3.30.98 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\) [2998]

Optimal. Leaf size=218 \[ \frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1523260 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac {99425780 \sqrt {1-2 x} \sqrt {2+3 x}}{15065589 \sqrt {3+5 x}}-\frac {19885156 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}}-\frac {609304 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}} \]

[Out]

-19885156/15065589*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-609304/15065589*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/231/(1-2*x)^(3/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)+456/5929/(3
+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)+5034/41503*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-1523260/1369599*(
1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)[phone]/15065589*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, 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 {609304 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}}-\frac {19885156 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}}+\frac {99425780 \sqrt {1-2 x} \sqrt {3 x+2}}{15065589 \sqrt {5 x+3}}-\frac {1523260 \sqrt {1-2 x} \sqrt {3 x+2}}{1369599 (5 x+3)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + 456/(5929*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))
+ (5034*Sqrt[1 - 2*x])/(41503*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1523260*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599
*(3 + 5*x)^(3/2)) + (99425780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15065589*Sqrt[3 + 5*x]) - (19885156*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(456533*Sqrt[33]) - (609304*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(456533*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)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {237}{2}-105 x}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4 \int \frac {\frac {36717}{4}+12825 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {8 \int \frac {\frac {80265}{2}-\frac {113265 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{124509}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1523260 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 (3+5 x)^{3/2}}-\frac {16 \int \frac {\frac {10801065}{8}-\frac {3427335 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{4108797}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1523260 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac {99425780 \sqrt {1-2 x} \sqrt {2+3 x}}{15065589 \sqrt {3+5 x}}+\frac {32 \int \frac {\frac {35441235}{2}+\frac {223708005 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{45196767}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1523260 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac {99425780 \sqrt {1-2 x} \sqrt {2+3 x}}{15065589 \sqrt {3+5 x}}+\frac {304652 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{456533}+\frac {19885156 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5021863}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {456}{5929 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {5034 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1523260 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 (3+5 x)^{3/2}}+\frac {99425780 \sqrt {1-2 x} \sqrt {2+3 x}}{15065589 \sqrt {3+5 x}}-\frac {19885156 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}}-\frac {609304 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{456533 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 7.76, size = 109, normalized size = 0.50 \begin {gather*} \frac {2 \left (\frac {283144937-211488180 x-1802210526 x^2+694871080 x^3+2982773400 x^4}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}+2 \sqrt {2} \left (4971289 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2457910 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{15065589} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((283144937 - 211488180*x - 1802210526*x^2 + 694871080*x^3 + 2982773400*x^4)/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]
*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(4971289*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2457910*EllipticF[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/15065589

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

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {6101}{889350}+\frac {1229 x}{88935}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (\frac {6580681}{30131178}-\frac {6384815 x}{15065589}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}+\frac {-\frac {4860}{343} x^{2}-\frac {486}{343} x +\frac {1458}{343}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {63006640 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{105459123 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {99425780 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{105459123 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(262\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (50267580 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-99425780 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+5026758 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-9942578 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-15080274 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2982773400 x^{4}-694871080 x^{3}[phone] x^{2}[phone] x -283144937\right )}{15065589 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15065589*(1-2*x)^(1/2)*(50267580*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)-99425780*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)+5026758*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)-9942578*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)-15080274*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]*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))
-2982773400*x^4-694871080*x^3+1802210526*x^2+211488180*x-283144937)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*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)^(3/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (2982773400 \, x^{4} + 694871080 \, x^{3} - 1802210526 \, x^{2} - 211488180 \, x + 283144937\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{15065589 \, {\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15065589*(2982773400*x^4 + 694871080*x^3 - 1802210526*x^2 - 211488180*x + 283144937)*sqrt(5*x + 3)*sqrt(3*x
+ 2)*sqrt(-2*x + 1)/(300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)

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

[Out]

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

[Out]

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

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