\(\int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx\) [37]

Optimal result

Integrand size = 33, antiderivative size = 193 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=-\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {954811 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{756 \sqrt {-5+2 x}} \]

[Out]

5/28*(-5+2*x)^(3/2)*(1+4*x)^(3/2)*(2-3*x)^(1/2)+72479/4536*EllipticF(1/11*33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*
66^(1/2)*(5-2*x)^(1/2)/(-5+2*x)^(1/2)+136/105*(1+4*x)^(3/2)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)-954811/22680*Elliptic
E(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*11^(1/2)*(-5+2*x)^(1/2)/(5-2*x)^(1/2)-20911/3780*(2-3*x)^(1/2)*(-
5+2*x)^(1/2)*(1+4*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {159, 164, 115, 114, 122, 120} \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{756 \sqrt {2 x-5}}-\frac {954811 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {5}{28} \sqrt {2-3 x} (2 x-5)^{3/2} (4 x+1)^{3/2}+\frac {136}{105} \sqrt {2-3 x} \sqrt {2 x-5} (4 x+1)^{3/2}-\frac {20911 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{3780} \]

[In]

Int[Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x),x]

[Out]

(-20911*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/3780 + (136*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*(1 + 4*x)^(3/2))/
105 + (5*Sqrt[2 - 3*x]*(-5 + 2*x)^(3/2)*(1 + 4*x)^(3/2))/28 - (954811*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin
[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(22680*Sqrt[5 - 2*x]) + (72479*Sqrt[11/6]*Sqrt[5 - 2*x]*EllipticF[ArcSin[
Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(756*Sqrt[-5 + 2*x])

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 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x
]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d*x]*Sqrt[b*((e + f*x)/(b*e - a*f))])), Int[Sqrt[b*(e/(b*e - a*f)
) + b*f*(x/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]), x], 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]

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 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

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 {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {1}{28} \int \frac {\left (\frac {1249}{2}-1088 x\right ) \sqrt {-5+2 x} \sqrt {1+4 x}}{\sqrt {2-3 x}} \, dx \\ & = \frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {1}{840} \int \frac {(38731-41822 x) \sqrt {1+4 x}}{\sqrt {2-3 x} \sqrt {-5+2 x}} \, dx \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {\int \frac {-787710+1909622 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{7560} \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {954811 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{7560}+\frac {797269 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1512} \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}+\frac {\left (72479 \sqrt {\frac {11}{2}} \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{756 \sqrt {-5+2 x}}+\frac {\left (954811 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{7560 \sqrt {5-2 x}} \\ & = -\frac {20911 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{3780}+\frac {136}{105} \sqrt {2-3 x} \sqrt {-5+2 x} (1+4 x)^{3/2}+\frac {5}{28} \sqrt {2-3 x} (-5+2 x)^{3/2} (1+4 x)^{3/2}-\frac {954811 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{22680 \sqrt {5-2 x}}+\frac {72479 \sqrt {\frac {11}{6}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{756 \sqrt {-5+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.65 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\frac {24 \sqrt {2-3 x} \sqrt {1+4 x} \left (48475-37975 x-6066 x^2+5400 x^3\right )-954811 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )+724790 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{45360 \sqrt {-5+2 x}} \]

[In]

Integrate[Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x),x]

[Out]

(24*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(48475 - 37975*x - 6066*x^2 + 5400*x^3) - 954811*Sqrt[66]*Sqrt[5 - 2*x]*Ellipt
icE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3] + 724790*Sqrt[66]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1
+ 4*x]], 1/3])/(45360*Sqrt[-5 + 2*x])

Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (-1555200 x^{5}+264748 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-954811 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+2395008 x^{4}[phone] x^{3}-18808968 x^{2}+3994200 x +2326800\right )}{45360 \left (24 x^{3}-70 x^{2}+21 x +10\right )}\)	\(144\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {59 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{30}-\frac {277 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{54}-\frac {31 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{36 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {7891 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{3780 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {10 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{7}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(228\)
risch	\(-\frac {\left (2700 x^{2}+3717 x -9695\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{1890 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {31 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{108 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {7891 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{11340 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(252\)

[In]

int((7+5*x)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/45360*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(-1555200*x^5+264748*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*
(5-2*x)^(1/2)*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2))-954811*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2
)*EllipticE(1/11*(11+44*x)^(1/2),3^(1/2))+2395008*x^4+10468080*x^3-18808968*x^2+3994200*x+2326800)/(24*x^3-70*
x^2+21*x+10)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.31 \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\frac {1}{1890} \, {\left (2700 \, x^{2} + 3717 \, x - 9695\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} - \frac {549703}{23328} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) + \frac {954811}{22680} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]

[In]

integrate((7+5*x)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2),x, algorithm="fricas")

[Out]

1/1890*(2700*x^2 + 3717*x - 9695)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2) - 549703/23328*sqrt(-6)*weierstra
ssPInverse(847/108, 6655/2916, x - 35/36) + 954811/22680*sqrt(-6)*weierstrassZeta(847/108, 6655/2916, weierstr
assPInverse(847/108, 6655/2916, x - 35/36))

Sympy [F]

\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int \sqrt {2 - 3 x} \sqrt {2 x - 5} \sqrt {4 x + 1} \cdot \left (5 x + 7\right )\, dx \]

[In]

integrate((7+5*x)*(2-3*x)**(1/2)*(-5+2*x)**(1/2)*(1+4*x)**(1/2),x)

[Out]

Integral(sqrt(2 - 3*x)*sqrt(2*x - 5)*sqrt(4*x + 1)*(5*x + 7), x)

Maxima [F]

\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int { {\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]

[In]

integrate((7+5*x)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2), x)

Giac [F]

\[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int { {\left (5 \, x + 7\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} \,d x } \]

[In]

integrate((7+5*x)*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2),x, algorithm="giac")

[Out]

integrate((5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x) \, dx=\int \sqrt {2-3\,x}\,\sqrt {4\,x+1}\,\sqrt {2\,x-5}\,\left (5\,x+7\right ) \,d x \]

[In]

int((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2)*(5*x + 7),x)

[Out]

int((2 - 3*x)^(1/2)*(4*x + 1)^(1/2)*(2*x - 5)^(1/2)*(5*x + 7), x)