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

Optimal. Leaf size=122 \[ \frac {18083 \sqrt {1-2 x} \sqrt {5 x+3}}{1176 (3 x+2)}+\frac {173 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}-\frac {68959 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \]

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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 = {99, 151, 12, 93, 204} \begin {gather*} \frac {18083 \sqrt {1-2 x} \sqrt {5 x+3}}{1176 (3 x+2)}+\frac {173 \sqrt {1-2 x} \sqrt {5 x+3}}{84 (3 x+2)^2}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^3}-\frac {68959 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (173*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^2) + (18083*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(1176*(2 + 3*x)) - (68959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[
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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 {\sqrt {1-2 x}}{(2+3 x)^4 \sqrt {3+5 x}} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}-\frac {1}{3} \int \frac {-\frac {31}{2}+20 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}+\frac {173 \sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^2}-\frac {1}{42} \int \frac {-\frac {3721}{4}+865 x}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}+\frac {173 \sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^2}+\frac {18083 \sqrt {1-2 x} \sqrt {3+5 x}}{1176 (2+3 x)}-\frac {1}{294} \int -\frac {206877}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}+\frac {173 \sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^2}+\frac {18083 \sqrt {1-2 x} \sqrt {3+5 x}}{1176 (2+3 x)}+\frac {68959}{784} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}+\frac {173 \sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^2}+\frac {18083 \sqrt {1-2 x} \sqrt {3+5 x}}{1176 (2+3 x)}+\frac {68959}{392} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^3}+\frac {173 \sqrt {1-2 x} \sqrt {3+5 x}}{84 (2+3 x)^2}+\frac {18083 \sqrt {1-2 x} \sqrt {3+5 x}}{1176 (2+3 x)}-\frac {68959 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 74, normalized size = 0.61 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (54249 x^2+74754 x+25856\right )}{(3 x+2)^3}-68959 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25856 + 74754*x + 54249*x^2))/(2 + 3*x)^3 - 68959*Sqrt[7]*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/2744

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IntegrateAlgebraic [C]  time = 0.64, size = 143, normalized size = 1.17 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (54249 \sqrt {5} (5 x+3)^{5/2}+48276 \sqrt {5} (5 x+3)^{3/2}+13331 \sqrt {5} \sqrt {5 x+3}\right )}{392 (3 (5 x+3)+1)^3}-\frac {68959 i \tanh ^{-1}\left (3 \sqrt {\frac {2}{35}} (5 x+3)+\frac {3 i \sqrt {11-2 (5 x+3)} \sqrt {5 x+3}}{\sqrt {35}}+\sqrt {\frac {2}{35}}\right )}{392 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(13331*Sqrt[5]*Sqrt[3 + 5*x] + 48276*Sqrt[5]*(3 + 5*x)^(3/2) + 54249*Sqrt[5]*(3 + 5*x)
^(5/2)))/(392*(1 + 3*(3 + 5*x))^3) - (((68959*I)/392)*ArcTanh[Sqrt[2/35] + 3*Sqrt[2/35]*(3 + 5*x) + ((3*I)*Sqr
t[3 + 5*x]*Sqrt[11 - 2*(3 + 5*x)])/Sqrt[35]])/Sqrt[7]

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fricas [A]  time = 1.30, size = 101, normalized size = 0.83 \begin {gather*} -\frac {68959 \, \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 ) - 14 \, {\left (54249 \, x^{2} + 74754 \, x + 25856\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5488 \, {\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*x)^(1/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/5488*(68959*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)) - 14*(54249*x^2 + 74754*x + 25856)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x
+ 8)

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giac [B]  time = 1.78, size = 315, normalized size = 2.58 \begin {gather*} \frac {11}{54880} \, \sqrt {5} {\left (6269 \, \sqrt {70} \sqrt {2} {\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 {280 \, \sqrt {2} {\left (13331 \, {\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} + 4674880 \, {\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 {491489600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {1965958400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\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}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/54880*sqrt(5)*(6269*sqrt(70)*sqrt(2)*(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)))) + 280*sqrt(2)*(13331*((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 + 4674880*((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)))^3 + 49148960
0*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1965958400*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(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)

