3.28.27 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\) [2727]

Optimal. Leaf size=249 \[ -\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac {3316711588 \sqrt {1-2 x} \sqrt {3+5 x}}{673876665 \sqrt {2+3 x}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {3316711588 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}}-\frac {103970992 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}} \]

[Out]

-2/33*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2)-3316711588/2021629995*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/
33*1155^(1/2))*33^(1/2)-103970992/2021629995*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-13
292/43659*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)+362/891*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-1366496/
4584195*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/96268095*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+
3316711588/673876665*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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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 = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {103970992 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}}-\frac {3316711588 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}}+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {13292 \sqrt {1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{7/2}}+\frac {3316711588 \sqrt {1-2 x} \sqrt {5 x+3}}{673876665 \sqrt {3 x+2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {5 x+3}}{96268095 (3 x+2)^{3/2}}-\frac {1366496 \sqrt {1-2 x} \sqrt {5 x+3}}{4584195 (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1366496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4584195*(2 + 3*x)^(5/2)) + (45748292*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(962
68095*(2 + 3*x)^(3/2)) + (3316711588*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(673876665*Sqrt[2 + 3*x]) - (13292*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(43659*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (3
62*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) - (3316711588*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(61261515*Sqrt[33]) - (103970992*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*Sqrt[
33])

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {(3+5 x)^{3/2} \left (-\frac {1993}{2}+\frac {1995 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {8 \int \frac {\sqrt {3+5 x} \left (-\frac {107577}{2}+\frac {189705 x}{4}\right )}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{130977}\\ &=-\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {9802263}{8}+\frac {2452215 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{13752585}\\ &=-\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {3+5 x}}{96268095 (2+3 x)^{3/2}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {600436437}{16}+\frac {171556095 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{288804285}\\ &=-\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac {3316711588 \sqrt {1-2 x} \sqrt {3+5 x}}{673876665 \sqrt {2+3 x}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {64 \int \frac {-\frac {7891481415}{16}-\frac {12437668455 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2021629995}\\ &=-\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac {3316711588 \sqrt {1-2 x} \sqrt {3+5 x}}{673876665 \sqrt {2+3 x}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {51985496 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{61261515}+\frac {3316711588 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{673876665}\\ &=-\frac {1366496 \sqrt {1-2 x} \sqrt {3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac {45748292 \sqrt {1-2 x} \sqrt {3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac {3316711588 \sqrt {1-2 x} \sqrt {3+5 x}}{673876665 \sqrt {2+3 x}}-\frac {13292 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {3316711588 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}}-\frac {103970992 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{61261515 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.28, size = 112, normalized size = 0.45 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (55875107717+415681177941 x+1234133449713 x^2+1829570010885 x^3+1356237833922 x^4+402980457942 x^5\right )}{(2+3 x)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-25619043520 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{16173039960 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(55875107717 + 415681177941*x + 1234133449713*x^2 + 1829570010885*x^3 + 13562
37833922*x^4 + 402980457942*x^5))/(2 + 3*x)^(11/2) + 53067385408*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2] - 25619043520*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(16173039960*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(185)=370\).
time = 0.10, size = 587, normalized size = 2.36

method	result	size
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {1658 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1948617 \left (\frac {2}{3}+x \right )^{5}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{649539 \left (\frac {2}{3}+x \right )^{6}}+\frac {5986462 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{371319795 \left (\frac {2}{3}+x \right )^{3}}-\frac {309494 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31827411 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {6633423176}{134775333} x^{2}-\frac {3316711588}{673876665} x +\frac {3316711588}{224625555}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {45748292 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{866412855 \left (\frac {2}{3}+x \right )^{2}}+\frac {2104395044 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2830281993 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {3316711588 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2830281993 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(332\)
default	\(\frac {2 \left (402980457942 \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}-208435846212 \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}-694786154040 \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}-926381538720 \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}-617587692480 \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}-205862564160 \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 )-27448341888 \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}-293294410596 x^{3}-8183904282084 x^{2}-3573505278318 x -502875969453\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2021629995 \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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x,method=_RETURNVERBOSE)

[Out]

2/2021629995*(402980457942*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)-208435846212*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)-694786154040*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)-926381538720*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)-617587692480*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+398005390560*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)-205862564160*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+53067385408*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))-27448341888*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+30306573018747*x^4-2932944
10596*x^3-8183904282084*x^2-3573505278318*x-502875969453)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(11
/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)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.31, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (402980457942 \, x^{5} + 1356237833922 \, x^{4} + 1829570010885 \, x^{3} + 1234133449713 \, x^{2} + 415681177941 \, x + 55875107717\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{673876665 \, {\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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

2/673876665*(402980457942*x^5 + 1356237833922*x^4 + 1829570010885*x^3 + 1234133449713*x^2 + 415681177941*x + 5
5875107717)*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)

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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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**(13/2),x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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