3.30.97 \(\int \frac {1}{(1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\) [2997]

Optimal. Leaf size=187 \[ \frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}-\frac {119732 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}-\frac {7388 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}} \]

[Out]

-119732/2152227*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-7388/2152227*EllipticF(1/7*21^(
1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/231*(2+3*x)^(1/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2)+368/5929*(2+3*x)^
(1/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-18470/195657*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+598660/2152227*(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 = 187, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {7388 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}-\frac {119732 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}+\frac {598660 \sqrt {1-2 x} \sqrt {3 x+2}}{2152227 \sqrt {5 x+3}}-\frac {18470 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 (5 x+3)^{3/2}}+\frac {368 \sqrt {3 x+2}}{5929 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {4 \sqrt {3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[2 + 3*x])/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (368*Sqrt[2 + 3*x])/(5929*Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2)) - (18470*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657*(3 + 5*x)^(3/2)) + (598660*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21
52227*Sqrt[3 + 5*x]) - (119732*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(65219*Sqrt[33]) - (7388*Ell
ipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(65219*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} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2}{231} \int \frac {-\frac {201}{2}-75 x}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {4 \int \frac {\frac {20445}{4}+6210 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {19965}{2}-\frac {83115 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{586971}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {1799235}{8}+\frac {1346985 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6456681}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}+\frac {3694 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{65219}+\frac {119732 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{717409}\\ &=\frac {4 \sqrt {2+3 x}}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {368 \sqrt {2+3 x}}{5929 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {18470 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 (3+5 x)^{3/2}}+\frac {598660 \sqrt {1-2 x} \sqrt {2+3 x}}{2152227 \sqrt {3+5 x}}-\frac {119732 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}-\frac {7388 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65219 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.01, size = 103, normalized size = 0.55 \begin {gather*} \frac {2 \left (\frac {\sqrt {2+3 x} \left (881831-1822554 x-2800980 x^2+5986600 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+\sqrt {2} \left (59866 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+1085 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2152227} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[2 + 3*x]*(881831 - 1822554*x - 2800980*x^2 + 5986600*x^3))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + Sqrt[
2]*(59866*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1085*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2])))/2152227

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(139)=278\).
time = 0.10, size = 305, normalized size = 1.63

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {163}{127050}+\frac {37 x}{12705}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (\frac {169982}{10761135}-\frac {59866 x}{2152227}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}+\frac {399830 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15065589 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {598660 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{15065589 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)
default	\(-\frac {2 \sqrt {1-2 x}\, \left (609510 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-598660 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+60951 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-59866 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-182853 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+179598 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-17959800 x^{4}-3570260 x^{3}[phone] x^{2}+999615 x -1763662\right )}{2152227 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2152227*(1-2*x)^(1/2)*(609510*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)-598660*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)+60951*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)-59866*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)-182853*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))+1795
98*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))-17959800*x^4
-3570260*x^3+11069622*x^2+999615*x-1763662)/(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)/(3+5*x)^(5/2)/(2+3*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.28, size = 60, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (5986600 \, x^{3} - 2800980 \, x^{2} - 1822554 \, x + 881831\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2152227 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2152227*(5986600*x^3 - 2800980*x^2 - 1822554*x + 881831)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(100*x^4
+ 20*x^3 - 59*x^2 - 6*x + 9)

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

[Out]

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

[Out]

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

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