3.24.26 \(\int \frac {(1+2 x)^{3/2}}{(2+3 x+5 x^2)^3} \, dx\) [2326]

Optimal. Leaf size=300 \[ -\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \sqrt {35}\right )} \tan ^{-1}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922} \]

[Out]

-1/62*(5-4*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)^2+1/1922*(67+120*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)-3/834148*ln(5+10*x+3
5^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-6545588+1173970*35^(1/2))^(1/2)+3/834148*ln(5+10*x+35^(1/2)+(1
+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-6545588+1173970*35^(1/2))^(1/2)-3/417074*arctan((-10*(1+2*x)^(1/2)+(20+1
0*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(6545588+1173970*35^(1/2))^(1/2)+3/417074*arctan((10*(1+2*x)^(1/2)
+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(6545588+1173970*35^(1/2))^(1/2)

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Rubi [A]
time = 0.30, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {752, 836, 840, 1183, 648, 632, 210, 642} \begin {gather*} -\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \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 )+\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \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 )-\frac {\sqrt {2 x+1} (5-4 x)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {\sqrt {2 x+1} (120 x+67)}{1922 \left (5 x^2+3 x+2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (2705 \sqrt {35}-15082\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1922} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/62*((5 - 4*x)*Sqrt[1 + 2*x])/(2 + 3*x + 5*x^2)^2 + (Sqrt[1 + 2*x]*(67 + 120*x))/(1922*(2 + 3*x + 5*x^2)) -
(3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35
])]])/961 + (3*Sqrt[(15082 + 2705*Sqrt[35])/434]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(
-2 + Sqrt[35])]])/961 - (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 +
2*x] + 5*(1 + 2*x)])/1922 + (3*Sqrt[(-15082 + 2705*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[
1 + 2*x] + 5*(1 + 2*x)])/1922

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 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 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)^{3/2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {1}{62} \int \frac {17+20 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\int \frac {1239+840 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{13454}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1638+840 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{6727}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (1638-168 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {1638 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (1638-168 \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 )}{13454 \sqrt {14 \left (2+\sqrt {35}\right )}}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}+\frac {\left (3 \left (140+39 \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 )}{67270}+\frac {\left (3 \left (140+39 \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 )}{67270}-\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \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 )}{1922}+\frac {\left (3 \sqrt {\frac {1}{434} \left (-15082+2705 \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 )}{1922}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}-\frac {\left (3 \left (140+39 \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 )}{33635}-\frac {\left (3 \left (140+39 \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 )}{33635}\\ &=-\frac {(5-4 x) \sqrt {1+2 x}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {\sqrt {1+2 x} (67+120 x)}{1922 \left (2+3 x+5 x^2\right )}-\frac {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \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 {3}{961} \sqrt {\frac {1}{434} \left (15082+2705 \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 {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}+\frac {3 \sqrt {\frac {1}{434} \left (-15082+2705 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1922}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.37, size = 143, normalized size = 0.48 \begin {gather*} \frac {\frac {217 \sqrt {1+2 x} \left (-21+565 x+695 x^2+600 x^3\right )}{2 \left (2+3 x+5 x^2\right )^2}+3 \sqrt {217 \left (15082+961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+3 \sqrt {217 \left (15082-961 i \sqrt {31}\right )} \tan ^{-1}\left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{208537} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((217*Sqrt[1 + 2*x]*(-21 + 565*x + 695*x^2 + 600*x^3))/(2*(2 + 3*x + 5*x^2)^2) + 3*Sqrt[217*(15082 + (961*I)*S
qrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + 3*Sqrt[217*(15082 - (961*I)*Sqrt[31])]*ArcTan[Sqrt
[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/208537