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maple [B]  time = 0.02, size = 202, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1861893 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3723786 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+759486 \sqrt {-10 x^{2}-x +3}\, x^{2}+2482524 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1046556 \sqrt {-10 x^{2}-x +3}\, x +551672 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+361984 \sqrt {-10 x^{2}-x +3}\right )}{5488 \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((-2*x+1)^(1/2)/(3*x+2)^4/(5*x+3)^(1/2),x)

[Out]

1/5488*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(1861893*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+37
23786*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+2482524*7^(1/2)*x*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))+759486*(-10*x^2-x+3)^(1/2)*x^2+551672*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))+1046556*(-10*x^2-x+3)^(1/2)*x+361984*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^3

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maxima [A]  time = 1.05, size = 107, normalized size = 0.88 \begin {gather*} \frac {68959}{5488} \, \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}}{3 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {173 \, \sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {18083 \, \sqrt {-10 \, x^{2} - x + 3}}{1176 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

68959/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/3*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*
x^2 + 36*x + 8) + 173/84*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 18083/1176*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [B]  time = 13.19, size = 1273, normalized size = 10.43

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((41325734*((1 - 2*x)^(1/2) - 1)^5)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^5) - (16452204*((1 - 2*x)^(1/2) - 1)^3
)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^3) - (275444*((1 - 2*x)^(1/2) - 1))/(765625*(3^(1/2) - (5*x + 3)^(1/2)))
- (20662867*((1 - 2*x)^(1/2) - 1)^7)/(153125*(3^(1/2) - (5*x + 3)^(1/2))^7) + (4113051*((1 - 2*x)^(1/2) - 1)^
9)/(12250*(3^(1/2) - (5*x + 3)^(1/2))^9) + (68861*((1 - 2*x)^(1/2) - 1)^11)/(1960*(3^(1/2) - (5*x + 3)^(1/2))^
11) + (2068378*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (1863097*3^(1/2)*((1
- 2*x)^(1/2) - 1)^4)/(109375*(3^(1/2) - (5*x + 3)^(1/2))^4) - (56261214*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(7656
25*(3^(1/2) - (5*x + 3)^(1/2))^6) + (1863097*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(17500*(3^(1/2) - (5*x + 3)^(1/2
))^8) + (1034189*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(9800*(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) - (68959*7^(1/2)
*atan(((68959*7^(1/2)*((206877*3^(1/2))/6125 + (206877*((1 - 2*x)^(1/2) - 1))/(12250*(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)*68959i)/5488 - (206877*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/
(2450*(3^(1/2) - (5*x + 3)^(1/2))^2)))/5488 + (68959*7^(1/2)*((206877*3^(1/2))/6125 + (206877*((1 - 2*x)^(1/2)
- 1))/(12250*(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)*68959i)/5488 - (2
06877*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(2450*(3^(1/2) - (5*x + 3)^(1/2))^2)))/5488)/((7^(1/2)*((206877*3^(1/2)
)/6125 + (206877*((1 - 2*x)^(1/2) - 1))/(12250*(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)*68959i)/5488 - (206877*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(2450*(3^(1/2) - (5*x + 3)^(1/2))^2))*
68959i)/5488 - (7^(1/2)*((206877*3^(1/2))/6125 + (206877*((1 - 2*x)^(1/2) - 1))/(12250*(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)*68959i)/5488 - (206877*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2
)/(2450*(3^(1/2) - (5*x + 3)^(1/2))^2))*68959i)/5488 + (4755343681*((1 - 2*x)^(1/2) - 1)^2)/(1920800*(3^(1/2)
- (5*x + 3)^(1/2))^2) + 4755343681/4802000)))/2744

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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