3.2964 \(\int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac {91}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {137 \sqrt {3 x+2} (5 x+3)^{5/2}}{33 \sqrt {1-2 x}}-\frac {817}{66} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-91 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

1/3*(2+3*x)^(3/2)*(3+5*x)^(5/2)/(1-2*x)^(3/2)-12101/60*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*3
3^(1/2)-91/15*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-137/33*(3+5*x)^(5/2)*(2+3*x)^(1/2
)/(1-2*x)^(1/2)-817/66*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)-91*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {137 \sqrt {3 x+2} (5 x+3)^{5/2}}{33 \sqrt {1-2 x}}-\frac {817}{66} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-91 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {91}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-91*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (817*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/66 - (137*Sq
rt[2 + 3*x]*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (121
01*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20 - (91*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/5

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 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 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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {77}{2}+60 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {8033}{2}-\frac {12255 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{495} \int \frac {\sqrt {3+5 x} \left (\frac {1053855}{4}+405405 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {-\frac {34129755}{4}-\frac {53909955 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4455}\\ &=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1001}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {12101}{20} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{5} \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 = 0.29, size = 125, normalized size = 0.66 \[ -\frac {-6095 \sqrt {2-4 x} (2 x-1) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (90 x^3+438 x^2-2579 x+957\right )+12101 \sqrt {2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{60 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(957 - 2579*x + 438*x^2 + 90*x^3) + 12101*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellip
ticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6095*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2])/(1 - 2*x)^(3/2)

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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(75*x^3 + 140*x^2 + 87*x + 18)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(8*x^3 - 12*x^2 + 6*x - 1)
, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 238, normalized size = 1.26 \[ \frac {\left (-13500 x^{5}-82800 x^{4}+298230 x^{3}+320180 x^{2}-24202 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+12190 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-27090 x +12101 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-6095 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-57420\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}}{60 \left (2 x -1\right )^{2} \left (15 x^{2}+19 x +6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12190*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)-24202*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)-
6095*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))+12101*
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))-13500*x^5-8
2800*x^4+298230*x^3+320180*x^2-27090*x-57420)*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)/(2*x-1)^2/(15*x^2+19*
x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((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 = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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