3.2753 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac {33232}{35} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {301304 \sqrt {1-2 x} \sqrt {3 x+2}}{21 \sqrt {5 x+3}}-\frac {16616 \sqrt {1-2 x} \sqrt {3 x+2}}{7 (5 x+3)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

-33232/385*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-301304/105*EllipticE(1/7*21^(1/2)*(1
-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/15*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)+536/45*(1-2*x)^(1/2)/(2+
3*x)^(3/2)/(3+5*x)^(3/2)+111884/315*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)-16616/7*(1-2*x)^(1/2)*(2+3*x)^(1
/2)/(3+5*x)^(3/2)+301304/21*(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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {98, 152, 158, 113, 119} \[ \frac {301304 \sqrt {1-2 x} \sqrt {3 x+2}}{21 \sqrt {5 x+3}}-\frac {16616 \sqrt {1-2 x} \sqrt {3 x+2}}{7 (5 x+3)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {33232}{35} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*Sqrt[1 - 2*x])/(15*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (536*Sqrt[1 - 2*x])/(45*(2 + 3*x)^(3/2)*(3 + 5*x)^(3
/2)) + (111884*Sqrt[1 - 2*x])/(315*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (16616*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(7*(3
+ 5*x)^(3/2)) + (301304*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(21*Sqrt[3 + 5*x]) - (301304*Sqrt[11/3]*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35 - (33232*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/35

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 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 {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {156-235 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {4}{315} \int \frac {\frac {33999}{2}-23450 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {8 \int \frac {1277955-\frac {2936955 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{2205}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}-\frac {16 \int \frac {\frac {209379555}{4}-\frac {64771245 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{72765}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {32 \int \frac {681610545+\frac {4306575735 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{800415}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {49848}{35} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {301304}{35} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {536 \sqrt {1-2 x}}{45 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {111884 \sqrt {1-2 x}}{315 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16616 \sqrt {1-2 x} \sqrt {2+3 x}}{7 (3+5 x)^{3/2}}+\frac {301304 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {301304}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {33232}{35} \sqrt {\frac {3}{11}} 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 = 0.35, size = 109, normalized size = 0.49 \[ \frac {2}{105} \left (4 \sqrt {2} \left (37663 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-18970 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {1-2 x} \left (101690100 x^4+261029520 x^3+251053266 x^2+107221804 x+17157169\right )}{(3 x+2)^{5/2} (5 x+3)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(17157169 + 107221804*x + 251053266*x^2 + 261029520*x^3 + 101690100*x^4))/((2 + 3*x)^(5/2)*
(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(37663*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 18970*EllipticF[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/105

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{10125 \, x^{7} + 45225 \, x^{6} + 86535 \, x^{5} + 91947 \, x^{4} + 58592 \, x^{3} + 22392 \, x^{2} + 4752 \, x + 432}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)/(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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 406, normalized size = 1.83 \[ -\frac {2 \sqrt {-2 x +1}\, \left (-203380200 x^{5}-420368940 x^{4}+6779340 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-3414600 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-241077012 x^{3}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-6601560 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{2}+8436512 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4249280 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x +1807824 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-910560 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone]\right )}{105 \left (3 x +2\right )^{\frac {5}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(-2*x+1)^(1/2)*(6779340*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)-3414600*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)-6601560*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)+8436512*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)-4249280*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)+1807824*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))-910560*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))-203380200*x^5-420368940*x^4-241077012*x^3+36609658*x^2+72907466*x+17157169)/(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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-2*x + 1)^(3/2)/((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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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