3.30.52 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^{5/2}} \, dx\) [2952]

Optimal. Leaf size=187 \[ \frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}-\frac {338 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}-\frac {992 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}} \]

[Out]

-338/79233*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-992/79233*EllipticF(1/7*21^(1/2)*(1-
2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/21*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2)+326/1617*(3+5*x)^(1/2)/(2+
3*x)^(3/2)/(1-2*x)^(1/2)-458/3773*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+338/26411*(1-2*x)^(1/2)*(3+5*x)^(1
/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {992 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}-\frac {338 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}+\frac {338 \sqrt {1-2 x} \sqrt {5 x+3}}{26411 \sqrt {3 x+2}}-\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{3773 (3 x+2)^{3/2}}+\frac {326 \sqrt {5 x+3}}{1617 \sqrt {1-2 x} (3 x+2)^{3/2}}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + (326*Sqrt[3 + 5*x])/(1617*Sqrt[1 - 2*x]*(2 + 3*x)^(3/
2)) - (458*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3773*(2 + 3*x)^(3/2)) + (338*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26411*Sqrt
[2 + 3*x]) - (338*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2401*Sqrt[33]) - (992*EllipticF[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2401*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)^{5/2} (2+3 x)^{5/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}-\frac {2}{21} \int \frac {-22-\frac {75 x}{2}}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}+\frac {4 \int \frac {\frac {4203}{4}+\frac {7335 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {14247}{8}+\frac {10305 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{33957}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {29115}{8}+\frac {7605 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{237699}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}+\frac {338 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{26411}+\frac {496 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2401}\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{3/2}}+\frac {326 \sqrt {3+5 x}}{1617 \sqrt {1-2 x} (2+3 x)^{3/2}}-\frac {458 \sqrt {1-2 x} \sqrt {3+5 x}}{3773 (2+3 x)^{3/2}}+\frac {338 \sqrt {1-2 x} \sqrt {3+5 x}}{26411 \sqrt {2+3 x}}-\frac {338 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}-\frac {992 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2401 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 7.84, size = 103, normalized size = 0.55 \begin {gather*} \frac {2 \left (\frac {\sqrt {3+5 x} \left (7965+727 x-21264 x^2+6084 x^3\right )}{(1-2 x)^{3/2} (2+3 x)^{3/2}}+\sqrt {2} \left (169 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+8015 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{79233} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[3 + 5*x]*(7965 + 727*x - 21264*x^2 + 6084*x^3))/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)) + Sqrt[2]*(169*Ell
ipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 8015*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/7
9233

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(139)=278\).
time = 0.10, size = 305, normalized size = 1.63

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {1}{2646}+\frac {2 x}{441}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right )^{2}}-\frac {2 \left (-18-30 x \right ) \left (\frac {3713}{475398}-\frac {169 x}{79233}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}+\frac {6470 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{554631 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1690 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{554631 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)
default	\(-\frac {2 \left (49104 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1014 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+8184 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-169 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-16368 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+338 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-30420 x^{4}+88068 x^{3}+60157 x^{2}-42006 x -23895\right ) \sqrt {1-2 x}}{79233 \left (2+3 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/79233*(49104*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(
1/2)-1014*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+8
184*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)-169*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)-16368*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))+338*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))-30420*x^4+88068*x^3+60157*x^2-42006*x-
23895)*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(-1+2*x)^2/(3+5*x)^(1/2)

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

[Out]

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

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Fricas [A]
time = 0.22, size = 60, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (6084 \, x^{3} - 21264 \, x^{2} + 727 \, x + 7965\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{79233 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/79233*(6084*x^3 - 21264*x^2 + 727*x + 7965)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(36*x^4 + 12*x^3 - 23
*x^2 - 4*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((3+5*x)**(1/2)/(1-2*x)**(5/2)/(2+3*x)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4372 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((3+5*x)^(1/2)/(1-2*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="giac")

[Out]

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

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

[Out]

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

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