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(210)=420\).
time = 1.83, size = 435, normalized size = 1.45 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1600*(3/7688*(2*x+1)^(7/2)-41/153760*(2*x+1)^(5/2)+4/4805*(2*x+1)^(3/2)-819/768800*(2*x+1)^(1/2))/(5*(2*x+1)^2
-8*x+3)^2-3/4170740*(1645*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-1090*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)-3/208537*(-2418*5^(1/2)*7^(1/2)+1/10*(
1645*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-1090*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))+3/4170740*(1645*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-1090*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))+3/208537*(2418*5^(1/2)*7^
(1/2)-1/10*(1645*(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-1090*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))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (213) = 426\).
time = 2.85, size = 625, normalized size = 2.08 \begin {gather*} -\frac {357492 \cdot 256095875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{971794421886819908125} \cdot 256095875^{\frac {3}{4}} \sqrt {3787} \sqrt {217} \sqrt {256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 5640925850 \, x + 564092585 \, \sqrt {35} + 2820462925} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} - \frac {1}{53405465484875} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (39 \, \sqrt {35} - 140\right )} + \frac {1}{31} \, \sqrt {35} \sqrt {31} + \frac {2}{31} \, \sqrt {31}\right ) + 357492 \cdot 256095875^{\frac {1}{4}} \sqrt {217} \sqrt {35} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \arctan \left (\frac {1}{14576916328302298621875} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {-852075 \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 4806491893638750 \, x + 480649189363875 \, \sqrt {35} + 2403245946819375} {\left (39 \, \sqrt {35} \sqrt {31} - 140 \, \sqrt {31}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} - \frac {1}{53405465484875} \cdot 256095875^{\frac {3}{4}} \sqrt {217} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} {\left (39 \, \sqrt {35} - 140\right )} - \frac {1}{31} \, \sqrt {35} \sqrt {31} - \frac {2}{31} \, \sqrt {31}\right ) - 3 \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (\frac {852075}{31} \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 155048125601250 \, x + 15504812560125 \, \sqrt {35} + 77524062800625\right ) + 3 \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (15082 \, \sqrt {35} \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} - 94675 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}\right )} \sqrt {81593620 \, \sqrt {35} + 512191750} \log \left (-\frac {852075}{31} \cdot 256095875^{\frac {1}{4}} \sqrt {217} {\left (4 \, \sqrt {35} \sqrt {31} - 39 \, \sqrt {31}\right )} \sqrt {2 \, x + 1} \sqrt {81593620 \, \sqrt {35} + 512191750} + 155048125601250 \, x + 15504812560125 \, \sqrt {35} + 77524062800625\right ) - 1224080909450 \, {\left (600 \, x^{3} + 695 \, x^{2} + 565 \, x - 21\right )} \sqrt {2 \, x + 1}}{2352683507962900 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2352683507962900*(357492*256095875^(1/4)*sqrt(217)*sqrt(35)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*sqrt(8159
3620*sqrt(35) + 512191750)*arctan(1/971794421886819908125*256095875^(3/4)*sqrt(3787)*sqrt(217)*sqrt(256095875^
(1/4)*sqrt(217)*(4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750) + 564092
5850*x + 564092585*sqrt(35) + 2820462925)*(39*sqrt(35)*sqrt(31) - 140*sqrt(31))*sqrt(81593620*sqrt(35) + 51219
1750) - 1/53405465484875*256095875^(3/4)*sqrt(217)*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750)*(39*sqrt(
35) - 140) + 1/31*sqrt(35)*sqrt(31) + 2/31*sqrt(31)) + 357492*256095875^(1/4)*sqrt(217)*sqrt(35)*(25*x^4 + 30*
x^3 + 29*x^2 + 12*x + 4)*sqrt(81593620*sqrt(35) + 512191750)*arctan(1/14576916328302298621875*256095875^(3/4)*
sqrt(217)*sqrt(-852075*256095875^(1/4)*sqrt(217)*(4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(815936
20*sqrt(35) + 512191750) + 4806491893638750*x + 480649189363875*sqrt(35) + 2403245946819375)*(39*sqrt(35)*sqrt
(31) - 140*sqrt(31))*sqrt(81593620*sqrt(35) + 512191750) - 1/53405465484875*256095875^(3/4)*sqrt(217)*sqrt(2*x
+ 1)*sqrt(81593620*sqrt(35) + 512191750)*(39*sqrt(35) - 140) - 1/31*sqrt(35)*sqrt(31) - 2/31*sqrt(31)) - 3*25
6095875^(1/4)*sqrt(217)*(15082*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 94675*sqrt(31)*(25*x^
4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(81593620*sqrt(35) + 512191750)*log(852075/31*256095875^(1/4)*sqrt(217)*(
4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*sqrt(35) + 512191750) + 155048125601250*x + 155
04812560125*sqrt(35) + 77524062800625) + 3*256095875^(1/4)*sqrt(217)*(15082*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3
+ 29*x^2 + 12*x + 4) - 94675*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*sqrt(81593620*sqrt(35) + 5121917
50)*log(-852075/31*256095875^(1/4)*sqrt(217)*(4*sqrt(35)*sqrt(31) - 39*sqrt(31))*sqrt(2*x + 1)*sqrt(81593620*s
qrt(35) + 512191750) + 155048125601250*x + 15504812560125*sqrt(35) + 77524062800625) - 1224080909450*(600*x^3
+ 695*x^2 + 565*x - 21)*sqrt(2*x + 1))/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (252) = 504\).
time = 71.64, size = 527, normalized size = 1.76 \begin {gather*} \frac {1145600 \left (2 x + 1\right )^{\frac {7}{2}}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {8870400 \left (2 x + 1\right )^{\frac {7}{2}}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} - \frac {1295360 \left (2 x + 1\right )^{\frac {5}{2}}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {4701760 \left (2 x + 1\right )^{\frac {5}{2}}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} + \frac {3017984 \left (2 x + 1\right )^{\frac {3}{2}}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {6868736 \left (2 x + 1\right )^{\frac {3}{2}}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} + \frac {640 \left (2 x + 1\right )^{\frac {3}{2}}}{- 34720 x + 21700 \left (2 x + 1\right )^{2} + 13020} - \frac {974848 \sqrt {2 x + 1}}{- 120547840 x + 26908000 \left (2 x + 1\right )^{4} - 43052800 \left (2 x + 1\right )^{3} + 92563520 \left (2 x + 1\right )^{2} - 7534240} - \frac {27016640 \sqrt {2 x + 1}}{- 843834880 x + 188356000 \left (2 x + 1\right )^{4} - 301369600 \left (2 x + 1\right )^{3} + 647944640 \left (2 x + 1\right )^{2} - 52739680} + \frac {1728 \sqrt {2 x + 1}}{- 34720 x + 21700 \left (2 x + 1\right )^{2} + 13020} + 64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )} - \frac {448 \operatorname {RootSum} {\left (3697830642882758349886062592 t^{4} + 2111968303753265086464 t^{2} + 705698730253125, \left ( t \mapsto t \log {\left (- \frac {3459438283411209322496 t^{3}}{1377792122625} + \frac {251494140770688 t}{357205365125} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} - \frac {64 \operatorname {RootSum} {\left (75465931487403231630327808 t^{4} + 9053854476152406016 t^{2} + 333142578125, \left ( t \mapsto t \log {\left (\frac {21632117045402271744 t^{3}}{158378125} + \frac {10865340674816 t}{1108646875} + \sqrt {2 x + 1} \right )} \right )\right )}}{5} + \frac {64 \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1145600*(2*x + 1)**(7/2)/(-120547840*x + 26908000*(2*x + 1)**4 - 43052800*(2*x + 1)**3 + 92563520*(2*x + 1)**2
- 7534240) - 8870400*(2*x + 1)**(7/2)/(-843834880*x + 188356000*(2*x + 1)**4 - 301369600*(2*x + 1)**3 + 64794
4640*(2*x + 1)**2 - 52739680) - 1295360*(2*x + 1)**(5/2)/(-120547840*x + 26908000*(2*x + 1)**4 - 43052800*(2*x
+ 1)**3 + 92563520*(2*x + 1)**2 - 7534240) - 4701760*(2*x + 1)**(5/2)/(-843834880*x + 188356000*(2*x + 1)**4
- 301369600*(2*x + 1)**3 + 647944640*(2*x + 1)**2 - 52739680) + 3017984*(2*x + 1)**(3/2)/(-120547840*x + 26908
000*(2*x + 1)**4 - 43052800*(2*x + 1)**3 + 92563520*(2*x + 1)**2 - 7534240) - 6868736*(2*x + 1)**(3/2)/(-84383
4880*x + 188356000*(2*x + 1)**4 - 301369600*(2*x + 1)**3 + 647944640*(2*x + 1)**2 - 52739680) + 640*(2*x + 1)*
*(3/2)/(-34720*x + 21700*(2*x + 1)**2 + 13020) - 974848*sqrt(2*x + 1)/(-120547840*x + 26908000*(2*x + 1)**4 -
43052800*(2*x + 1)**3 + 92563520*(2*x + 1)**2 - 7534240) - 27016640*sqrt(2*x + 1)/(-843834880*x + 188356000*(2
*x + 1)**4 - 301369600*(2*x + 1)**3 + 647944640*(2*x + 1)**2 - 52739680) + 1728*sqrt(2*x + 1)/(-34720*x + 2170
0*(2*x + 1)**2 + 13020) + 64*RootSum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 + 3331425781
25, Lambda(_t, _t*log(21632117045402271744*_t**3/158378125 + 10865340674816*_t/1108646875 + sqrt(2*x + 1)))) -
448*RootSum(3697830642882758349886062592*_t**4 + 2111968303753265086464*_t**2 + 705698730253125, Lambda(_t, _
t*log(-3459438283411209322496*_t**3/1377792122625 + 251494140770688*_t/357205365125 + sqrt(2*x + 1))))/5 - 64*
RootSum(75465931487403231630327808*_t**4 + 9053854476152406016*_t**2 + 333142578125, Lambda(_t, _t*log(2163211
7045402271744*_t**3/158378125 + 10865340674816*_t/1108646875 + sqrt(2*x + 1))))/5 + 64*RootSum(199500603446394
88*_t**4 + 498437272576*_t**2 + 10878125, Lambda(_t, _t*log(-11049511452672*_t**3/2205125 + 307918256*_t/22051
25 + sqrt(2*x + 1))))/5

