3.29.5 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx\) [2805]

Optimal. Leaf size=253 \[ \frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {20549264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261} \]

[Out]

683150096/27783*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)[phone]/27783*EllipticF(1/7*21
^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/27*(1-2*x)^(3/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2)+652/81*(1-2*x)^
(1/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2)+11660/189*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+813208/1323*(1-2*x)^(1/2
)/(2+3*x)^(3/2)/(3+5*x)^(1/2)[phone]/9261*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-3415750480/27783*(1-2*x
)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 253, 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 {20549264 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}-\frac {3415750480 \sqrt {3 x+2} \sqrt {1-2 x}}{27783 \sqrt {5 x+3}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {813208 \sqrt {1-2 x}}{1323 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {11660 \sqrt {1-2 x}}{189 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {652 \sqrt {1-2 x}}{81 (3 x+2)^{7/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*(3 + 5*x)^(3/2)),x]

[Out]

(14*(1 - 2*x)^(3/2))/(27*(2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) + (652*Sqrt[1 - 2*x])/(81*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x
]) + (11660*Sqrt[1 - 2*x])/(189*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (813208*Sqrt[1 - 2*x])/(1323*(2 + 3*x)^(3/2)*
Sqrt[3 + 5*x]) + (113020952*Sqrt[1 - 2*x])/(9261*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (3415750480*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(27783*Sqrt[3 + 5*x]) + (683150096*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/92
61 + (20549264*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261

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)^{11/2} (3+5 x)^{3/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {2}{27} \int \frac {(229-227 x) \sqrt {1-2 x}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}-\frac {4}{567} \int \frac {-\frac {50897}{2}+38346 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {5572105}{2}+\frac {7651875 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{19845}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {842998695}{4}+\frac {480300975 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{416745}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {32 \int \frac {-8991074400+\frac {22250999925 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2917215}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}+\frac {64 \int \frac {-\frac {936620355825}{8}-\frac {739723463325 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{32089365}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}-\frac {113020952 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9261}-\frac {683150096 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{9261}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {20549264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}\\ \end {align*}

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Mathematica [A]
time = 8.99, size = 115, normalized size = 0.45 \begin {gather*} \frac {2 \left (-\frac {3 \sqrt {1-2 x} \left (17289178827+131099014240 x+397527527442 x^2+602551975428 x^3+456548966244 x^4+138337894440 x^5\right )}{(2+3 x)^{9/2} \sqrt {3+5 x}}-4 \sqrt {2} \left (85393762 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-43010905 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{27783} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-3*Sqrt[1 - 2*x]*(17289178827 + 131099014240*x + 397527527442*x^2 + 602551975428*x^3 + 456548966244*x^4 +
138337894440*x^5))/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) - 4*Sqrt[2]*(85393762*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 43010905*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/27783

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(185)=370\).
time = 0.11, size = 494, normalized size = 1.95

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {6050 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6561 \left (\frac {2}{3}+x \right )^{5}}-\frac {73970 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5103 \left (\frac {2}{3}+x \right )^{3}}-\frac {1114 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{4}}-\frac {515062946 \left (-30 x^{2}-3 x +9\right )}{27783 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {5028574 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11907 \left (\frac {2}{3}+x \right )^{2}}-\frac {2162471240 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {3415750480 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{194481 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(325\)
default	\(-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (27667578888 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-13732045668 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-36618788448 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-36618788448 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-16275017088 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2712502848 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}-405988496886 x^{2}-289561969758 x -51867536481\right )}{27783 \left (2+3 x \right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/27783*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(27667578888*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)-13732045668*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)-36618788448*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)-36618788448*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]*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)-16275017088*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+5465200768*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))-2712502848*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+2245664953836*x^4+577509238368*x^3-405988
496886*x^2-289561969758*x-51867536481)/(2+3*x)^(9/2)/(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)^(11/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)), x)

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Fricas [A]
time = 0.52, size = 80, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (138337894440 \, x^{5} + 456548966244 \, x^{4} + 602551975428 \, x^{3} + 397527527442 \, x^{2} + 131099014240 \, x + 17289178827\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9261 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-2/9261*(138337894440*x^5 + 456548966244*x^4 + 602551975428*x^3 + 397527527442*x^2 + 131099014240*x + 17289178
827)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x
+ 96)

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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)**(5/2)/(2+3*x)**(11/2)/(3+5*x)**(3/2),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)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)), 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 )}^{11/2}\,{\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)^(11/2)*(5*x + 3)^(3/2)),x)

[Out]

int((1 - 2*x)^(5/2)/((3*x + 2)^(11/2)*(5*x + 3)^(3/2)), x)

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