3.30.44 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\) [2944]

Optimal. Leaf size=249 \[ \frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {181551856 \sqrt {1-2 x} \sqrt {2+3 x}}{871563 (3+5 x)^{3/2}}+\frac {12071114168 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}-\frac {12071114168 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}}-\frac {363103712 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}} \]

[Out]

-12071114168/47935965*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-363103712/47935965*Ellipt
icF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/77/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)+138/26
95*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+19548/18865*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+4115652/132
055*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-181551856/871563*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+12071
114168/9587193*(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 = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \begin {gather*} -\frac {363103712 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}}-\frac {12071114168 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}}+\frac {12071114168 \sqrt {1-2 x} \sqrt {3 x+2}}{9587193 \sqrt {5 x+3}}-\frac {181551856 \sqrt {1-2 x} \sqrt {3 x+2}}{871563 (5 x+3)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)),x]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (138*Sqrt[1 - 2*x])/(2695*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/
2)) + (19548*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (4115652*Sqrt[1 - 2*x])/(132055*Sqrt[2 +
3*x]*(3 + 5*x)^(3/2)) - (181551856*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(871563*(3 + 5*x)^(3/2)) + (12071114168*Sqrt[
1 - 2*x]*Sqrt[2 + 3*x])/(9587193*Sqrt[3 + 5*x]) - (12071114168*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3
3])/(1452605*Sqrt[33]) - (363103712*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1452605*Sqrt[33])

Rule 106

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

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 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}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {203}{2}-135 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {4 \int \frac {-\frac {3277}{2}+\frac {2415 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx}{2695}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {8 \int \frac {-\frac {540213}{4}+\frac {366525 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{56595}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16 \int \frac {-\frac {40301295}{4}+\frac {46301085 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{396165}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {181551856 \sqrt {1-2 x} \sqrt {2+3 x}}{871563 (3+5 x)^{3/2}}+\frac {32 \int \frac {-\frac {3301192785}{8}+\frac {510614595 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{13073445}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {181551856 \sqrt {1-2 x} \sqrt {2+3 x}}{871563 (3+5 x)^{3/2}}+\frac {12071114168 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}-\frac {64 \int \frac {-\frac {42986714535}{8}-\frac {67900017195 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{143807895}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {181551856 \sqrt {1-2 x} \sqrt {2+3 x}}{871563 (3+5 x)^{3/2}}+\frac {12071114168 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}+\frac {181551856 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1452605}+\frac {12071114168 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{15978655}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {19548 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4115652 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {181551856 \sqrt {1-2 x} \sqrt {2+3 x}}{871563 (3+5 x)^{3/2}}+\frac {12071114168 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}-\frac {12071114168 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}}-\frac {363103712 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1452605 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.05, size = 114, normalized size = 0.46 \begin {gather*} \frac {2 \left (\frac {687365548973+2920885694212 x+1466692421066 x^2-9658241620704 x^3-16841199826980 x^4-8148002063400 x^5}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}+4 \sqrt {2} \left (1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-759987865 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{47935965} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*((687365548973 + 2920885694212*x + 1466692421066*x^2 - 9658241620704*x^3 - 16841199826980*x^4 - 81480020634
00*x^5)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(1508889271*EllipticE[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2] - 759987865*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/47935965

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(397\) vs. \(2(185)=370\).
time = 0.11, size = 398, normalized size = 1.60

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {4412500}{1331} x^{2}-\frac {2206250}{3993} x +\frac {4412500}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {64 \left (-30 x^{2}-38 x -12\right )}{3195731 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{245 \left (\frac {2}{3}+x \right )^{3}}+\frac {-\frac {10178676}{2401} x^{2}-\frac {5089338}{12005} x +\frac {15268014}{12005}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {3048 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}+\frac {7642082584 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{67110351 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12071114168 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{67110351 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(329\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (134802253080 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-271600068780 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-525093466308 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-337991196704 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-72426685008 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-8148002063400 x^{5}-16841199826980 x^{4}-9658241620704 x^{3}[phone] x^{2}[phone] x [phone]\right )}{47935965 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\)	\(398\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/47935965*(1-2*x)^(1/2)*(134802253080*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)-271600068780*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)-525093466308*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)-337991196704*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)[phone]*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))-72426685008*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))-8148002063400*x^5-16841199826980*x^4-9658241620704*x^3+1466692421066*x^2+29208856
94212*x+687365548973)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)

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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/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

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Fricas [A]
time = 0.18, size = 80, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (8148002063400 \, x^{5} + 16841199826980 \, x^{4} + 9658241620704 \, x^{3} - 1466692421066 \, x^{2} - 2920885694212 \, x - 687365548973\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{47935965 \, {\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

2/47935965*(8148002063400*x^5 + 16841199826980*x^4 + 9658241620704*x^3 - 1466692421066*x^2 - 2920885694212*x -
687365548973)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2
- 420*x - 72)

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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/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(5/2),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/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(3/2)*(3*x + 2)^(7/2)*(5*x + 3)^(5/2)),x)

[Out]

int(1/((1 - 2*x)^(3/2)*(3*x + 2)^(7/2)*(5*x + 3)^(5/2)), x)

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