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

Optimal. Leaf size=151 \[ \frac {625115 \sqrt {1-2 x} \sqrt {5 x+3}}{197568 (3 x+2)}+\frac {6005 \sqrt {1-2 x} \sqrt {5 x+3}}{14112 (3 x+2)^2}+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{504 (3 x+2)^3}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}-\frac {794365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \]

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Rubi [A]  time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {97, 151, 12, 93, 204} \begin {gather*} \frac {625115 \sqrt {1-2 x} \sqrt {5 x+3}}{197568 (3 x+2)}+\frac {6005 \sqrt {1-2 x} \sqrt {5 x+3}}{14112 (3 x+2)^2}+\frac {37 \sqrt {1-2 x} \sqrt {5 x+3}}{504 (3 x+2)^3}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}-\frac {794365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(504*(2 + 3*x)^3) + (6005*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (625115*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (7
94365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*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 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 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} \sqrt {3+5 x}}{(2+3 x)^5} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {-\frac {1}{2}-10 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {1}{252} \int \frac {\frac {1015}{4}-370 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {6005 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {\int \frac {\frac {128305}{8}-\frac {30025 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{3528}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {6005 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {625115 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {\int \frac {7149285}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{24696}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {6005 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {625115 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {794365 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {6005 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {625115 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}+\frac {794365 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{12 (2+3 x)^4}+\frac {37 \sqrt {1-2 x} \sqrt {3+5 x}}{504 (2+3 x)^3}+\frac {6005 \sqrt {1-2 x} \sqrt {3+5 x}}{14112 (2+3 x)^2}+\frac {625115 \sqrt {1-2 x} \sqrt {3+5 x}}{197568 (2+3 x)}-\frac {794365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.52 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (1875345 x^3+3834760 x^2+2617388 x+594416\right )}{(3 x+2)^4}-794365 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(594416 + 2617388*x + 3834760*x^2 + 1875345*x^3))/(2 + 3*x)^4 - 794365*Sqrt[7]
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/153664

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IntegrateAlgebraic [A]  time = 0.30, size = 122, normalized size = 0.81 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {6565 (1-2 x)^3}{(5 x+3)^3}-\frac {197365 (1-2 x)^2}{(5 x+3)^2}-\frac {1171149 (1-2 x)}{5 x+3}-2251795\right )}{21952 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^4}-\frac {794365 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*Sqrt[1 - 2*x]*(-2251795 + (6565*(1 - 2*x)^3)/(3 + 5*x)^3 - (197365*(1 - 2*x)^2)/(3 + 5*x)^2 - (1171149*(
1 - 2*x))/(3 + 5*x)))/(21952*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^4) - (794365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7
]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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fricas [A]  time = 1.70, size = 116, normalized size = 0.77 \begin {gather*} -\frac {794365 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (1875345 \, x^{3} + 3834760 \, x^{2} + 2617388 \, x + 594416\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/307328*(794365*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1875345*x^3 + 3834760*x^2 + 2617388*x + 594416)*sqrt(5*x + 3)*sqrt(-
2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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giac [B]  time = 2.34, size = 368, normalized size = 2.44 \begin {gather*} \frac {158873}{614656} \, \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 {605 \, \sqrt {10} {\left (1313 \, {\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 )}^{7} - 1578920 \, {\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} - 374767680 \, {\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 {28822976000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {115291904000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{10976 \, {\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 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

158873/614656*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)))) - 605/10976*sqrt(10)*(1313*((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)))^7 - 1578920*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 374767680*
((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
- 28822976000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 115291904000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*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)^4

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maple [B]  time = 0.02, size = 250, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (64343565 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8321824 \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/307328*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(64343565*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4
[phone]*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^2*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+76259040*7^(1/2)*x*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2+12709840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x+8321824*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^4

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maxima [A]  time = 1.16, size = 157, normalized size = 1.04 \begin {gather*} \frac {794365}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {32825}{16464} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {185 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{392 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {19695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{10976 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {242905 \, \sqrt {-10 \, x^{2} - x + 3}}{65856 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

794365/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 32825/16464*sqrt(-10*x^2 - x + 3) +
3/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 185/392*(-10*x^2 - x + 3)^(3/2)/(27*x^
3 + 54*x^2 + 36*x + 8) + 19695/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 242905/65856*sqrt(-10*x^2 -
x + 3)/(3*x + 2)

