3.28.95 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx\) [2795]

Optimal. Leaf size=249 \[ \frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {697995152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}} \]

[Out]

-23204503328/6417873*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-697995152/6417873*Elliptic
F(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/33*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(11/2)+4508/8
91*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+171004/6237*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+7173272/436
59*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/305613*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+232045
03328/2139291*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \begin {gather*} -\frac {697995152 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {23204503328 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {23204503328 \sqrt {5 x+3} \sqrt {1-2 x}}{2139291 \sqrt {3 x+2}}+\frac {333930952 \sqrt {5 x+3} \sqrt {1-2 x}}{305613 (3 x+2)^{3/2}}+\frac {7173272 \sqrt {5 x+3} \sqrt {1-2 x}}{43659 (3 x+2)^{5/2}}+\frac {171004 \sqrt {5 x+3} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {4508 \sqrt {5 x+3} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(13/2)*Sqrt[3 + 5*x]),x]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (4508*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(891*(2 + 3*x)^(
9/2)) + (171004*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6237*(2 + 3*x)^(7/2)) + (7173272*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4
3659*(2 + 3*x)^(5/2)) + (333930952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(3/2)) + (23204503328*Sqrt[1
- 2*x]*Sqrt[3 + 5*x])/(2139291*Sqrt[2 + 3*x]) - (23204503328*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(194481*Sqrt[33]) - (697995152*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(194481*Sqrt[33])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 114

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

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {(227-223 x) \sqrt {1-2 x}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {-\frac {49835}{2}+37438 x}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {5473405}{2}+\frac {7481425 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{43659}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}-\frac {16 \int \frac {-\frac {833286615}{4}+\frac {470745975 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1528065}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}-\frac {32 \int \frac {-9037592970+\frac {21914218725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{32089365}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {64 \int \frac {-\frac {964063843575}{8}-\frac {380698882725 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{224625555}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}+\frac {348997576 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{194481}+\frac {23204503328 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2139291}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{33 (2+3 x)^{11/2}}+\frac {4508 \sqrt {1-2 x} \sqrt {3+5 x}}{891 (2+3 x)^{9/2}}+\frac {171004 \sqrt {1-2 x} \sqrt {3+5 x}}{6237 (2+3 x)^{7/2}}+\frac {7173272 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{5/2}}+\frac {333930952 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{3/2}}+\frac {23204503328 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}}-\frac {23204503328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}-\frac {697995152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{194481 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 8.61, size = 115, normalized size = 0.46 \begin {gather*} \frac {\frac {12 \sqrt {1-2 x} \sqrt {3+5 x} \left (391506734113+2903435279352 x+8615827181322 x^2+12787628716260 x^3+9492493272732 x^4+2819347154352 x^5\right )}{(2+3 x)^{11/2}}+16 \sqrt {2} \left (2900562916 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-1460947915 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{12835746} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(13/2)*Sqrt[3 + 5*x]),x]

[Out]

((12*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(391506734113 + 2903435279352*x + 8615827181322*x^2 + 12787628716260*x^3 + 94
92493272732*x^4 + 2819347154352*x^5))/(2 + 3*x)^(11/2) + 16*Sqrt[2]*(2900562916*EllipticE[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2] - 1460947915*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/12835746

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(185)=370\).
time = 0.11, size = 587, normalized size = 2.36

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {4256 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{5}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{72171 \left (\frac {2}{3}+x \right )^{6}}+\frac {7173272 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1178793 \left (\frac {2}{3}+x \right )^{3}}+\frac {171004 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{505197 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {232045033280}{2139291} x^{2}-\frac {23204503328}{2139291} x +\frac {23204503328}{713097}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {333930952 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2750517 \left (\frac {2}{3}+x \right )^{2}}+\frac {73452483320 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{44925111 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {116022516640 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{44925111 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(321\)
default	\(\frac {2 \left (2819347154352 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1399305780972 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4664352603240 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6219136804320 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4146091202880 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{7}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1382030400960 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-184270720128 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}-2138118521814 x^{3}-57086936770452 x^{2}-24956397311829 x -3523560607017\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{6417873 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\)	\(587\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^(13/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/6417873*(2819347154352*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*
(1-2*x)^(1/2)-1399305780972*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(-3-5*x)^(1/
2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^
(1/2)*(1-2*x)^(1/2)-4664352603240*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*
x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-
3-5*x)^(1/2)*(1-2*x)^(1/2)-6219136804320*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)
*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1
/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-4146091202880*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)
^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^7+2784540399360*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2
*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-1382030400960*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/
2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^6+371272053248*2^(1/2)*(2+3*x)^(1/2
)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-184270720128*2^(1/2)*(2+3*x)^(1/2)*
(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))[phone]*x^5+211405262133852*x
^4-2138118521814*x^3-57086936770452*x^2-24956397311829*x-3523560607017)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-
3)/(2+3*x)^(11/2)

________________________________________________________________________________________

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)^(5/2)/(2+3*x)^(13/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.21, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (2819347154352 \, x^{5} + 9492493272732 \, x^{4} + 12787628716260 \, x^{3} + 8615827181322 \, x^{2} + 2903435279352 \, x + 391506734113\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2139291 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(13/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

2/2139291*(2819347154352*x^5 + 9492493272732*x^4 + 12787628716260*x^3 + 8615827181322*x^2 + 2903435279352*x +
391506734113)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)/(2+3*x)**(13/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(13/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(13/2)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)/((3*x + 2)^(13/2)*(5*x + 3)^(1/2)),x)

[Out]

int((1 - 2*x)^(5/2)/((3*x + 2)^(13/2)*(5*x + 3)^(1/2)), x)

________________________________________________________________________________________