3.24.58 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\)
Optimal. Leaf size=122 \[ \frac {19415 \sqrt {1-2 x} \sqrt {5 x+3}}{2744 (3 x+2)}+\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}-\frac {222185 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 122, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used =
{103, 151, 12, 93, 204} \begin {gather*} \frac {19415 \sqrt {1-2 x} \sqrt {5 x+3}}{2744 (3 x+2)}+\frac {185 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}-\frac {222185 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x]),x]
[Out]
(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (185*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^2) + (19415*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (222185*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sq
rt[7])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
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] && IntegerQ[m]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {1}{21} \int \frac {\frac {105}{2}-60 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {185 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {1}{294} \int \frac {\frac {12015}{4}-2775 x}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {185 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {19415 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}+\frac {\int \frac {666555}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2058}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {185 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {19415 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}+\frac {222185 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {185 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {19415 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}+\frac {222185 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {185 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {19415 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {222185 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 74, normalized size = 0.61 \begin {gather*} \frac {\frac {63 \sqrt {1-2 x} \sqrt {5 x+3} \left (19415 x^2+26750 x+9248\right )}{(3 x+2)^3}-222185 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{19208} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x]),x]
[Out]
((63*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9248 + 26750*x + 19415*x^2))/(2 + 3*x)^3 - 222185*Sqrt[7]*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/19208
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.24, size = 106, normalized size = 0.87 \begin {gather*} \frac {99 \sqrt {1-2 x} \left (\frac {4685 (1-2 x)^2}{(5 x+3)^2}+\frac {41720 (1-2 x)}{5 x+3}+109907\right )}{2744 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^3}-\frac {222185 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x]),x]
[Out]
(99*Sqrt[1 - 2*x]*(109907 + (4685*(1 - 2*x)^2)/(3 + 5*x)^2 + (41720*(1 - 2*x))/(3 + 5*x)))/(2744*Sqrt[3 + 5*x]
*(7 + (1 - 2*x)/(3 + 5*x))^3) - (222185*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])
________________________________________________________________________________________
fricas [A] time = 1.59, size = 101, normalized size = 0.83 \begin {gather*} -\frac {222185 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 126 \, {\left (19415 \, x^{2} + 26750 \, x + 9248\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="fricas")
[Out]
-1/38416*(222185*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 126*(19415*x^2 + 26750*x + 9248)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*
x + 8)
________________________________________________________________________________________
giac [B] time = 1.84, size = 310, normalized size = 2.54 \begin {gather*} \frac {44437}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {495 \, \sqrt {10} {\left (937 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 333760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {35170240 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {140680960 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1372 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="giac")
[Out]
44437/76832*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22
))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 495/1372*sqrt(10)*(937*((sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 333760*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 35170240*(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 140680960*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)^3
________________________________________________________________________________________
maple [B] time = 0.02, size = 202, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (5998995 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+11997990 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2446290 \sqrt {-10 x^{2}-x +3}\, x^{2}+7998660 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3370500 \sqrt {-10 x^{2}-x +3}\, x +1777480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1165248 \sqrt {-10 x^{2}-x +3}\right )}{38416 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3*x+2)^4/(-2*x+1)^(1/2)/(5*x+3)^(1/2),x)
