3.24.16 \(\int \frac {(1+2 x)^{7/2}}{(2+3 x+5 x^2)^2} \, dx\) [2316]

Optimal. Leaf size=296 \[ \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

-8/155*(1+2*x)^(3/2)-1/31*(5-4*x)*(1+2*x)^(5/2)/(5*x^2+3*x+2)+604/775*(1+2*x)^(1/2)+1/120125*arctan((-10*(1+2*
x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-1761642580+300382250*35^(1/2))^(1/2)-1/120125*arct
an((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-1761642580+300382250*35^(1/2))^(1/2)+1
/240250*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(1761642580+300382250*35^(1/2))^(1/2)-1/24025
0*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(1761642580+300382250*35^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.30, 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 = {752, 838, 840, 1183, 648, 632, 210, 642} \begin {gather*} \frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\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 )-\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\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 )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} (2 x+1)^{3/2}+\frac {604}{775} \sqrt {2 x+1}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/(31*(2 + 3*x + 5*x^2)) + (Sqrt
[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[
35])]])/775 - (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]
*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775

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 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), 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] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 838

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

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 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+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {(29-12 x) (1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx\\ &=-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \int \frac {\sqrt {1+2 x} (193+302 x)}{2+3 x+5 x^2} \, dx\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \int \frac {-243+1628 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {2}{775} \text {Subst}\left (\int \frac {-2114+1628 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {-2114 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-2114-1628 \sqrt {\frac {7}{5}}\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 )}{775 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-2114 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-2114-1628 \sqrt {\frac {7}{5}}\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 )}{775 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {1460631-245828 \sqrt {35}} \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 )}{3875}-\frac {\sqrt {1460631-245828 \sqrt {35}} \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 )}{3875}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\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 )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\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 )\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )+\frac {\left (2 \sqrt {1460631-245828 \sqrt {35}}\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 )}{3875}+\frac {\left (2 \sqrt {1460631-245828 \sqrt {35}}\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 )}{3875}\\ &=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \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 )-\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \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 )+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 1.14, size = 136, normalized size = 0.46 \begin {gather*} \frac {2 \left (\frac {155 \sqrt {1+2 x} \left (1003+1132 x+2480 x^2\right )}{4+6 x+10 x^2}-\sqrt {155 \left (-5682718+135439 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {155 \left (-5682718-135439 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{120125} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(2*((155*Sqrt[1 + 2*x]*(1003 + 1132*x + 2480*x^2))/(4 + 6*x + 10*x^2) - Sqrt[155*(-5682718 + (135439*I)*Sqrt[3
1])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - Sqrt[155*(-5682718 - (135439*I)*Sqrt[31])]*ArcTan[Sqrt[
(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/120125

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(423\) vs. \(2(202)=404\).
time = 1.84, size = 424, normalized size = 1.43 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)^(7/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)

[Out]

16/25*(2*x+1)^(1/2)+16/25*(-89/310*(2*x+1)^(3/2)+189/620*(2*x+1)^(1/2))/((2*x+1)^2-8/5*x+3/5)-1/240250*(-3657*
(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-2560*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)-2/24025*(9362*5^(1/2)*7^(1/2)+1/10*(-3657*(2*5^(1/2)*7^(1/2)+4)^
(1/2)*5^(1/2)-2560*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))+1/2
40250*(-3657*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-2560*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))+2/24025*(-9362*5^(1/2)*7^(1/2)-1/10*(-3657*(2*5^(1/2
)*7^(1/2)+4)^(1/2)*5^(1/2)-2560*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))

________________________________________________________________________________________

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+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (205) = 410\).
time = 2.84, size = 541, normalized size = 1.83 \begin {gather*} \frac {16794436 \cdot 21898835^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \arctan \left (\frac {1}{60332699662225359002939375} \cdot 21898835^{\frac {3}{4}} \sqrt {4369} \sqrt {3955} \sqrt {155} \sqrt {21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 40683471557750 \, x + 4068347155775 \, \sqrt {35} + 20341735778875} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} - \frac {1}{3218062600218025} \cdot 21898835^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) + 16794436 \cdot 21898835^{\frac {1}{4}} \sqrt {155} \sqrt {35} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \arctan \left (\frac {1}{4223288976355775130205756250} \cdot 21898835^{\frac {3}{4}} \sqrt {4369} \sqrt {155} \sqrt {-19379500 \cdot 21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 788425337053416125000 \, x + 78842533705341612500 \, \sqrt {35} + 394212668526708062500} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} - \frac {1}{3218062600218025} \cdot 21898835^{\frac {3}{4}} \sqrt {155} \sqrt {2 \, x + 1} {\left (151 \, \sqrt {35} + 814\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) - 21898835^{\frac {1}{4}} \sqrt {155} {\left (5682718 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} + 33914125 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \log \left (\frac {19379500}{4369} \cdot 21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 180458992230125000 \, x + 18045899223012500 \, \sqrt {35} + 90229496115062500\right ) + 21898835^{\frac {1}{4}} \sqrt {155} {\left (5682718 \, \sqrt {35} \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} + 33914125 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )}\right )} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} \log \left (-\frac {19379500}{4369} \cdot 21898835^{\frac {1}{4}} \sqrt {155} {\left (814 \, \sqrt {35} \sqrt {31} + 5285 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {-11012823348100 \, \sqrt {35} + 65723878543750} + 180458992230125000 \, x + 18045899223012500 \, \sqrt {35} + 90229496115062500\right ) + 1261187618290250 \, {\left (2480 \, x^{2} + 1132 \, x + 1003\right )} \sqrt {2 \, x + 1}}{977420404174943750 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")