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (213) = 426\).
time = 2.37, size = 640, normalized size = 2.13 \begin {gather*} \frac {3}{3576409550} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 9555 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 19110 \, \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 {3}{3576409550} \, \sqrt {31} {\left (210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} - \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} + 9555 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {-140 \, \sqrt {35} + 2450} + 19110 \, \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 {3}{7152819100} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 9555 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 19110 \, \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 {3}{7152819100} \, \sqrt {31} {\left (\sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 210 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {3}{4}} \sqrt {140 \, \sqrt {35} + 2450} {\left (2 \, \sqrt {35} - 35\right )} - 420 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (2 \, \sqrt {35} + 35\right )} \sqrt {-140 \, \sqrt {35} + 2450} + 2 \, \left (\frac {7}{5}\right )^{\frac {3}{4}} {\left (-140 \, \sqrt {35} + 2450\right )}^{\frac {3}{2}} + 9555 \, \sqrt {31} \left (\frac {7}{5}\right )^{\frac {1}{4}} \sqrt {140 \, \sqrt {35} + 2450} - 19110 \, \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 {2 \, {\left (300 \, {\left (2 \, x + 1\right )}^{\frac {7}{2}} - 205 \, {\left (2 \, x + 1\right )}^{\frac {5}{2}} + 640 \, {\left (2 \, x + 1\right )}^{\frac {3}{2}} - 819 \, \sqrt {2 \, x + 1}\right )}}{961 \, {\left (5 \, {\left (2 \, x + 1\right )}^{2} - 8 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/3576409550*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - sqrt(31)*(7/5)^
(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)*sqrt(140*sqrt
(35) + 2450)*(2*sqrt(35) - 35) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 19110*(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)) + 3/3576409550*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2
450) - sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/
5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) +
19110*(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)) + 3/7152819100*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)
^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) +
35)*sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(1
40*sqrt(35) + 2450) - 19110*(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) - 3/7152819100*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/
2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*
sqrt(-140*sqrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 9555*sqrt(31)*(7/5)^(1/4)*sqrt(140*s
qrt(35) + 2450) - 19110*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450))*log(-2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqr
t(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 2/961*(300*(2*x + 1)^(7/2) - 205*(2*x + 1)^(5/2) + 640*(2*x + 1)^(3/2) -
819*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)^2