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mupad [B]  time = 17.40, size = 1509, normalized size = 9.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2430615287*((1 - 2*x)^(1/2) - 1)^7)/(76562500*(3^(1/2) - (5*x + 3)^(1/2))^7) - (4227392*((1 - 2*x)^(1/2) - 1
)^3)/(2734375*(3^(1/2) - (5*x + 3)^(1/2))^3) - (29304073*((1 - 2*x)^(1/2) - 1)^5)/(7656250*(3^(1/2) - (5*x + 3
)^(1/2))^5) - (1572266*((1 - 2*x)^(1/2) - 1))/(133984375*(3^(1/2) - (5*x + 3)^(1/2))) - (2430615287*((1 - 2*x)
^(1/2) - 1)^9)/(30625000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (29304073*((1 - 2*x)^(1/2) - 1)^11)/(490000*(3^(1/2)
- (5*x + 3)^(1/2))^11) + (132106*((1 - 2*x)^(1/2) - 1)^13)/(875*(3^(1/2) - (5*x + 3)^(1/2))^13) + (786133*((1
- 2*x)^(1/2) - 1)^15)/(109760*(3^(1/2) - (5*x + 3)^(1/2))^15) + (474659*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(382
8125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (7936034*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(2734375*(3^(1/2) - (5*x + 3)^
(1/2))^4) - (402724691*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(38281250*(3^(1/2) - (5*x + 3)^(1/2))^6) + (6732597583
*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(267968750*(3^(1/2) - (5*x + 3)^(1/2))^8) - (402724691*3^(1/2)*((1 - 2*x)^(1
/2) - 1)^10)/(6125000*(3^(1/2) - (5*x + 3)^(1/2))^10) + (3968017*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(35000*(3^(
1/2) - (5*x + 3)^(1/2))^12) + (474659*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(15680*(3^(1/2) - (5*x + 3)^(1/2))^14)
)/((45056*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (294784*((1 - 2*x)^(1/2) - 1)^4)/(
390625*(3^(1/2) - (5*x + 3)^(1/2))^4) - (1921024*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^
6) + (5828656*((1 - 2*x)^(1/2) - 1)^8)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^8) - (480256*((1 - 2*x)^(1/2) - 1)^
10)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^10) + (18424*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2))
^12) + (704*((1 - 2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/2))^14) + ((1 - 2*x)^(1/2) - 1)^16/(3^(1/2)
- (5*x + 3)^(1/2))^16 - (21504*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^3) + (48384
*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^5) - (4992*3^(1/2)*((1 - 2*x)^(1/2) - 1)^
7)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^7) + (2496*3^(1/2)*((1 - 2*x)^(1/2) - 1)^9)/(78125*(3^(1/2) - (5*x + 3)
^(1/2))^9) - (6048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^11)/(625*(3^(1/2) - (5*x + 3)^(1/2))^11) + (672*3^(1/2)*((1 -
2*x)^(1/2) - 1)^13)/(25*(3^(1/2) - (5*x + 3)^(1/2))^13) + (24*3^(1/2)*((1 - 2*x)^(1/2) - 1)^15)/(5*(3^(1/2) -
(5*x + 3)^(1/2))^15) - (3072*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(390625*(3^(1/2) - (5*x + 3)^(1/2))) + 256/390625
) - (794365*7^(1/2)*atan(((794365*7^(1/2)*((476619*3^(1/2))/68600 + (476619*((1 - 2*x)^(1/2) - 1))/(137200*(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)*794365i)/307328 - (476619*3^(1/2)*
((1 - 2*x)^(1/2) - 1)^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2)))/307328 + (794365*7^(1/2)*((476619*3^(1/2))/68
600 + (476619*((1 - 2*x)^(1/2) - 1))/(137200*(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)*794365i)/307328 - (476619*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2)
))/307328)/((7^(1/2)*((476619*3^(1/2))/68600 + (476619*((1 - 2*x)^(1/2) - 1))/(137200*(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)*794365i)/307328 - (476619*3^(1/2)*((1 - 2*x)^(1/2) - 1)
^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2))*794365i)/307328 - (7^(1/2)*((476619*3^(1/2))/68600 + (476619*((1 -
2*x)^(1/2) - 1))/(137200*(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)*794365
i)/307328 - (476619*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(27440*(3^(1/2) - (5*x + 3)^(1/2))^2))*794365i)/307328 +
(25240630129*((1 - 2*x)^(1/2) - 1)^2)/(240945152*(3^(1/2) - (5*x + 3)^(1/2))^2) + 25240630129/602362880)))/153
664

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

[Out]

Timed out

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