3.28.79 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx\) [2779]

Optimal. Leaf size=222 \[ \frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {452399 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}+\frac {135334 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515} \]

[Out]

-2/9*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(3/2)-452399/76545*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/
2))*33^(1/2)+135334/76545*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+370/27*(1-2*x)^(3/2)*
(3+5*x)^(5/2)/(2+3*x)^(1/2)-31298/567*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)+5260/567*(3+5*x)^(5/2)*(1-2*x)
^(1/2)*(2+3*x)^(1/2)+135334/5103*(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 {135334 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}-\frac {452399 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt {3 x+2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {135334 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{5103} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5103 - (31298*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/
567 - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(3/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(27*Sqrt
[2 + 3*x]) + (5260*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/567 - (452399*Sqrt[11/3]*EllipticE[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/25515 + (135334*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/25
515

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} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {\left (\frac {235}{2}-\frac {6575 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (83425-\frac {1173675 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}\\ &=-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\left (\frac {1656225}{8}-\frac {5075025 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42525}\\ &=\frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {2298225}{2}-\frac {33929925 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{382725}\\ &=\frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {452399 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{25515}-\frac {744337 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{25515}\\ &=\frac {135334 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}-\frac {31298}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{27 \sqrt {2+3 x}}+\frac {5260}{567} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {452399 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}+\frac {135334 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25515}\\ \end {align*}

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Mathematica [A]
time = 9.04, size = 112, normalized size = 0.50 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (56963+108285 x+5949 x^2-25110 x^3+24300 x^4\right )}{(2+3 x)^{3/2}}+452399 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2685410 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{76545} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(56963 + 108285*x + 5949*x^2 - 25110*x^3 + 24300*x^4))/(2 + 3*x)^(3/2) + 4523
99*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2685410*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sq
rt[3 + 5*x]], -33/2])/76545

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

method	result	size
default	\(\frac {\left (6699033 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1357197 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7290000 x^{6}+4466022 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+904798 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6804000 x^{5}-1155600 x^{4}[phone] x^{3}[phone] x^{2}-8036760 x -5126670\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{76545 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\)	\(230\)
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {200 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}-\frac {1420 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {15962 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5103}+\frac {122572 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{107163 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {452399 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{107163 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {36260}{729} x^{2}-\frac {3626}{729} x +\frac {3626}{243}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6561 \left (\frac {2}{3}+x \right )^{2}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(297\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76545*(6699033*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)+1357197*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)+7
290000*x^6+4466022*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))+904798*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))-6804
000*x^5-1155600*x^4+34923870*x^3+19802040*x^2-8036760*x-5126670)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3
*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)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.28, size = 55, normalized size = 0.25 \begin {gather*} \frac {2 \, {\left (24300 \, x^{4} - 25110 \, x^{3} + 5949 \, x^{2} + 108285 \, x + 56963\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{5103 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5103*(24300*x^4 - 25110*x^3 + 5949*x^2 + 108285*x + 56963)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(9*x^2
+ 12*x + 4)

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

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(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 (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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