\(\int \frac {(3-4 x^2)^{5/2} (4+6 x-2 x^2)}{(5-3 x) \sqrt {1+2 x}} \, dx\) [1]

Optimal result

Integrand size = 38, antiderivative size = 243 \[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=-\frac {64 \sqrt {1+2 x} (5232011+1719891 x) \sqrt {3-4 x^2}}{10945935}+\frac {2 \sqrt {1+2 x} (200681+143542 x) \left (3-4 x^2\right )^{3/2}}{243243}-\frac {2 (70-33 x) \sqrt {1+2 x} \left (3-4 x^2\right )^{5/2}}{1287}-\frac {610834712 \sqrt {1+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{2189187}-\frac {24360445832 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{32837805 \sqrt {1+\sqrt {3}}}+\frac {405004 \sqrt {2 \left (53+67 \sqrt {3}\right )} \operatorname {EllipticPi}\left (-\frac {6}{73} \left (9+10 \sqrt {3}\right ),\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{6561} \] Output:

-64/10945935*(1+2*x)^(1/2)*(5232011+1719891*x)*(-4*x^2+3)^(1/2)+2/243243*( 
1+2*x)^(1/2)*(200681+143542*x)*(-4*x^2+3)^(3/2)-2/1287*(70-33*x)*(1+2*x)^( 
1/2)*(-4*x^2+3)^(5/2)-610834712/2189187*(1+3^(1/2))^(1/2)*EllipticE(1/6*(3 
-2*x*3^(1/2))^(1/2)*6^(1/2),(3-3^(1/2))^(1/2))-24360445832/32837805*Ellipt 
icF(1/6*(3-2*x*3^(1/2))^(1/2)*6^(1/2),(3-3^(1/2))^(1/2))/(1+3^(1/2))^(1/2) 
+405004/6561*(106+134*3^(1/2))^(1/2)*EllipticPi(1/6*(3-2*x*3^(1/2))^(1/2)* 
6^(1/2),-54/73-60/73*3^(1/2),(3-3^(1/2))^(1/2))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.88 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {2 \left (9 \sqrt {1+2 x} \sqrt {3-4 x^2} \left (-145690567-33132357 x-21834180 x^2-32573520 x^3-9525600 x^4+4490640 x^5\right )+\frac {12 i \left (1145315085 \left (1+\sqrt {3}\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right )|-2+\sqrt {3}\right )-\left (4190370814+1145315085 \sqrt {3}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )[phone] \operatorname {EllipticPi}\left (-\frac {3}{13} \left (-1+\sqrt {3}\right ),i \text {arcsinh}\left (\frac {\sqrt {1+2 x}}{\sqrt {-1+\sqrt {3}}}\right ),-2+\sqrt {3}\right )\right )}{\sqrt {1+\sqrt {3}}}\right )}{98513415} \] Input:

Integrate[((3 - 4*x^2)^(5/2)*(4 + 6*x - 2*x^2))/((5 - 3*x)*Sqrt[1 + 2*x]), 
x]
 

Output:

(2*(9*Sqrt[1 + 2*x]*Sqrt[3 - 4*x^2]*(-145690567 - 33132357*x - 21834180*x^ 
2 - 32573520*x^3 - 9525600*x^4 + 4490640*x^5) + ((12*I)*(1145315085*(1 + S 
qrt[3])*EllipticE[I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3 
]] - (4190370814 + 1145315085*Sqrt[3])*EllipticF[I*ArcSinh[Sqrt[1 + 2*x]/S 
qrt[-1 + Sqrt[3]]], -2 + Sqrt[3]] + 2845659355*EllipticPi[(-3*(-1 + Sqrt[3 
]))/13, I*ArcSinh[Sqrt[1 + 2*x]/Sqrt[-1 + Sqrt[3]]], -2 + Sqrt[3]]))/Sqrt[ 
1 + Sqrt[3]]))/98513415
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (3-4 x^2\right )^{5/2} \left (-2 x^2+6 x+4\right )}{(5-3 x) \sqrt {2 x+1}} \, dx\)

