3.2877 \(\int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac {598 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1375 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{165 (5 x+3)^{3/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3 x+2}}{9075 \sqrt {5 x+3}}-\frac {2797 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375 \sqrt {33}} \]

[Out]

-2797/45375*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-598/45375*EllipticF(1/7*21^(1/2)*(1
-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/165*(2+3*x)^(3/2)*(1-2*x)^(1/2)/(3+5*x)^(3/2)-404/9075*(1-2*x)^(1/2)*(
2+3*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 125, 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} \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^{3/2}}{165 (5 x+3)^{3/2}}-\frac {404 \sqrt {1-2 x} \sqrt {3 x+2}}{9075 \sqrt {5 x+3}}-\frac {598 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375 \sqrt {33}}-\frac {2797 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(165*(3 + 5*x)^(3/2)) - (404*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9075*Sqrt[3 + 5*
x]) - (2797*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1375*Sqrt[33]) - (598*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/(1375*Sqrt[33])

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 {(2+3 x)^{5/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {2}{165} \int \frac {\left (-\frac {215}{2}-\frac {291 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x} \sqrt {2+3 x}}{9075 \sqrt {3+5 x}}-\frac {4 \int \frac {-1752-\frac {8391 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x} \sqrt {2+3 x}}{9075 \sqrt {3+5 x}}+\frac {2797 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{15125}+\frac {299 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {404 \sqrt {1-2 x} \sqrt {2+3 x}}{9075 \sqrt {3+5 x}}-\frac {2797 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375 \sqrt {33}}-\frac {598 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1375 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 97, normalized size = 0.78 \[ \frac {7070 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2} (1175 x+716)}{(5 x+3)^{3/2}}+2797 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{45375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(716 + 1175*x))/(3 + 5*x)^(3/2) + 2797*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] + 7070*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/45375

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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{250 \, x^{4} + 325 \, x^{3} + 45 \, x^{2} - 81 \, x - 27}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(9*x^2 + 12*x + 4)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(250*x^4 + 325*x^3 + 45*x^2 - 81*x - 2
7), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 219, normalized size = 1.75 \[ -\frac {\left (70500 x^{3}+54710 x^{2}+13985 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+35350 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-16340 x +8391 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+21210 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-14320\right ) \sqrt {-2 x +1}\, \sqrt {3 x +2}}{45375 \left (6 x^{2}+x -2\right ) \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45375*(35350*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)+13985*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)+21210*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))+8
391*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))+70500*x
^3+54710*x^2-16340*x-14320)*(-2*x+1)^(1/2)*(3*x+2)^(1/2)/(6*x^2+x-2)/(5*x+3)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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