} {\left (499 \, \sqrt {35} - 8190\right )} - \frac {1}{58486518937462105} \cdot 161637035^{\frac {3}{4}} \sqrt {4298} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {12504542 \, \sqrt {35} + 92138375} {\left (499 \, \sqrt {35} - 8190\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 3 \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} {\left (12504542 \, \sqrt {35} \sqrt {31} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} - 92138375 \, \sqrt {31} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )}\right )} \sqrt {12504542 \, \sqrt {35} + 92138375} \log \left (\frac {11286950937500}{53789} \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} {\left (234 \, \sqrt {35} \sqrt {31} - 499 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {12504542 \, \sqrt {35} + 92138375} + 1631678084376528125000 \, x + 163167808437652812500 \, \sqrt {35} + 815839042188264062500\right ) - 3 \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} {\left (12504542 \, \sqrt {35} \sqrt {31} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )} - 92138375 \, \sqrt {31} {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )}\right )} \sqrt {12504542 \, \sqrt {35} + 92138375} \log \left (-\frac {11286950937500}{53789} \cdot 161637035^{\frac {1}{4}} \sqrt {4298} \sqrt {217} {\left (234 \, \sqrt {35} \sqrt {31} - 499 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {12504542 \, \sqrt {35} + 92138375} + 1631678084376528125000 \, x + 163167808437652812500 \, \sqrt {35} + 815839042188264062500\right ) + 337474562505598 \, {\left (140400 \, x^{3} + 183140 \, x^{2} + 112560 \, x + 34121\right )} \sqrt {2 \, x + 1}}{10765101069366070602 \, {\left (20 \, x^{4} + 32 \, x^{3} + 25 \, x^{2} + 11 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10765101069366070602*(620294748*161637035^(1/4)*sqrt(4298)*sqrt(217)*sqrt(35)*(20*x^4 + 32*x^3 + 25*x^2 + 1
1*x + 2)*sqrt(12504542*sqrt(35) + 92138375)*arctan(1/55152316249116723757744225*161637035^(3/4)*sqrt(4298)*sqr
t(1535)*sqrt(217)*sqrt(149)*sqrt(161637035^(1/4)*sqrt(4298)*sqrt(217)*(234*sqrt(35)*sqrt(31) - 499*sqrt(31))*s
qrt(2*x + 1)*sqrt(12504542*sqrt(35) + 92138375) + 7775911578470*x + 777591157847*sqrt(35) + 3887955789235)*sqr
t(12504542*sqrt(35) + 92138375)*(499*sqrt(35) - 8190) - 1/58486518937462105*161637035^(3/4)*sqrt(4298)*sqrt(21
7)*sqrt(2*x + 1)*sqrt(12504542*sqrt(35) + 92138375)*(499*sqrt(35) - 8190) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt
(31)) + 620294748*161637035^(1/4)*sqrt(4298)*sqrt(217)*sqrt(35)*(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2)*sqrt(125
04542*sqrt(35) + 92138375)*arctan(1/4729311118361759062226567293750*161637035^(3/4)*sqrt(4298)*sqrt(217)*sqrt(
149)*sqrt(-11286950937500*161637035^(1/4)*sqrt(4298)*sqrt(217)*(234*sqrt(35)*sqrt(31) - 499*sqrt(31))*sqrt(2*x
+ 1)*sqrt(12504542*sqrt(35) + 92138375) + 87766332480529071315625000*x + 8776633248052907131562500*sqrt(35) +
43883166240264535657812500)*sqrt(12504542*sqrt(35) + 92138375)*(499*sqrt(35) - 8190) - 1/58486518937462105*16
1637035^(3/4)*sqrt(4298)*sqrt(217)*sqrt(2*x + 1)*sqrt(12504542*sqrt(35) + 92138375)*(499*sqrt(35) - 8190) - 1/
31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 3*161637035^(1/4)*sqrt(4298)*sqrt(217)*(12504542*sqrt(35)*sqrt(31)*(20
*x^4 + 32*x^3 + 25*x^2 + 11*x + 2) - 92138375*sqrt(31)*(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2))*sqrt(12504542*sq
rt(35) + 92138375)*log(11286950937500/53789*161637035^(1/4)*sqrt(4298)*sqrt(217)*(234*sqrt(35)*sqrt(31) - 499*
sqrt(31))*sqrt(2*x + 1)*sqrt(12504542*sqrt(35) + 92138375) + 1631678084376528125000*x + 163167808437652812500*
sqrt(35) + 815839042188264062500) - 3*161637035^(1/4)*sqrt(4298)*sqrt(217)*(12504542*sqrt(35)*sqrt(31)*(20*x^4
+ 32*x^3 + 25*x^2 + 11*x + 2) - 92138375*sqrt(31)*(20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2))*sqrt(12504542*sqrt(3
5) + 92138375)*log(-11286950937500/53789*161637035^(1/4)*sqrt(4298)*sqrt(217)*(234*sqrt(35)*sqrt(31) - 499*sqr
t(31))*sqrt(2*x + 1)*sqrt(12504542*sqrt(35) + 92138375) + 1631678084376528125000*x + 163167808437652812500*sqr
t(35) + 815839042188264062500) + 337474562505598*(140400*x^3 + 183140*x^2 + 112560*x + 34121)*sqrt(2*x + 1))/(
20*x^4 + 32*x^3 + 25*x^2 + 11*x + 2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (205) = 410\).
