3.2954 \(\int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac {722133 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3500}+\frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac {56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}-\frac {1341}{154} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac {140289 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{3850}-\frac {2166399 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{7700}-\frac {6547351 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500} \]

[Out]

1/3*(2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2)-722133/38500*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2
))*33^(1/2)-6547351/10500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-56/11*(2+3*x)^(5/2)*(
3+5*x)^(3/2)/(1-2*x)^(1/2)-1341/154*(2+3*x)^(3/2)*(3+5*x)^(3/2)*(1-2*x)^(1/2)-140289/3850*(3+5*x)^(3/2)*(1-2*x
)^(1/2)*(2+3*x)^(1/2)-2166399/7700*(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 = 222, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac {56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt {1-2 x}}-\frac {1341}{154} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac {140289 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{3850}-\frac {2166399 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{7700}-\frac {722133 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}-\frac {6547351 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2166399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7700 - (140289*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
))/3850 - (1341*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/154 - (56*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/(11*
Sqrt[1 - 2*x]) + ((2 + 3*x)^(7/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (6547351*Sqrt[11/3]*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500 - (722133*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])
/3500

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)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x} \left (\frac {93}{2}+75 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-6285-\frac {20115 x}{2}\right ) (2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (\frac {2664975}{4}+\frac {2104335 x}{2}\right )}{\sqrt {1-2 x}} \, dx}{1155}\\ &=-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {\left (-\frac {190065795}{4}-\frac {292463865 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{28875}\\ &=-\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {\frac {12310799985}{8}+\frac {9722816235 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{259875}\\ &=-\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {2166399 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7000}+\frac {6547351 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3500}\\ &=-\frac {2166399 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7700}-\frac {140289 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{3850}-\frac {1341}{154} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {56 (2+3 x)^{5/2} (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {6547351 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}-\frac {722133 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3500}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 130, normalized size = 0.59 \[ -\frac {-6595505 \sqrt {2-4 x} (2 x-1) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (40500 x^4+198180 x^3+567906 x^2-2751916 x+1041609\right )[phone] \sqrt {2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{21000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21000*(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1041609 - 2751916*x + 567906*x^2 + 198180*x^3 + 40500*x^4) + 1309470
2*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 6595505*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 = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 243, normalized size = 1.09 \[ \frac {\left (-6075000 x^{6}-37422000 x^{5}-125270100 x^{4}[phone] x^{3}[phone] x^{2}-26189404 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-32790750 x [phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-6595505 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-62496540\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}}{21000 \left (2 x -1\right )^{2} \left (15 x^{2}+19 x +6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21000*(13191010*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)-26189404*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)-6075000*x^6-6595505*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))[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))-37422000*x^5-125270100*x^4+292994460*x^3+332548330*x^2-32790750*x-62496540)*(-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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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