[Out]

1/977420404174943750*(16794436*21898835^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(-11012823348100*sqrt(3
5) + 65723878543750)*arctan(1/60332699662225359002939375*21898835^(3/4)*sqrt(4369)*sqrt(3955)*sqrt(155)*sqrt(2
1898835^(1/4)*sqrt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) +
65723878543750) + 40683471557750*x + 4068347155775*sqrt(35) + 20341735778875)*(151*sqrt(35) + 814)*sqrt(-11012
823348100*sqrt(35) + 65723878543750) - 1/3218062600218025*21898835^(3/4)*sqrt(155)*sqrt(2*x + 1)*(151*sqrt(35)
+ 814)*sqrt(-11012823348100*sqrt(35) + 65723878543750) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) + 16794436*2
1898835^(1/4)*sqrt(155)*sqrt(35)*(5*x^2 + 3*x + 2)*sqrt(-11012823348100*sqrt(35) + 65723878543750)*arctan(1/42
23288976355775130205756250*21898835^(3/4)*sqrt(4369)*sqrt(155)*sqrt(-19379500*21898835^(1/4)*sqrt(155)*(814*sq
rt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 7884253370534
16125000*x + 78842533705341612500*sqrt(35) + 394212668526708062500)*(151*sqrt(35) + 814)*sqrt(-11012823348100*
sqrt(35) + 65723878543750) - 1/3218062600218025*21898835^(3/4)*sqrt(155)*sqrt(2*x + 1)*(151*sqrt(35) + 814)*sq
rt(-11012823348100*sqrt(35) + 65723878543750) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) - 21898835^(1/4)*sqrt(
155)*(5682718*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) + 33914125*sqrt(31)*(5*x^2 + 3*x + 2))*sqrt(-11012823348100*
sqrt(35) + 65723878543750)*log(19379500/4369*21898835^(1/4)*sqrt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*
sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750) + 180458992230125000*x + 18045899223012500*sqrt(
35) + 90229496115062500) + 21898835^(1/4)*sqrt(155)*(5682718*sqrt(35)*sqrt(31)*(5*x^2 + 3*x + 2) + 33914125*sq
rt(31)*(5*x^2 + 3*x + 2))*sqrt(-11012823348100*sqrt(35) + 65723878543750)*log(-19379500/4369*21898835^(1/4)*sq
rt(155)*(814*sqrt(35)*sqrt(31) + 5285*sqrt(31))*sqrt(2*x + 1)*sqrt(-11012823348100*sqrt(35) + 65723878543750)
+ 180458992230125000*x + 18045899223012500*sqrt(35) + 90229496115062500) + 1261187618290250*(2480*x^2 + 1132*x
+ 1003)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (240) = 480\).
time = 164.74, size = 566, normalized size = 1.91 \begin {gather*} - \frac {17280 \left (2 x + 1\right )^{\frac {3}{2}}}{- 124000 x + 77500 \left (2 x + 1\right )^{2} + 46500} + \frac {21280 \left (2 x + 1\right )^{\frac {3}{2}}}{- 868000 x + 542500 \left (2 x + 1\right )^{2} + 325500} + \frac {16 \sqrt {2 x + 1}}{25} + \frac {6912 \sqrt {2 x + 1}}{- 124000 x + 77500 \left (2 x + 1\right )^{2} + 46500} + \frac {57456 \sqrt {2 x + 1}}{- 868000 x + 542500 \left (2 x + 1\right )^{2} + 325500} + 16 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {1632 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {624 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} - \frac {336 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {288 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {112 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{25} - \frac {48 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {48 \operatorname {RootSum} {\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log {\left (- \frac {27776 t^{3}}{5} + \frac {108 t}{5} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {48 \operatorname {RootSum} {\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log {\left (9920 t^{3} + 8 t + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {448 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} + \frac {2016 \operatorname {RootSum} {\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log {\left (- \frac {11049511452672 t^{3}}{2205125} + \frac {307918256 t}{2205125} + \sqrt {2 x + 1} \right )} \right )\right )}}{25} + \frac {3904 \operatorname {RootSum} {\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log {\left (\frac {33312534528 t^{3}}{235} + \frac {166784 t}{235} + \sqrt {2 x + 1} \right )} \right )\right )}}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)

[Out]

