3.28.97 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\) [2797]

Optimal. Leaf size=222 \[ -\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {196499 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{590625}+\frac {167228 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}-\frac {1972 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{4725}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {1509007 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2953125}-\frac {299863 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2953125} \]

[Out]

-1509007/8859375*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-299863/8859375*EllipticF(1/7*2
1^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/5*(1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(1/2)-8/45*(1-2*x)^(3/
2)*(2+3*x)^(5/2)*(3+5*x)^(1/2)+167228/118125*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-1972/4725*(2+3*x)^(5/2)
*(1-2*x)^(1/2)*(3+5*x)^(1/2)+196499/590625*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.06, 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 = {99, 159, 164, 114, 120} \begin {gather*} -\frac {299863 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2953125}-\frac {1509007 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2953125}-\frac {8}{45} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {1972 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{4725}-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}+\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{118125}+\frac {196499 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{590625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (196499*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/59
0625 + (167228*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/118125 - (1972*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[
3 + 5*x])/4725 - (8*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/45 - (1509007*Sqrt[11/3]*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2953125 - (299863*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/2953125

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 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)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (-\frac {5}{2}-30 x\right ) (1-2 x)^{3/2} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {4}{675} \int \frac {\left (\frac {465}{4}-\frac {7395 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}}{5 \sqrt {3+5 x}}-\frac {1972 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{4725}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {8 \int \frac {\left (\frac {684795}{8}-\frac {627105 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{70875}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {167228 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}-\frac {1972 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{4725}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {8 \int \frac {\sqrt {2+3 x} \left (-\frac {721125}{2}+\frac {8842455 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1771875}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {196499 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{590625}+\frac {167228 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}-\frac {1972 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{4725}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {8 \int \frac {\frac {111172815}{16}+\frac {67905315 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26578125}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {196499 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{590625}+\frac {167228 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}-\frac {1972 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{4725}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {1509007 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2953125}+\frac {3298493 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5906250}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {196499 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{590625}+\frac {167228 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{118125}-\frac {1972 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{4725}-\frac {8}{45} (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {1509007 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2953125}-\frac {299863 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2953125}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 8.79, size = 112, normalized size = 0.50 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (443337+650155 x-844650 x^2-382500 x^3+945000 x^4\right )}{\sqrt {3+5 x}}+3018014 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+6877465 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{17718750} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(443337 + 650155*x - 844650*x^2 - 382500*x^3 + 945000*x^4))/Sqrt[3 + 5*x] + 3
018014*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 6877465*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]], -33/2])/17718750

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 153, normalized size = 0.69

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-170100000 x^{6}+9895479 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3018014 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}-114638400 x^{3}-149984310 x^{2}[phone] x [phone]\right )}{17718750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(153\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}-\frac {844 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2625}-\frac {1222 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{13125}+\frac {6521 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{23625}+\frac {2470507 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{24806250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1509007 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{12403125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{15625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(284\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/17718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-170100000*x^6+9895479*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))-3018014*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))[phone]*x^5+220212000*x^4-114638400*x^3-149984310*x^2
[phone]*x+26600220)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 43, normalized size = 0.19 \begin {gather*} \frac {{\left (945000 \, x^{4} - 382500 \, x^{3} - 844650 \, x^{2} + 650155 \, x + 443337\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{590625 \, \sqrt {5 \, x + 3}} \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)^(3/2),x, algorithm="fricas")

[Out]

1/590625*(945000*x^4 - 382500*x^3 - 844650*x^2 + 650155*x + 443337)*sqrt(3*x + 2)*sqrt(-2*x + 1)/sqrt(5*x + 3)

________________________________________________________________________________________

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)**(3/2),x)

[Out]

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

________________________________________________________________________________________

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)^(3/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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 )}^{3/2}} \,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)^(3/2),x)

[Out]

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

________________________________________________________________________________________