[Out]
1/38416*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(5998995*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1
1997990*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+7998660*7^(1/2)*x*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))+2446290*(-10*x^2-x+3)^(1/2)*x^2+1777480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))+3370500*(-10*x^2-x+3)^(1/2)*x+1165248*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^3
________________________________________________________________________________________
maxima [A] time = 1.23, size = 107, normalized size = 0.88 \begin {gather*} \frac {222185}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {185 \, \sqrt {-10 \, x^{2} - x + 3}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {19415 \, \sqrt {-10 \, x^{2} - x + 3}}{2744 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="maxima")
[Out]
222185/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/7*sqrt(-10*x^2 - x + 3)/(27*x^3 + 5
4*x^2 + 36*x + 8) + 185/196*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 19415/2744*sqrt(-10*x^2 - x + 3)/(3*x +
2)
________________________________________________________________________________________
mupad [B] time = 13.94, size = 1273, normalized size = 10.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)^(1/2)),x)
[Out]
((133036146*((1 - 2*x)^(1/2) - 1)^5)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^5) - (52971444*((1 - 2*x)^(1/2) - 1)
^3)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^3) - (885996*((1 - 2*x)^(1/2) - 1))/(5359375*(3^(1/2) - (5*x + 3)^(1/
2))) - (66518073*((1 - 2*x)^(1/2) - 1)^7)/(1071875*(3^(1/2) - (5*x + 3)^(1/2))^7) + (13242861*((1 - 2*x)^(1/2)
- 1)^9)/(85750*(3^(1/2) - (5*x + 3)^(1/2))^9) + (221499*((1 - 2*x)^(1/2) - 1)^11)/(13720*(3^(1/2) - (5*x + 3)
^(1/2))^11) + (6657318*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^2) + (6000759*3^(
1/2)*((1 - 2*x)^(1/2) - 1)^4)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^4) - (181223838*3^(1/2)*((1 - 2*x)^(1/2) - 1
)^6)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^6) + (6000759*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(122500*(3^(1/2) - (5
*x + 3)^(1/2))^8) + (3328659*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(68600*(3^(1/2) - (5*x + 3)^(1/2))^10))/((5856*
((1 - 2*x)^(1/2) - 1)^2)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (4224*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1/2
) - (5*x + 3)^(1/2))^4) - (14776*((1 - 2*x)^(1/2) - 1)^6)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^6) - (1056*((1 -
2*x)^(1/2) - 1)^8)/(625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (366*((1 - 2*x)^(1/2) - 1)^10)/(25*(3^(1/2) - (5*x +
3)^(1/2))^10) + ((1 - 2*x)^(1/2) - 1)^12/(3^(1/2) - (5*x + 3)^(1/2))^12 - (7776*3^(1/2)*((1 - 2*x)^(1/2) - 1)^
3)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^3) + (34704*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(15625*(3^(1/2) - (5*x + 3)
^(1/2))^5) - (17352*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^7) + (972*3^(1/2)*((1 -
2*x)^(1/2) - 1)^9)/(125*(3^(1/2) - (5*x + 3)^(1/2))^9) + (18*3^(1/2)*((1 - 2*x)^(1/2) - 1)^11)/(5*(3^(1/2) -
(5*x + 3)^(1/2))^11) - (576*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(15625*(3^(1/2) - (5*x + 3)^(1/2))) + 64/15625) - (
222185*7^(1/2)*atan(((222185*7^(1/2)*((133311*3^(1/2))/8575 + (133311*((1 - 2*x)^(1/2) - 1))/(17150*(3^(1/2) -
(5*x + 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)
*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*222185i)/38416 - (133311*3^(1/2)*((1 - 2*
x)^(1/2) - 1)^2)/(3430*(3^(1/2) - (5*x + 3)^(1/2))^2)))/38416 + (222185*7^(1/2)*((133311*3^(1/2))/8575 + (1333
11*((1 - 2*x)^(1/2) - 1))/(17150*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3
^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125
)*222185i)/38416 - (133311*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430*(3^(1/2) - (5*x + 3)^(1/2))^2)))/38416)/((7^
(1/2)*((133311*3^(1/2))/8575 + (133311*((1 - 2*x)^(1/2) - 1))/(17150*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*(
(212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3
^(1/2) - (5*x + 3)^(1/2))) - 536/125)*222185i)/38416 - (133311*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430*(3^(1/2)
- (5*x + 3)^(1/2))^2))*222185i)/38416 - (7^(1/2)*((133311*3^(1/2))/8575 + (133311*((1 - 2*x)^(1/2) - 1))/(171
50*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) +
(888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*222185i)/38416 - (133311*3^(
1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430*(3^(1/2) - (5*x + 3)^(1/2))^2))*222185i)/38416 + (1974646969*((1 - 2*x)^(1
/2) - 1)^2)/(3764768*(3^(1/2) - (5*x + 3)^(1/2))^2) + 1974646969/9411920)))/19208
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \sqrt {5 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(2+3*x)**4/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)
[Out]
Integral(1/(sqrt(1 - 2*x)*(3*x + 2)**4*sqrt(5*x + 3)), x)
________________________________________________________________________________________