3.30.88 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [2988]

Optimal. Leaf size=218 \[ \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {694 \sqrt {1-2 x}}{41503 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {336536 \sqrt {1-2 x}}{290521 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {113693540 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}+\frac {22738708 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{290521 \sqrt {33}}+\frac {673072 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{290521 \sqrt {33}} \]

[Out]

22738708/9587193*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+673072/9587193*EllipticF(1/7*2
1^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/231/(1-2*x)^(3/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+1352/17787/(2+
3*x)^(3/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)+694/41503*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+336536/290521*(1-2*
x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-113693540/9587193*(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 = 218, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {673072 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{290521 \sqrt {33}}+\frac {22738708 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{290521 \sqrt {33}}-\frac {113693540 \sqrt {1-2 x} \sqrt {3 x+2}}{9587193 \sqrt {5 x+3}}+\frac {336536 \sqrt {1-2 x}}{290521 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {694 \sqrt {1-2 x}}{41503 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {1352}{17787 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + 1352/(17787*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x
]) + (694*Sqrt[1 - 2*x])/(41503*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (336536*Sqrt[1 - 2*x])/(290521*Sqrt[2 + 3*x]*
Sqrt[3 + 5*x]) - (113693540*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*Sqrt[3 + 5*x]) + (22738708*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(290521*Sqrt[33]) + (673072*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(290521*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)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {233}{2}-105 x}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {34841}{4}+12675 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {694 \sqrt {1-2 x}}{41503 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \int \frac {55293-\frac {46845 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{373527}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {694 \sqrt {1-2 x}}{41503 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {336536 \sqrt {1-2 x}}{290521 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {12510795}{8}-\frac {1893015 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2614689}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {694 \sqrt {1-2 x}}{41503 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {336536 \sqrt {1-2 x}}{290521 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {113693540 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}-\frac {32 \int \frac {\frac {161815545}{8}+\frac {255810465 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{28761579}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {694 \sqrt {1-2 x}}{41503 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {336536 \sqrt {1-2 x}}{290521 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {113693540 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}-\frac {336536 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{290521}-\frac {22738708 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3195731}\\ &=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1352}{17787 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {694 \sqrt {1-2 x}}{41503 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {336536 \sqrt {1-2 x}}{290521 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {113693540 \sqrt {1-2 x} \sqrt {2+3 x}}{9587193 \sqrt {3+5 x}}+\frac {22738708 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{290521 \sqrt {33}}+\frac {673072 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{290521 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.45, size = 109, normalized size = 0.50 \begin {gather*} \frac {2 \left (\frac {-215753865+198573504 x+1285584962 x^2-615527112 x^3-2046483720 x^4}{(1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (5684677 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2908255 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{9587193} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-215753865 + 198573504*x + 1285584962*x^2 - 615527112*x^3 - 2046483720*x^4)/((1 - 2*x)^(3/2)*(2 + 3*x)^(3
/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(5684677*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2908255*EllipticF
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/9587193

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Maple [A]
time = 0.11, size = 305, normalized size = 1.40

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {3251}{320166}+\frac {1093 x}{53361}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (-\frac {20076583}{57523158}+\frac {6867479 x}{9587193}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {1250 \left (-30 x^{2}-5 x +10\right )}{1331 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {71918020 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{67110351 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {113693540 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{67110351 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(262\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (68216124 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-33317064 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5552844 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-22738708 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{4}[phone] x^{3}-1285584962 x^{2}-198573504 x [phone]\right )}{9587193 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9587193*(1-2*x)^(1/2)*(68216124*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)-33317064*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)-5552844*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)-22738708*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))[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))
[phone]*x^4+615527112*x^3-1285584962*x^2-198573504*x+215753865)/(2+3*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

[Out]

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

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Fricas [A]
time = 0.21, size = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (2046483720 \, x^{4} + 615527112 \, x^{3} - 1285584962 \, x^{2} - 198573504 \, x + 215753865\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9587193 \, {\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9587193*(2046483720*x^4 + 615527112*x^3 - 1285584962*x^2 - 198573504*x + 215753865)*sqrt(5*x + 3)*sqrt(3*x
+ 2)*sqrt(-2*x + 1)/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3879 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/(1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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