3.30.32 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\) [2932]

Optimal. Leaf size=160 \[ \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {23180 \sqrt {1-2 x} \sqrt {2+3 x}}{5929 \sqrt {3+5 x}}+\frac {4636}{539} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {124}{539} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

4636/5929*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+124/5929*EllipticF(1/7*21^(1/2)*(1-2*
x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/77/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)+186/539*(1-2*x)^(1/2)/(2+3*x
)^(1/2)/(3+5*x)^(1/2)-23180/5929*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {124}{539} \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4636}{539} \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {23180 \sqrt {1-2 x} \sqrt {3 x+2}}{5929 \sqrt {5 x+3}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (186*Sqrt[1 - 2*x])/(539*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (23
180*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(5929*Sqrt[3 + 5*x]) + (4636*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2
*x]], 35/33])/539 + (124*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/539

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)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {91}{2}-45 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4}{539} \int \frac {-440+\frac {465 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {23180 \sqrt {1-2 x} \sqrt {2+3 x}}{5929 \sqrt {3+5 x}}+\frac {8 \int \frac {-\frac {21885}{4}-\frac {17385 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5929}\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {23180 \sqrt {1-2 x} \sqrt {2+3 x}}{5929 \sqrt {3+5 x}}-\frac {186}{539} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {13908 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5929}\\ &=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {23180 \sqrt {1-2 x} \sqrt {2+3 x}}{5929 \sqrt {3+5 x}}+\frac {4636}{539} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {124}{539} \sqrt {\frac {3}{11}} 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 = 6.22, size = 98, normalized size = 0.61 \begin {gather*} \frac {2 \left (\frac {-22003+9544 x+69540 x^2}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\sqrt {2} \left (-2318 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+1295 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{5929} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-22003 + 9544*x + 69540*x^2)/(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + Sqrt[2]*(-2318*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1295*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/5929

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Maple [A]
time = 0.21, size = 133, normalized size = 0.83

method	result	size
default	\(\frac {2 \left (1023 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2318 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-69540 x^{2}-9544 x +22003\right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}}{5929 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\frac {139080}{5929} x^{2}+\frac {19088}{5929} x -\frac {44006}{5929}}{\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14590 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{41503 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {23180 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{41503 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5929*(1023*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))-23
18*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))-69540*x^2-95
44*x+22003)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(30*x^3+23*x^2-7*x-6)

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

[Out]

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

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Fricas [A]
time = 0.31, size = 50, normalized size = 0.31 \begin {gather*} -\frac {2 \, {\left (69540 \, x^{2} + 9544 \, x - 22003\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{5929 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5929*(69540*x^2 + 9544*x - 22003)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(30*x^3 + 23*x^2 - 7*x - 6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\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)^(3/2)*(3*x + 2)^(3/2)*(5*x + 3)^(3/2)),x)

[Out]

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

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