\(\int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\) [2973]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {4451 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}-\frac {67 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{55 \sqrt {33}} \]

[Out]

-4451/3630*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-67/1815*EllipticF(1/7*21^(1/2)*(1-2*
x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/33*(2+3*x)^(3/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)-448/363*(2+3*x)^(1/2)*(3+5*x
)^(1/2)/(1-2*x)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 155, 164, 114, 120} \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {67 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{55 \sqrt {33}}-\frac {4451 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}-\frac {448 \sqrt {5 x+3} \sqrt {3 x+2}}{363 \sqrt {1-2 x}} \]

[In]

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

[Out]

(-448*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)
) - (4451*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(110*Sqrt[33]) - (67*EllipticF[ArcSin[Sqrt[3/7]*S
qrt[1 - 2*x]], 35/33])/(55*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 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*} \text {integral}& = \frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\sqrt {2+3 x} \left (\frac {247}{2}+201 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{363} \int \frac {-4227-\frac {13353 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {67}{110} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {4451 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1210} \\ & = -\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {4451 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}-\frac {67 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{55 \sqrt {33}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {490 \sqrt {2+3 x} \sqrt {3+5 x} (-6+23 x)-4451 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4585 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{3630 (1-2 x)^{3/2}} \]

[In]

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

[Out]

(490*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-6 + 23*x) - (4451*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticE[I*ArcSinh[Sqrt[9
+ 15*x]], -2/33] + (4585*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(3630*(1 -
2*x)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(93)=186\).

Time = 1.39 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {49 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{264 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {5635}{242} x^{2}-\frac {21413}{726} x -\frac {1127}{121}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {2818 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4451 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(219\)
default	\(-\frac {\left (8646 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8902 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4323 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+4451 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-169050 x^{3}-170030 x^{2}-11760 x +17640\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{3630 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\)	\(228\)

[In]

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

[Out]

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)*(49/264*(-30*x^3-23*x^2+7*x+6)^(1/
2)/(x-1/2)^2+1127/1452*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)+2818/12705*(10+15*x)^(1/2)*(21-42*x
)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+4451/12705*(10+1
5*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*7
0^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {44100 \, {\left (23 \, x - 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 151247 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 400590 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{326700 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

[In]

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

[Out]

1/326700*(44100*(23*x - 6)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 151247*sqrt(-30)*(4*x^2 - 4*x + 1)*wei
erstrassPInverse(1159/675, 38998/91125, x + 23/90) + 400590*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/6
75, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 - 4*x + 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]

[In]

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

[Out]

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