-17280*(2*x + 1)**(3/2)/(-124000*x + 77500*(2*x + 1)**2 + 46500) + 21280*(2*x + 1)**(3/2)/(-868000*x + 542500*
(2*x + 1)**2 + 325500) + 16*sqrt(2*x + 1)/25 + 6912*sqrt(2*x + 1)/(-124000*x + 77500*(2*x + 1)**2 + 46500) + 5
7456*sqrt(2*x + 1)/(-868000*x + 542500*(2*x + 1)**2 + 325500) + 16*RootSum(407144088666112*_t**4 + 3325152256*
_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_t/235 + sqrt(2*x + 1)))) - 1632*RootSum(40714
4088666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_t/235 + sqrt(2*
x + 1))))/25 - 624*RootSum(1722112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(
2*x + 1))))/125 - 336*RootSum(19950060344639488*_t**4 + 498437272576*_t**2 + 10878125, Lambda(_t, _t*log(-1104
9511452672*_t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1))))/5 - 288*RootSum(1722112*_t**4 + 1984*_t**2
+ 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/25 - 112*RootSum(1230080*_t**4 + 1984*_t**
2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/25 - 48*RootSum(407144088666112*_t**4 + 33251522
56*_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_t/235 + sqrt(2*x + 1))))/5 + 48*RootSum(17
22112*_t**4 + 1984*_t**2 + 5, Lambda(_t, _t*log(-27776*_t**3/5 + 108*_t/5 + sqrt(2*x + 1))))/5 + 48*RootSum(12
30080*_t**4 + 1984*_t**2 + 7, Lambda(_t, _t*log(9920*_t**3 + 8*_t + sqrt(2*x + 1))))/5 + 448*RootSum(199500603
44639488*_t**4 + 498437272576*_t**2 + 10878125, Lambda(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t
/2205125 + sqrt(2*x + 1))))/125 + 2016*RootSum(19950060344639488*_t**4 + 498437272576*_t**2 + 10878125, Lambda
(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t/2205125 + sqrt(2*x + 1))))/25 + 3904*RootSum(40714408
8666112*_t**4 + 3325152256*_t**2 + 11045, Lambda(_t, _t*log(33312534528*_t**3/235 + 166784*_t/235 + sqrt(2*x +
1))))/125

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (205) = 410\).
time = 1.74, size = 633, normalized size = 2.14 \begin {gather*} \frac {1}{14421006250} \, \sqrt {31} {\left (85470 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 407 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 814 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 170940 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 2589650 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 5179300 \, \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}{14421006250} \, \sqrt {31} {\left (85470 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - 407 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 814 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 170940 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 2589650 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} - 5179300 \, \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}{28842012500} \, \sqrt {31} {\left (407 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 85470 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 170940 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 814 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 2589650 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 5179300 \, \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}{28842012500} \, \sqrt {31} {\left (407 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 85470 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 170940 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 814 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} - 2589650 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} + 5179300 \, \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 {16}{25} \, \sqrt {2 \, x + 1} - \frac {4 \, {\left (178 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 189 \, \sqrt {2 \, x + 1}\right )}}{775 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")

[Out]

1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 407*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 170940*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 5179300
*(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)) + 1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 407*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(3
5) + 2450)^(3/2) + 170940*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/
4)*sqrt(-140*sqrt(35) + 2450) - 5179300*(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)) + 1/28842012500*sqrt(31)*(407*sqr
t(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(3
5) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/5)^(3/4)*(-140*sqrt(35) +
2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 5179300*(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) - 1/28842012500*sqrt
(31)*(407*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 24
50)*(2*sqrt(35) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/5)^(3/4)*(-14
0*sqrt(35) + 2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 5179300*(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) + 16/2
5*sqrt(2*x + 1) - 4/775*(178*(2*x + 1)^(3/2) - 189*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

________________________________________________________________________________________

Mupad [B]
time = 0.15, size = 216, normalized size = 0.73 \begin {gather*} \frac {16\,\sqrt {2\,x+1}}{25}+\frac {\frac {756\,\sqrt {2\,x+1}}{3875}-\frac {712\,{\left (2\,x+1\right )}^{3/2}}{3875}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2)^2,x)

[Out]

(16*(2*x + 1)^(1/2))/25 + ((756*(2*x + 1)^(1/2))/3875 - (712*(2*x + 1)^(3/2))/3875)/((2*x + 1)^2 - (8*x)/5 + 3
/5) - (155^(1/2)*atan((155^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2)*559232i)/(46923828125*((31
^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)) + (1118464*31^(1/2)*155^(1/2)*(5682718 - 31^(1/2)*1354
39i)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*((31^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)))*(56827
18 - 31^(1/2)*135439i)^(1/2)*2i)/120125 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x +
1)^(1/2)*559232i)/(46923828125*((31^(1/2)*591108224i)/9384765625 + 2004287488/9384765625)) - (1118464*31^(1/2
)*155^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*((31^(1/2)*591108224i)/93847656
25 + 2004287488/9384765625)))*(31^(1/2)*135439i + 5682718)^(1/2)*2i)/120125

________________________________________________________________________________________