3.2698 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ -\frac {310208 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{245 \sqrt {33}}+\frac {10312712 \sqrt {1-2 x} \sqrt {3 x+2}}{1617 \sqrt {5 x+3}}-\frac {155104 \sqrt {1-2 x} \sqrt {3 x+2}}{147 (5 x+3)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {2 \sqrt {1-2 x}}{5 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {10312712 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}} \]

[Out]

-10312712/8085*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-310208/8085*EllipticF(1/7*21^(1/
2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/5*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+556/105*(1-2*x)^(1/2)
/(2+3*x)^(3/2)/(3+5*x)^(3/2)+116044/735*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-155104/147*(1-2*x)^(1/2)*(2+
3*x)^(1/2)/(3+5*x)^(3/2)[phone]/1617*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.08, 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 = {99, 152, 158, 113, 119} \[ \frac {10312712 \sqrt {1-2 x} \sqrt {3 x+2}}{1617 \sqrt {5 x+3}}-\frac {155104 \sqrt {1-2 x} \sqrt {3 x+2}}{147 (5 x+3)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {2 \sqrt {1-2 x}}{5 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {310208 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}}-\frac {10312712 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - 2*x])/(5*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (556*Sqrt[1 - 2*x])/(105*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/
2)) + (116044*Sqrt[1 - 2*x])/(735*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (155104*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(147*(
3 + 5*x)^(3/2)) + (10312712*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1617*Sqrt[3 + 5*x]) - (10312712*EllipticE[ArcSin[Sqr
t[3/7]*Sqrt[1 - 2*x]], 35/33])/(245*Sqrt[33]) - (310208*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(24
5*Sqrt[33])

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*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))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(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 152

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 158

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 {\sqrt {1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {2}{5} \int \frac {-23+35 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {4}{105} \int \frac {-\frac {5037}{2}+3475 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {8}{735} \int \frac {-\frac {378705}{2}+\frac {435165 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {155104 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {16 \int \frac {-\frac {31023465}{4}+4798530 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{24255}\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {155104 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 \sqrt {3+5 x}}-\frac {32 \int \frac {-\frac {403972965}{4}-\frac {638099055 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{266805}\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {155104 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 \sqrt {3+5 x}}+\frac {155104}{245} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {10312712 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2695}\\ &=\frac {2 \sqrt {1-2 x}}{5 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x}}{105 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {116044 \sqrt {1-2 x}}{735 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {155104 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 \sqrt {3+5 x}}-\frac {10312712 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}}-\frac {310208 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{245 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 109, normalized size = 0.50 \[ \frac {2 \left (4 \sqrt {2} \left (1289089 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-649285 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {1-2 x} \left (3480540300 x^4+8934240060 x^3+8592783498 x^2+3669873602 x+587237237\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}\right )}{8085} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(587237237 + 3669873602*x + 8592783498*x^2 + 8934240060*x^3 + 3480540300*x^4))/((2 + 3*x)^(
5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(1289089*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 649285*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/8085

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fricas [F]  time = 1.37, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10125 \, x^{7} + 45225 \, x^{6} + 86535 \, x^{5} + 91947 \, x^{4} + 58592 \, x^{3} + 22392 \, x^{2} + 4752 \, x + 432}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(10125*x^7 + 45225*x^6 + 86535*x^5 + 91947*x^4 + 58592*x^3
+ 22392*x^2 + 4752*x + 432), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 406, normalized size = 1.86 \[ -\frac {2 \sqrt {-2 x +1}\, \left (-6961080600 x^{5}-14387939820 x^{4}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-116871300 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-8251326936 x^{3}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-225951180 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{2}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-145439840 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x [phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-31165680 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone]\right )}{8085 \left (3 x +2\right )^{\frac {5}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/8085*(-2*x+1)^(1/2)*(232036020*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^3*(5*x+3)^(1/2)*(3
*x+2)^(1/2)*(-2*x+1)^(1/2)-116871300*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^3*(5*x+3)^(1/2)
*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1
/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-225951180*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)
^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3
)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-145439840*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+
3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)[phone]*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/
11*(110*x+66)^(1/2),1/2*I*66^(1/2))-31165680*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/11
*(110*x+66)^(1/2),1/2*I*66^(1/2))-6961080600*x^5-14387939820*x^4-8251326936*x^3+1253036294*x^2+2495399128*x+58
7237237)/(3*x+2)^(5/2)/(5*x+3)^(3/2)/(2*x-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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