\(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx\) [2912]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {67}{63} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]

[Out]

4451/378*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+67/189*EllipticF(1/7*21^(1/2)*(1-2*x)^
(1/2),1/33*1155^(1/2))*33^(1/2)+11/7*(3+5*x)^(3/2)*(2+3*x)^(1/2)/(1-2*x)^(1/2)+335/63*(1-2*x)^(1/2)*(2+3*x)^(1
/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 129, 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, 159, 164, 114, 120} \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {67}{63} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {11 \sqrt {3 x+2} (5 x+3)^{3/2}}{7 \sqrt {1-2 x}}+\frac {335}{63} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]

[In]

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

[Out]

(335*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/63 + (11*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]) + (4
451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/126 + (67*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3
/7]*Sqrt[1 - 2*x]], 35/33])/63

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 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*} \text {integral}& = \frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {1}{7} \int \frac {\sqrt {3+5 x} \left (\frac {435}{2}+335 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {1}{63} \int \frac {-7045-\frac {22255 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}-\frac {737}{126} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {4451}{126} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {335}{63} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {11 \sqrt {2+3 x} (3+5 x)^{3/2}}{7 \sqrt {1-2 x}}+\frac {4451}{126} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {67}{63} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.39 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {6 (632-175 x) \sqrt {2+3 x} \sqrt {3+5 x}-4451 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4585 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{378 \sqrt {1-2 x}} \]

[In]

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

[Out]

(6*(632 - 175*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (4451*I)*Sqrt[33 - 66*x]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -
2/33] + (4585*I)*Sqrt[33 - 66*x]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(378*Sqrt[1 - 2*x])

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09

method	result	size
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (4323 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4451 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+15750 x^{3}-36930 x^{2}-65748 x -22752\right )}{11340 x^{3}+8694 x^{2}-2646 x -2268}\)	\(140\)
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {25 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18}-\frac {2818 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4451 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {121 \left (-30 x^{2}-38 x -12\right )}{28 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(214\)

[In]

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

[Out]

1/378*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(4323*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/
2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))-4451*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*El
lipticE((10+15*x)^(1/2),1/35*70^(1/2))+15750*x^3-36930*x^2-65748*x-22752)/(30*x^3+23*x^2-7*x-6)

Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx=\frac {540 \, {\left (175 \, x - 632\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 151247 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 400590 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{34020 \, {\left (2 \, x - 1\right )}} \]

[In]

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

[Out]

1/34020*(540*(175*x - 632)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 151247*sqrt(-30)*(2*x - 1)*weierstrass
PInverse(1159/675, 38998/91125, x + 23/90) - 400590*sqrt(-30)*(2*x - 1)*weierstrassZeta(1159/675, 38998/91125,
weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2*x - 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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