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Mupad [B]
time = 1.04, size = 245, normalized size = 0.82 \begin {gather*} \frac {\frac {1638\,\sqrt {2\,x+1}}{24025}-\frac {256\,{\left (2\,x+1\right )}^{3/2}}{4805}+\frac {82\,{\left (2\,x+1\right )}^{5/2}}{4805}-\frac {24\,{\left (2\,x+1\right )}^{7/2}}{961}}{\frac {112\,x}{25}-\frac {86\,{\left (2\,x+1\right )}^2}{25}+\frac {8\,{\left (2\,x+1\right )}^3}{5}-{\left (2\,x+1\right )}^4+\frac {7}{25}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}+\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082-\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}\,432{}\mathrm {i}}{5656566125\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}-\frac {864\,\sqrt {31}\,\sqrt {217}\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,\sqrt {2\,x+1}}{175353549875\,\left (-\frac {94176}{808080875}+\frac {\sqrt {31}\,16848{}\mathrm {i}}{808080875}\right )}\right )\,\sqrt {-15082+\sqrt {31}\,961{}\mathrm {i}}\,3{}\mathrm {i}}{208537} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1638*(2*x + 1)^(1/2))/24025 - (256*(2*x + 1)^(3/2))/4805 + (82*(2*x + 1)^(5/2))/4805 - (24*(2*x + 1)^(7/2))/
961)/((112*x)/25 - (86*(2*x + 1)^2)/25 + (8*(2*x + 1)^3)/5 - (2*x + 1)^4 + 7/25) + (217^(1/2)*atan((217^(1/2)*
(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(5656566125*((31^(1/2)*16848i)/808080875 + 94176/8080808
75)) + (864*31^(1/2)*217^(1/2)*(- 31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(175353549875*((31^(1/2)*16848
i)/808080875 + 94176/808080875)))*(- 31^(1/2)*961i - 15082)^(1/2)*3i)/208537 - (217^(1/2)*atan((217^(1/2)*(31^
(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2)*432i)/(5656566125*((31^(1/2)*16848i)/808080875 - 94176/808080875)) -
(864*31^(1/2)*217^(1/2)*(31^(1/2)*961i - 15082)^(1/2)*(2*x + 1)^(1/2))/(175353549875*((31^(1/2)*16848i)/80808
0875 - 94176/808080875)))*(31^(1/2)*961i - 15082)^(1/2)*3i)/208537

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