\(\Big \downarrow \) 2349

\(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx+\int \frac {\left (\frac {2 x}{3}-\frac {8}{9}\right ) \left (3-4 x^2\right )^{5/2}}{\sqrt {2 x+1}}dx\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {5}{572} \int \frac {8 (82 x+321) \left (3-4 x^2\right )^{3/2}}{9 \sqrt {2 x+1}}dx+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {10 \int \frac {(82 x+321) \left (3-4 x^2\right )^{3/2}}{\sqrt {2 x+1}}dx}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {10 \left (\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}-\frac {1}{84} \int -\frac {32 (1711 x+2136) \sqrt {3-4 x^2}}{\sqrt {2 x+1}}dx\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {10 \left (\frac {8}{21} \int \frac {(1711 x+2136) \sqrt {3-4 x^2}}{\sqrt {2 x+1}}dx+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \sqrt {2 x+1} (5133 x+7258) \sqrt {3-4 x^2}-\frac {1}{60} \int -\frac {4 (59830 x+58947)}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \int \frac {59830 x+58947}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 600

\(\displaystyle -\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \left (29032 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx+29915 \int \frac {\sqrt {2 x+1}}{\sqrt {3-4 x^2}}dx\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \left (29032 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {29915 \sqrt {3+\sqrt {3}} \int \frac {\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}}}{\sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \left (29032 \int \frac {1}{\sqrt {2 x+1} \sqrt {3-4 x^2}}dx-\frac {29915 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \left (-\frac {29032 \sqrt [4]{3} \int \frac {1}{\sqrt {1-\frac {3-2 \sqrt {3} x}{3+\sqrt {3}}} \sqrt {\frac {1}{6} \left (2 \sqrt {3} x-3\right )+1}}d\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}}{\sqrt {3+\sqrt {3}}}-\frac {29915 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}+\frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \left (-\frac {29032 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {29915 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

\(\Big \downarrow \) 744

\(\displaystyle \frac {76}{9} \int \frac {\left (3-4 x^2\right )^{5/2}}{(5-3 x) \sqrt {2 x+1}}dx-\frac {10 \left (\frac {8}{21} \left (\frac {1}{15} \left (-\frac {29032 \sqrt [4]{3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right ),3-\sqrt {3}\right )}{\sqrt {3+\sqrt {3}}}-\frac {29915 \sqrt {3+\sqrt {3}} E\left (\arcsin \left (\frac {\sqrt {3-2 \sqrt {3} x}}{\sqrt {6}}\right )|3-\sqrt {3}\right )}{\sqrt [4]{3}}\right )+\frac {1}{15} \sqrt {2 x+1} \sqrt {3-4 x^2} (5133 x+7258)\right )+\frac {1}{63} \sqrt {2 x+1} (574 x+2561) \left (3-4 x^2\right )^{3/2}\right )}{1287}-\frac {2 (70-33 x) \sqrt {2 x+1} \left (3-4 x^2\right )^{5/2}}{1287}\)

Input:

Int[((3 - 4*x^2)^(5/2)*(4 + 6*x - 2*x^2))/((5 - 3*x)*Sqrt[1 + 2*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 744

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, 
 x] /; FreeQ[{a, c, d, e, f, g, m, n, p}, x]
 

rule 2349

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d 
*x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c 
+ d*x, x]   Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] &&  !IntegerQ[n 
] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(181)=362\).

