3.30.94 \(\int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [2994]

Optimal. Leaf size=187 \[ \frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}+\frac {2209 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}-\frac {494 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}} \]

[Out]

7/33*(2+3*x)^(3/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2)+2209/219615*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2
))*33^(1/2)-494/219615*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/121*(2+3*x)^(1/2)/(3+
5*x)^(3/2)/(1-2*x)^(1/2)-247/3993*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)-2209/43923*(1-2*x)^(1/2)*(2+3*x)^(
1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157, 164, 114, 120} \begin {gather*} -\frac {494 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}+\frac {2209 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}+\frac {7 (3 x+2)^{3/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {3 x+2}}{43923 \sqrt {5 x+3}}-\frac {247 \sqrt {1-2 x} \sqrt {3 x+2}}{3993 (5 x+3)^{3/2}}+\frac {14 \sqrt {3 x+2}}{121 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*Sqrt[2 + 3*x])/(121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (247*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(3993*(3 + 5*x)^(3/
2)) + (7*(2 + 3*x)^(3/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (2209*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(43923*Sqr
t[3 + 5*x]) + (2209*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6655*Sqrt[33]) - (494*EllipticF[ArcSin
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6655*Sqrt[33])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 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 {(2+3 x)^{5/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {33}{2}-9 x\right ) \sqrt {2+3 x}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {-552-\frac {1593 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {2 \int \frac {\frac {3993}{4}+\frac {2223 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{11979}\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}-\frac {4 \int \frac {\frac {3519}{2}+\frac {19881 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{131769}\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}-\frac {2209 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{73205}+\frac {247 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6655}\\ &=\frac {14 \sqrt {2+3 x}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {247 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{3/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2209 \sqrt {1-2 x} \sqrt {2+3 x}}{43923 \sqrt {3+5 x}}+\frac {2209 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}-\frac {494 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6655 \sqrt {33}}\\ \end {align*}

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Mathematica [A]
time = 8.25, size = 104, normalized size = 0.56 \begin {gather*} \frac {-\frac {10 \sqrt {2+3 x} \left (-7186-22059 x-3402 x^2+22090 x^3\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+\sqrt {2} \left (-2209 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+10360 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{219615} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-10*Sqrt[2 + 3*x]*(-7186 - 22059*x - 3402*x^2 + 22090*x^3))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + Sqrt[2]*(-22
09*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 10360*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2]))/219615

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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 = 0.12, size = 305, normalized size = 1.63

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {733}{181500}+\frac {1229 x}{181500}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {5611}{4392300}+\frac {2209 x}{439230}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}-\frac {782 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{307461 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2209 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{307461 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)
default	\(-\frac {\sqrt {1-2 x}\, \left (81510 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+22090 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+8151 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2209 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-24453 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6627 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+662700 x^{4}+339740 x^{3}-729810 x^{2}-656760 x -143720\right )}{219615 \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2} \sqrt {2+3 x}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/219615*(1-2*x)^(1/2)*(81510*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)+22090*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)+8151*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)+2209*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)-2
4453*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))-6627*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))+662700*x^4+339740*x^
3-729810*x^2-656760*x-143720)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*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((2+3*x)^(5/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.27, size = 60, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (22090 \, x^{3} - 3402 \, x^{2} - 22059 \, x - 7186\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{43923 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/43923*(22090*x^3 - 3402*x^2 - 22059*x - 7186)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(100*x^4 + 20*x^3
- 59*x^2 - 6*x + 9)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 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((2+3*x)^(5/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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