3.2791 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=129 \[ -\frac {164}{135} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}+\frac {812 \sqrt {5 x+3} \sqrt {1-2 x}}{27 \sqrt {3 x+2}}-\frac {3896}{135} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-3896/405*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-164/405*EllipticF(1/7*21^(1/2)*(1-2*x
)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/9*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+812/27*(1-2*x)^(1/2)*(3+5*x)^
(1/2)/(2+3*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ \frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{9 (3 x+2)^{3/2}}+\frac {812 \sqrt {5 x+3} \sqrt {1-2 x}}{27 \sqrt {3 x+2}}-\frac {164}{135} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {3896}{135} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + (812*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27*Sqrt[2 + 3*x])
- (3896*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/135 - (164*Sqrt[11/3]*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/135

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 150

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && 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 {(1-2 x)^{5/2}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\sqrt {1-2 x} (95+41 x)}{(2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {812 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {-\frac {1259}{2}-974 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {812 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}+\frac {902}{135} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {3896}{135} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{9 (2+3 x)^{3/2}}+\frac {812 \sqrt {1-2 x} \sqrt {3+5 x}}{27 \sqrt {2+3 x}}-\frac {3896}{135} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {164}{135} \sqrt {\frac {11}{3}} 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 = 0.22, size = 97, normalized size = 0.75 \[ \frac {2}{405} \left (-595 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {735 \sqrt {1-2 x} \sqrt {5 x+3} (24 x+17)}{(3 x+2)^{3/2}}+1948 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((735*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(17 + 24*x))/(2 + 3*x)^(3/2) + 1948*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 595*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/405

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fricas [F]  time = 1.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 2
4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 219, normalized size = 1.70 \[ \frac {2 \left (176400 x^{3}+142590 x^{2}-5844 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1785 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-40425 x -3896 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1190 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-37485\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{405 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/405*(1785*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)-5844*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)+1
190*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))-3896*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))+176400*x^3+14
2590*x^2-40425*x-37485)*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(10*x^2+x-3)/(3*x+2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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