3.29.96 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx\) [2896]

Optimal. Leaf size=125 \[ \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}+\frac {4}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {62 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}} \]

[Out]

-62/1617*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/49*EllipticE(1/7*21^(1/2)*(1-2*x)^(1
/2),1/33*1155^(1/2))*33^(1/2)+2/7*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)-12/49*(1-2*x)^(1/2)*(3+5*x)^(1/2)/
(2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 157, 164, 114, 120} \begin {gather*} -\frac {62 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {4}{49} \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {12 \sqrt {1-2 x} \sqrt {5 x+3}}{49 \sqrt {3 x+2}}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) - (12*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(49*Sqrt[2 + 3*x]) + (4*S
qrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/49 - (62*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(49*Sqrt[33])

Rule 101

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 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 157

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 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 {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {2}{7} \int \frac {-4-\frac {15 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}-\frac {4}{49} \int \frac {\frac {5}{4}+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}-\frac {12}{49} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {31}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}+\frac {4}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {62 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 5.60, size = 122, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {2+3 x} \sqrt {3+5 x} (1+12 x)-4 \sqrt {2-4 x} (2+3 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+35 \sqrt {2-4 x} (2+3 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{49 \sqrt {1-2 x} (2+3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1 + 12*x) - 4*Sqrt[2 - 4*x]*(2 + 3*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] + 35*Sqrt[2 - 4*x]*(2 + 3*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(49*Sqrt[1 - 2*x]*
(2 + 3*x))

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 133, normalized size = 1.06

method	result	size
default	\(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (31 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+4 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+120 x^{2}+82 x +6\right )}{49 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-18-30 x \right ) \left (\frac {1}{294}+\frac {2 x}{49}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {5 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1029 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {20 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/49*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(31*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+4*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))+120*x^2+82*x+6)/(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((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.21, size = 38, normalized size = 0.30 \begin {gather*} -\frac {2 \, {\left (12 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{49 \, {\left (6 \, x^{2} + x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/49*(12*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(6*x^2 + x - 2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________