time = 1.94, size = 638, normalized size = 2.16 \begin {gather*} -\frac {1}{7914248230} \, \sqrt {31} {\left (24570 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 117 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 234 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 49140 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 244510 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 489020 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) - \frac {1}{7914248230} \, \sqrt {31} {\left (24570 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 117 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 234 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 49140 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 244510 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 489020 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450}\right )} \arctan \left (-\frac {5 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (\left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} - \sqrt {2 \, x + 1}\right )}}{7 \, \sqrt {-\frac {1}{35} \, \sqrt {35} + \frac {1}{2}}}\right ) - \frac {1}{15828496460} \, \sqrt {31} {\left (117 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 24570 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 49140 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 234 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 244510 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 489020 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) + \frac {1}{15828496460} \, \sqrt {31} {\left (117 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 24570 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 49140 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 234 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 244510 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 489020 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450}\right )} \log \left (-2 \, \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {2 \, x + 1} \sqrt {\frac {1}{35} \, \sqrt {35} + \frac {1}{2}} + 2 \, x + \sqrt {\frac {7}{5}} + 1\right ) - \frac {4 \, {\left (890 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} + 233 \, \sqrt {2 \, x + 1}\right )}}{10633 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}} - \frac {16 \, {\left (48 \, x + 31\right )}}{1029 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7914248230*sqrt(31)*(24570*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 117*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 234*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) - 49140*(7/5)^(3/4)*sq
rt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 244510*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 489020*(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)) - 1/7914248230*sqrt(31)*(24570*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-
140*sqrt(35) + 2450) - 117*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 234*(7/5)^(3/4)*(140*sqrt(35) +
2450)^(3/2) - 49140*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 244510*sqrt(31)*(7/5)^(1/4)*sqr
t(-140*sqrt(35) + 2450) - 489020*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*s
qrt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/15828496460*sqrt(31)*(117*sqrt(31)*(
7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 24570*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35
) + 49140*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 234*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3
/2) + 244510*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 489020*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*l
og(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 1/15828496460*sqrt(31)*(117*
sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 24570*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqr
t(35) - 35) + 49140*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 234*(7/5)^(3/4)*(-140*sqrt(35)
+ 2450)^(3/2) + 244510*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 489020*(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) - 4/10633*(890*(2*x
+ 1)^(3/2) + 233*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3) - 16/1029*(48*x + 31)/(2*x + 1)^(3/2)

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Mupad [B]
time = 0.17, size = 226, normalized size = 0.76 \begin {gather*} -\frac {\frac {128\,x}{147}-\frac {5492\,{\left (2\,x+1\right )}^2}{31899}+\frac {4680\,{\left (2\,x+1\right )}^3}{10633}+\frac {144}{245}}{\frac {7\,{\left (2\,x+1\right )}^{3/2}}{5}-\frac {4\,{\left (2\,x+1\right )}^{5/2}}{5}+{\left (2\,x+1\right )}^{7/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}\,6884992{}\mathrm {i}}{1900211000023\,\left (-\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}-\frac {13769984\,\sqrt {31}\,\sqrt {217}\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}}{58906541000713\,\left (-\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}\right )\,\sqrt {-12504542-\sqrt {31}\,1667459{}\mathrm {i}}\,10{}\mathrm {i}}{2307361}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}\,6884992{}\mathrm {i}}{1900211000023\,\left (\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}+\frac {13769984\,\sqrt {31}\,\sqrt {217}\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,\sqrt {2\,x+1}}{58906541000713\,\left (\frac {63259306496}{271458714289}+\frac {\sqrt {31}\,3435611008{}\mathrm {i}}{271458714289}\right )}\right )\,\sqrt {-12504542+\sqrt {31}\,1667459{}\mathrm {i}}\,10{}\mathrm {i}}{2307361} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*1667459i - 12504542)^(1/2)*(2*x + 1)^(1/2)*6884992i)/(1900211000023*((3
1^(1/2)*3435611008i)/271458714289 - 63259306496/271458714289)) - (13769984*31^(1/2)*217^(1/2)*(- 31^(1/2)*1667
459i - 12504542)^(1/2)*(2*x + 1)^(1/2))/(58906541000713*((31^(1/2)*3435611008i)/271458714289 - 63259306496/271
458714289)))*(- 31^(1/2)*1667459i - 12504542)^(1/2)*10i)/2307361 - ((128*x)/147 - (5492*(2*x + 1)^2)/31899 + (
4680*(2*x + 1)^3)/10633 + 144/245)/((7*(2*x + 1)^(3/2))/5 - (4*(2*x + 1)^(5/2))/5 + (2*x + 1)^(7/2)) - (217^(1
/2)*atan((217^(1/2)*(31^(1/2)*1667459i - 12504542)^(1/2)*(2*x + 1)^(1/2)*6884992i)/(1900211000023*((31^(1/2)*3
435611008i)/271458714289 + 63259306496/271458714289)) + (13769984*31^(1/2)*217^(1/2)*(31^(1/2)*1667459i - 1250
4542)^(1/2)*(2*x + 1)^(1/2))/(58906541000713*((31^(1/2)*3435611008i)/271458714289 + 63259306496/271458714289))
)*(31^(1/2)*1667459i - 12504542)^(1/2)*10i)/2307361

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