3.29.7 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx\) [2807]

Optimal. Leaf size=222 \[ -\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}+\frac {33778 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {22866 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}-\frac {284}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {49321 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {32836 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375} \]

[Out]

-2/15*(1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2)+49321/328125*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1
/2))*33^(1/2)-32836/328125*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-62/15*(1-2*x)^(3/2)*
(2+3*x)^(5/2)/(3+5*x)^(1/2)+22866/4375*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-284/175*(2+3*x)^(5/2)*(1-2*x)
^(1/2)*(3+5*x)^(1/2)+33778/21875*(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 = 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 = {99, 155, 159, 164, 114, 120} \begin {gather*} -\frac {32836 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}+\frac {49321 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {284}{175} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{4375}+\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{21875} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(15*Sqrt[3 +
5*x]) + (33778*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/21875 + (22866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3
+ 5*x])/4375 - (284*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/175 + (49321*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/109375 - (32836*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/10
9375

Rule 99

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-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 155

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 159

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 164

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}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (-\frac {5}{2}-30 x\right ) (1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (-15-\frac {3195 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}-\frac {284}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (\frac {134235}{4}-\frac {514485 x}{4}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{7875}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}+\frac {22866 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}-\frac {284}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {8 \int \frac {\sqrt {2+3 x} \left (-\frac {561375}{8}+\frac {2280015 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{196875}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}+\frac {33778 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {22866 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}-\frac {284}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {8 \int \frac {\frac {881145}{8}-\frac {6658335 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2953125}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}+\frac {33778 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {22866 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}-\frac {284}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {49321 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{109375}+\frac {180598 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{109375}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 \sqrt {3+5 x}}+\frac {33778 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}+\frac {22866 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{4375}-\frac {284}{175} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {49321 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}-\frac {32836 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{109375}\\ \end {align*}

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Mathematica [A]
time = 8.76, size = 112, normalized size = 0.50 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (-19087-23425 x-41025 x^2-47250 x^3+67500 x^4\right )}{(3+5 x)^{3/2}}-49321 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+591115 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{328125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-19087 - 23425*x - 41025*x^2 - 47250*x^3 + 67500*x^4))/(3 + 5*x)^(3/2) - 493
21*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 591115*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2])/328125

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Maple [A]
time = 0.10, size = 230, normalized size = 1.04

method	result	size
default	\(-\frac {\left (2708970 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+246605 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4050000 x^{6}+1625382 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+147963 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2160000 x^{5}+4284000 x^{4}+870750 x^{3}+558970 x^{2}-277630 x -381740\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{328125 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\)	\(230\)
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {72 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}-\frac {684 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {2362 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {6527 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {49321 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{234375 \left (x +\frac {3}{5}\right )^{2}}-\frac {4774 \left (-30 x^{2}-5 x +10\right )}{46875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\)	\(297\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/328125*(2708970*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^
(1/2)+246605*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-
4050000*x^6+1625382*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1
/2))+147963*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+216
0000*x^5+4284000*x^4+870750*x^3+558970*x^2-277630*x-381740)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3
/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.30, size = 55, normalized size = 0.25 \begin {gather*} \frac {2 \, {\left (67500 \, x^{4} - 47250 \, x^{3} - 41025 \, x^{2} - 23425 \, x - 19087\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{65625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

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)^(5/2),x, algorithm="fricas")

[Out]

2/65625*(67500*x^4 - 47250*x^3 - 41025*x^2 - 23425*x - 19087)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(25*x
^2 + 30*x + 9)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)^(5/2),x, algorithm="giac")

[Out]

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

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

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)^(5/2),x)

[Out]

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

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