3.29.4 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\) [2804]

Optimal. Leaf size=222 \[ \frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2332 \sqrt {1-2 x}}{21 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {324104 \sqrt {1-2 x}}{147 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {9795160 \sqrt {1-2 x} \sqrt {2+3 x}}{441 \sqrt {3+5 x}}+\frac {1959032}{147} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {58928}{147} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

1959032/441*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+58928/441*EllipticF(1/7*21^(1/2)*(1
-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/3*(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2)+104/9*(1-2*x)^(1/2)/(2+3*x
)^(5/2)/(3+5*x)^(1/2)+2332/21*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+324104/147*(1-2*x)^(1/2)/(2+3*x)^(1/2)
/(3+5*x)^(1/2)-9795160/441*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {58928}{147} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {1959032}{147} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}}-\frac {9795160 \sqrt {3 x+2} \sqrt {1-2 x}}{441 \sqrt {5 x+3}}+\frac {324104 \sqrt {1-2 x}}{147 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {2332 \sqrt {1-2 x}}{21 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {104 \sqrt {1-2 x}}{9 (3 x+2)^{5/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(9/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*(1 - 2*x)^(3/2))/(3*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x]) + (104*Sqrt[1 - 2*x])/(9*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])
+ (2332*Sqrt[1 - 2*x])/(21*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (324104*Sqrt[1 - 2*x])/(147*Sqrt[2 + 3*x]*Sqrt[3 +
5*x]) - (9795160*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(441*Sqrt[3 + 5*x]) + (1959032*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/147 + (58928*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147

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)^{9/2} (3+5 x)^{3/2}} \, dx &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {(196-161 x) \sqrt {1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {4}{315} \int \frac {-\frac {31955}{2}+21945 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2332 \sqrt {1-2 x}}{21 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {2417415}{2}+\frac {2754675 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{6615}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2332 \sqrt {1-2 x}}{21 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {324104 \sqrt {1-2 x}}{147 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {206265675}{4}+\frac {63807975 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{46305}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2332 \sqrt {1-2 x}}{21 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {324104 \sqrt {1-2 x}}{147 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {9795160 \sqrt {1-2 x} \sqrt {2+3 x}}{441 \sqrt {3+5 x}}+\frac {32 \int \frac {-\frac {1342947375}{2}-\frac {4242528675 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{509355}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2332 \sqrt {1-2 x}}{21 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {324104 \sqrt {1-2 x}}{147 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {9795160 \sqrt {1-2 x} \sqrt {2+3 x}}{441 \sqrt {3+5 x}}-\frac {324104}{147} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {1959032}{147} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {104 \sqrt {1-2 x}}{9 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2332 \sqrt {1-2 x}}{21 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {324104 \sqrt {1-2 x}}{147 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {9795160 \sqrt {1-2 x} \sqrt {2+3 x}}{441 \sqrt {3+5 x}}+\frac {1959032}{147} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {58928}{147} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

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Mathematica [A]
time = 8.75, size = 110, normalized size = 0.50 \begin {gather*} \frac {2}{441} \left (-\frac {3 \sqrt {1-2 x} \left (24789615+150788294 x+343801494 x^2+348250356 x^3+132234660 x^4\right )}{(2+3 x)^{7/2} \sqrt {3+5 x}}-4 \sqrt {2} \left (244879 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-123340 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(9/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-3*Sqrt[1 - 2*x]*(24789615 + 150788294*x + 343801494*x^2 + 348250356*x^3 + 132234660*x^4))/((2 + 3*x)^(7/
2)*Sqrt[3 + 5*x]) - 4*Sqrt[2]*(244879*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 123340*EllipticF[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/441

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(162)=324\).
time = 0.11, size = 401, normalized size = 1.81

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {1210 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {170 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{3}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{243 \left (\frac {2}{3}+x \right )^{4}}-\frac {1425422 \left (-30 x^{2}-3 x +9\right )}{441 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {12946 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189 \left (\frac {2}{3}+x \right )^{2}}-\frac {6201200 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3087 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9795160 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3087 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(301\)
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (13126212 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-26446932 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-52893864 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-35262576 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3889248 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-7836128 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-793407960 x^{5}-1692798156 x^{4}-1018057896 x^{3}[phone] x^{2}[phone] x [phone]\right )}{441 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/441*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13126212*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)-26446932*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)[phone]*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)-52893864*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)[phone]*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)-35262576*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)+3889248*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))-7836128*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))-793407960*x^5-1692798156*x^4-1018057896*x^3+126674718*x^2+303627192*x+74368845)/(2+3*x)^(7/2)/(10*x^2+x-3
)

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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)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)), x)

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Fricas [A]
time = 0.45, size = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (132234660 \, x^{4} + 348250356 \, x^{3} + 343801494 \, x^{2} + 150788294 \, x + 24789615\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{147 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-2/147*(132234660*x^4 + 348250356*x^3 + 343801494*x^2 + 150788294*x + 24789615)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq
rt(-2*x + 1)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)/(2+3*x)**(9/2)/(3+5*x)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7317 deep

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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)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(9/2)), 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 )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)/((3*x + 2)^(9/2)*(5*x + 3)^(3/2)),x)

[Out]

int((1 - 2*x)^(5/2)/((3*x + 2)^(9/2)*(5*x + 3)^(3/2)), x)

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