Time = 1.28 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.95

method	result	size
risch	\(-\frac {2 \left (4490640 x^{5}-9525600 x^{4}-32573520 x^{3}-21834180 x^{2}-33132357 x -145690567\right ) \left (4 x^{2}-3\right ) \sqrt {1+2 x}\, \sqrt {\left (-4 x^{2}+3\right ) \left (1+2 x \right )}}{10945935 \sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \sqrt {-4 x^{2}+3}}-\frac {2 \left (-\frac {4788995216 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{14073345 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {1221669424 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{6567561 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}-\frac {29565292 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{59049 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right ) \sqrt {\left (-4 x^{2}+3\right ) \left (1+2 x \right )}}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\)	\(473\)
elliptic	\(\frac {\sqrt {-\left (4 x^{2}-3\right ) \left (1+2 x \right )}\, \left (\frac {32 x^{5} \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{39}-\frac {2240 x^{4} \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{1287}-\frac {206816 x^{3} \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{34749}-\frac {970408 x^{2} \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{243243}-\frac {7362746 x \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{1216215}-\frac {26489194 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}{995085}+\frac {9577990432 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{14073345 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {2443338848 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \left (\left (-\frac {\sqrt {3}}{2}+\frac {1}{2}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )-\frac {\operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{2}\right )}{6567561 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}}+\frac {59130584 \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {-\frac {x +\frac {1}{2}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\, \sqrt {-3 \left (x -\frac {\sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\left (x +\frac {\sqrt {3}}{2}\right ) \sqrt {3}}}{3}, -\frac {\sqrt {3}}{-\frac {\sqrt {3}}{2}-\frac {5}{3}}, \sqrt {\frac {\sqrt {3}}{\frac {\sqrt {3}}{2}-\frac {1}{2}}}\right )}{59049 \sqrt {-8 x^{3}-4 x^{2}+6 x +3}\, \left (-\frac {\sqrt {3}}{2}-\frac {5}{3}\right )}\right )}{\sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}}\)	\(518\)
default	\(-\frac {2 \sqrt {-4 x^{2}+3}\, \sqrt {1+2 x}\, \left (3933645309+8761864257 x -2351572560 x^{5}-27535517163 \sqrt {3}-1155398904 x^{4}-524180160 x^{7}[phone] \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )-96206467140 \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right ) \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}-2866190274 x^{2}-9623954916 x^{3}-2930709600 x^{6}[phone] x^{8}-61333049799 \sqrt {3}\, x [phone] \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )[phone] \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right ) \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}-147974286460 \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}\, \sqrt {-\left (1+2 x \right ) \left (\sqrt {3}-1\right )}\, \sqrt {3}\, \sqrt {\left (-2 x +\sqrt {3}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {3}\right ) \sqrt {3}}}{6}, \frac {6 \sqrt {3}}{10+3 \sqrt {3}}, \frac {\sqrt {2}\, \sqrt {\sqrt {3}\, \left (\sqrt {3}-1\right )}}{\sqrt {3}-1}\right )-2263282560 \sqrt {3}\, x^{8}[phone] \sqrt {3}\, x^{7}[phone] \sqrt {3}\, x^{6}[phone] x^{5} \sqrt {3}[phone] x^{4} \sqrt {3}[phone] x^{3} \sqrt {3}[phone] x^{2} \sqrt {3}\right )}{98513415 \left (\sqrt {3}-1\right ) \left (8 x^{3}+4 x^{2}-6 x -3\right ) \left (10+3 \sqrt {3}\right )}\)	\(612\)

Input:

int((-4*x^2+3)^(5/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x,method=_RETURN 
VERBOSE)
 

Output:

-2/10945935*(4490640*x^5-9525600*x^4-32573520*x^3-21834180*x^2-33132357*x- 
145690567)*(4*x^2-3)*(1+2*x)^(1/2)/(-(4*x^2-3)*(1+2*x))^(1/2)*((-4*x^2+3)* 
(1+2*x))^(1/2)/(-4*x^2+3)^(1/2)-2*(-4788995216/14073345*((x+1/2*3^(1/2))*3 
^(1/2))^(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2))*3^(1/ 
2))^(1/2)/(-8*x^3-4*x^2+6*x+3)^(1/2)*EllipticF(1/3*3^(1/2)*((x+1/2*3^(1/2) 
)*3^(1/2))^(1/2),(3^(1/2)/(1/2*3^(1/2)-1/2))^(1/2))-1221669424/6567561*((x 
+1/2*3^(1/2))*3^(1/2))^(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2 
*3^(1/2))*3^(1/2))^(1/2)/(-8*x^3-4*x^2+6*x+3)^(1/2)*((-1/2*3^(1/2)+1/2)*El 
lipticE(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^(1/2))^(1/2),(3^(1/2)/(1/2*3^(1/2)- 
1/2))^(1/2))-1/2*EllipticF(1/3*3^(1/2)*((x+1/2*3^(1/2))*3^(1/2))^(1/2),(3^ 
(1/2)/(1/2*3^(1/2)-1/2))^(1/2)))-29565292/59049*((x+1/2*3^(1/2))*3^(1/2))^ 
(1/2)*(-(x+1/2)/(1/2*3^(1/2)-1/2))^(1/2)*(-3*(x-1/2*3^(1/2))*3^(1/2))^(1/2 
)/(-8*x^3-4*x^2+6*x+3)^(1/2)/(-1/2*3^(1/2)-5/3)*EllipticPi(1/3*3^(1/2)*((x 
+1/2*3^(1/2))*3^(1/2))^(1/2),-3^(1/2)/(-1/2*3^(1/2)-5/3),(3^(1/2)/(1/2*3^( 
1/2)-1/2))^(1/2)))*((-4*x^2+3)*(1+2*x))^(1/2)/(-4*x^2+3)^(1/2)/(1+2*x)^(1/ 
2)
 

