3.30.1 \(\int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\) [2901]

Optimal. Leaf size=157 \[ \frac {139}{10} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {4621}{50} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {139}{50} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

4621/150*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+139/150*EllipticF(1/7*21^(1/2)*(1-2*x)
^(1/2),1/33*1155^(1/2))*33^(1/2)+(2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(1/2)+9/5*(3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+
3*x)^(1/2)+139/10*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {139}{50} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4621}{50} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}+\frac {9}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {139}{10} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(139*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/10 + (9*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((2 +
3*x)^(3/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (4621*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/50 + (139*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/50

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 {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (\frac {57}{2}+45 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {9}{5} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{25} \int \frac {\left (-\frac {4065}{2}-\frac {6255 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {139}{10} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {1}{225} \int \frac {\frac {263295}{4}+\frac {207945 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {139}{10} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {1529}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {4621}{50} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {139}{10} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {9}{5} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {4621}{50} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {139}{50} \sqrt {\frac {11}{3}} 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 = 6.51, size = 110, normalized size = 0.70 \begin {gather*} \frac {-30 \sqrt {2+3 x} \sqrt {3+5 x} \left (-253+106 x+30 x^2\right )-9242 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+4655 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{300 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-253 + 106*x + 30*x^2) - 9242*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2] + 4655*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(300*Sqrt[1 - 2*x]
)

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Maple [A]
time = 0.10, size = 143, normalized size = 0.91

method	result	size
default	\(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4587 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-9242 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+13500 x^{4}+64800 x^{3}-48030 x^{2}-125130 x -45540\right )}{9000 x^{3}+6900 x^{2}-2100 x -1800}\)	\(143\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {3 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2}+\frac {121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{20}-\frac {5851 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{420 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4621 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{210 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {77 \left (-30 x^{2}-38 x -12\right )}{8 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\)	\(252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/300*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4587*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-9242*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))+13500*x^4+64800*x^3-48030*x^2-125130*x-45540)/(30*x^3+23*x^2-7*x-6)

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

[Out]

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

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Fricas [A]
time = 0.23, size = 40, normalized size = 0.25 \begin {gather*} \frac {{\left (30 \, x^{2} + 106 \, x - 253\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(30*x^2 + 106*x - 253)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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