3.1.45 \(\int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^2}{\sqrt {-5+2 x}} \, dx\) [45]

3.1.45.1 Optimal result

Integrand size = 35, antiderivative size = 205 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^2}{\sqrt {-5+2 x}} \, dx=\frac {73207 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1080}+\frac {173}{60} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)+\frac {1}{7} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2+\frac {8198333 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{9072 \sqrt {5-2 x}}-\frac {1679161 \sqrt {\frac {11}{6}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{756 \sqrt {-5+2 x}} \]

output

-1679161/4536*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)+8198333/9072*EllipticE(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)+73207/1080*(2-3 
*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)+173/60*(7+5*x)*(2-3*x)^(1/2)*(-5+2* 
x)^(1/2)*(1+4*x)^(1/2)+1/7*(7+5*x)^2*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^ 
(1/2)
 

3.1.45.2 Mathematica [A] (verified)

Time = 4.40 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^2}{\sqrt {-5+2 x}} \, dx=\frac {12 \sqrt {2-3 x} \sqrt {1+4 x} \left (-717955+102592 x+46836 x^2+10800 x^3\right )+8198333 \sqrt {66} \sqrt {5-2 x} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )-6716644 \sqrt {66} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{18144 \sqrt {-5+2 x}} \]

input

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

output

(12*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(-717955 + 102592*x + 46836*x^2 + 10800*x^ 
3) + 8198333*Sqrt[66]*Sqrt[5 - 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4 
*x]], 1/3] - 6716644*Sqrt[66]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sq 
rt[1 + 4*x]], 1/3])/(18144*Sqrt[-5 + 2*x])
 

3.1.45.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {180, 25, 2103, 27, 2118, 27, 176, 124, 123, 131, 27, 129}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {4 x+1} (5 x+7)^2}{\sqrt {2 x-5}} \, dx\)

\(\Big \downarrow \) 180

\(\displaystyle \frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2-\frac {1}{14} \int -\frac {(5 x+7) \left (-2422 x^2+175 x+543\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{14} \int \frac {(5 x+7) \left (-2422 x^2+175 x+543\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 2103

\(\displaystyle \frac {1}{14} \left (\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)-\frac {1}{120} \int -\frac {2 \left (-2049796 x^2-568915 x+527177\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \int \frac {-2049796 x^2-568915 x+527177}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 2118

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {1}{108} \int \frac {330 (368193-1490606 x)}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \int \frac {368193-1490606 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \left (-3358322 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-745303 \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx\right )+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \left (-\frac {745303 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{\sqrt {5-2 x}}-3358322 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \left (-3358322 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {745303 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \left (-\frac {305302 \sqrt {22} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {745303 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \left (-\frac {3358322 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {745303 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{14} \left (\frac {1}{60} \left (\frac {55}{18} \left (-\frac {305302 \sqrt {\frac {22}{3}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {745303 \sqrt {\frac {11}{6}} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )+\frac {512449}{9} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}\right )+\frac {1211}{30} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)\right )+\frac {1}{7} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2\)

input

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

output

(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^2)/7 + ((1211*Sqrt[2 
 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x))/30 + ((512449*Sqrt[2 - 3*x 
]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/9 + (55*((-745303*Sqrt[11/6]*Sqrt[-5 + 2*x 
]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[5 - 2*x] - (30530 
2*Sqrt[22/3]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3 
])/Sqrt[-5 + 2*x]))/18)/60)/14
 

3.1.45.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> 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] &&  !L 
tQ[-(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 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[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] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> 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] && PosQ[-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 131

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x]   Int[1/(Sq 
rt[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] && Simpler 
Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(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] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

rule 180

Int[(((a_.) + (b_.)*(x_))^(m_)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)* 
(x_)])/Sqrt[(c_.) + (d_.)*(x_)], x_] :> Simp[2*(a + b*x)^m*Sqrt[c + d*x]*Sq 
rt[e + f*x]*(Sqrt[g + h*x]/(d*(2*m + 3))), x] - Simp[1/(d*(2*m + 3))   Int[ 
((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*b*c* 
e*g*m + a*(c*(f*g + e*h) - 2*d*e*g*(m + 1)) - (b*(2*d*e*g - c*(f*g + e*h)*( 
2*m + 1)) - a*(2*c*f*h - d*(2*m + 1)*(f*g + e*h)))*x - (2*a*d*f*h*m + b*(d* 
(f*g + e*h) - 2*c*f*h*(m + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
 g, h, m}, x] && IntegerQ[2*m] && GtQ[m, 0]
 

rule 2103

Int[(((a_.) + (b_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[ 
(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_S 
ymbol] :> Simp[2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/( 
d*f*h*(2*m + 3))), x] + Simp[1/(d*f*h*(2*m + 3))   Int[((a + b*x)^(m - 1)/( 
Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a 
*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m + 3) - C* 
(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b 
*B*d*f*h*(2*m + 3) + 2*C*(a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x 
^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2 
*m] && GtQ[m, 0]
 

rule 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

3.1.45.4 Maple [A] (verified)

Time = 1.61 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (1555200 x^{5}+3753266 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-8198333 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+6096384 x^{4}[phone] x^{3}-110665104 x^{2}[phone] x [phone]\right )}{435456 x^{3}-1270080 x^{2}+381024 x +181440}\)	\(144\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {293 x \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{12}+\frac {20513 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{216}+\frac {17533 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{1584 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {745303 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{16632 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {25 x^{2} \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{7}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(228\)
risch	\(-\frac {\left (5400 x^{2}+36918 x +143591\right ) \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{1512 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (\frac {17533 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{4752 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {745303 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{49896 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(252\)

input

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

output

1/18144*(2-3*x)^(1/2)*(1+4*x)^(1/2)*(-5+2*x)^(1/2)*(1555200*x^5+3753266*(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))-8198333*(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))+6096384*x^4+11703888*x^3-110665104 
*x^2+40615092*x[phone])/(24*x^3-70*x^2+21*x+10)
 

3.1.45.5 Fricas [C] (verification not implemented)

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

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x} (7+5 x)^2}{\sqrt {-5+2 x}} \, dx=\frac {1}{1512} \, {\left (5400 \, x^{2} + 36918 \, x + 143591\right )} \sqrt {4 \, x + 1} \sqrt {2 \, x - 5} \sqrt {-3 \, x + 2} + \frac {30577063}{46656} \, \sqrt {-6} {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right ) - \frac {8198333}{9072} \, \sqrt {-6} {\rm weierstrassZeta}\left (\frac {847}{108}, \frac {6655}{2916}, {\rm weierstrassPInverse}\left (\frac {847}{108}, \frac {6655}{2916}, x - \frac {35}{36}\right )\right ) \]

input

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

output

1/1512*(5400*x^2 + 36918*x + 143591)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x 
 + 2) + 30577063/46656*sqrt(-6)*weierstrassPInverse(847/108, 6655/2916, x 
- 35/36) - 8198333/9072*sqrt(-6)*weierstrassZeta(847/108, 6655/2916, weier 
strassPInverse(847/108, 6655/2916, x - 35/36))
 

3.1.45.6 Sympy [F]

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

input

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

output

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

3.1.45.7 Maxima [F]

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

input

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

output

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

3.1.45.8 Giac [F]

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

input

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

output

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

3.1.45.9 Mupad [F(-1)]

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

input

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

output

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