Fricas [F]

\[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} {\left (-4 \, x^{2} + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:

integrate((-4*x^2+3)^(5/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit 
hm="fricas")
 

Output:

integral(2*(16*x^6 - 48*x^5 - 56*x^4 + 72*x^3 + 57*x^2 - 27*x - 18)*sqrt(- 
4*x^2 + 3)*sqrt(2*x + 1)/(6*x^2 - 7*x - 5), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\text {Timed out} \] Input:

integrate((-4*x**2+3)**(5/2)*(-2*x**2+6*x+4)/(5-3*x)/(1+2*x)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} {\left (-4 \, x^{2} + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:

integrate((-4*x^2+3)^(5/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit 
hm="maxima")
 

Output:

2*integrate((x^2 - 3*x - 2)*(-4*x^2 + 3)^(5/2)/((3*x - 5)*sqrt(2*x + 1)), 
x)
 

Giac [F]

\[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int { \frac {2 \, {\left (x^{2} - 3 \, x - 2\right )} {\left (-4 \, x^{2} + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x - 5\right )} \sqrt {2 \, x + 1}} \,d x } \] Input:

integrate((-4*x^2+3)^(5/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x, algorit 
hm="giac")
 

Output:

integrate(2*(x^2 - 3*x - 2)*(-4*x^2 + 3)^(5/2)/((3*x - 5)*sqrt(2*x + 1)), 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\int -\frac {{\left (3-4\,x^2\right )}^{5/2}\,\left (-2\,x^2+6\,x+4\right )}{\sqrt {2\,x+1}\,\left (3\,x-5\right )} \,d x \] Input:

int(-((3 - 4*x^2)^(5/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)),x)
 

Output:

int(-((3 - 4*x^2)^(5/2)*(6*x - 2*x^2 + 4))/((2*x + 1)^(1/2)*(3*x - 5)), x)
 

Reduce [F]

\[ \int \frac {\left (3-4 x^2\right )^{5/2} \left (4+6 x-2 x^2\right )}{(5-3 x) \sqrt {1+2 x}} \, dx=\frac {32 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{5}}{39}-\frac {2240 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{4}}{1287}-\frac {206816 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{3}}{34749}-\frac {970408 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{2}}{243243}-\frac {7362746 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{1216215}+\frac {30130187 \sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{3648645}-\frac {305417356 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x^{3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{243243}+\frac {2696716081 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}\, x}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{3648645}+\frac {18228295 \left (\int \frac {\sqrt {2 x +1}\, \sqrt {-4 x^{2}+3}}{24 x^{4}-28 x^{3}-38 x^{2}+21 x +15}d x \right )}{243243} \] Input:

int((-4*x^2+3)^(5/2)*(-2*x^2+6*x+4)/(5-3*x)/(1+2*x)^(1/2),x)
 

Output:

(2993760*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**5 - 6350400*sqrt(2*x + 1)*sq 
rt( - 4*x**2 + 3)*x**4 - 21715680*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**3 - 
 14556120*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**2 - 22088238*sqrt(2*x + 1)* 
sqrt( - 4*x**2 + 3)*x + 30130187*sqrt(2*x + 1)*sqrt( - 4*x**2 + 3) - 45812 
60340*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)*x**3)/(24*x**4 - 28*x**3 - 38 
*x**2 + 21*x + 15),x) + 2696716081*int((sqrt(2*x + 1)*sqrt( - 4*x**2 + 3)* 
x)/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x) + 273424425*int((sqrt(2*x 
+ 1)*sqrt( - 4*x**2 + 3))/(24*x**4 - 28*x**3 - 38*x**2 + 21*x + 15),x))/36 
48645