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

Optimal. Leaf size=91 \[ -\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}-\frac {2}{9} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {37 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9 \sqrt {7}} \]

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Rubi [A]  time = 0.03, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 157, 54, 216, 93, 204} \begin {gather*} -\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)}-\frac {2}{9} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {37 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)) - (2*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 - (37*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9*Sqrt[7])

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {-\frac {1}{2}-10 x}{\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)}-\frac {10}{9} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {37}{18} \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)}+\frac {37}{9} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {1}{9} \left (4 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {2}{9} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {37 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{9 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 120, normalized size = 1.32 \begin {gather*} \frac {-21 \sqrt {-(2 x-1)^2} \sqrt {5 x+3}-37 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+14 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{63 \sqrt {2 x-1} (3 x+2)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-21*Sqrt[-(-1 + 2*x)^2]*Sqrt[3 + 5*x] + 14*Sqrt[10 - 20*x]*(2 + 3*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 37*
(2 + 3*x)*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(63*Sqrt[-1 + 2*x]*(2 + 3*x))

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IntegrateAlgebraic [A]  time = 0.13, size = 110, normalized size = 1.21 \begin {gather*} -\frac {11 \sqrt {1-2 x}}{3 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )}+\frac {2}{9} \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {37 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{9 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Sqrt[1 - 2*x])/(3*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))) + (2*Sqrt[10]*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])
/Sqrt[3 + 5*x]])/9 - (37*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9*Sqrt[7])

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fricas [A]  time = 1.31, size = 116, normalized size = 1.27 \begin {gather*} -\frac {37 \, \sqrt {7} {\left (3 \, x + 2\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 \, \sqrt {10} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{126 \, {\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)^2,x, algorithm="fricas")

[Out]

-1/126*(37*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) -
14*sqrt(10)*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*sqrt
(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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giac [B]  time = 1.39, size = 260, normalized size = 2.86 \begin {gather*} \frac {37}{1260} \, \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 {1}{9} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {22 \, \sqrt {10} {\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 \, {\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 )}} \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)^2,x, algorithm="giac")

[Out]

37/1260*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)))) - 1/9*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 22/3*sqrt(10)*((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)))/(((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)

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maple [A]  time = 0.01, size = 131, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (42 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-111 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+28 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-74 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+42 \sqrt {-10 x^{2}-x +3}\right )}{126 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \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)^2,x)

[Out]

-1/126*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(42*10^(1/2)*arcsin(20/11*x+1/11)*x-111*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x+28*10^(1/2)*arcsin(20/11*x+1/11)-74*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))+42*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)

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maxima [A]  time = 1.16, size = 61, normalized size = 0.67 \begin {gather*} -\frac {1}{9} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {37}{126} \, \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 (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)^2,x, algorithm="maxima")

[Out]

-1/9*sqrt(10)*arcsin(20/11*x + 1/11) + 37/126*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/3*
sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [B]  time = 5.61, size = 752, normalized size = 8.26 \begin {gather*} \frac {37\,\sqrt {7}\,\mathrm {atan}\left (\frac {31139252096\,\sqrt {7}\,\left (\sqrt {1-2\,x}-1\right )}{38759765625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}-\frac {901743872\,\sqrt {3}\,\sqrt {7}}{4306640625\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}+\frac {450871936\,\sqrt {3}\,\sqrt {7}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{861328125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {1076274944\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1550390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {14849685376\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{12919921875\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2152549888}{7751953125}\right )}\right )}{63}-\frac {4\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{9}-\frac {2\,{\left (\sqrt {1-2\,x}-1\right )}^3}{3\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}+\frac {4\,\left (\sqrt {1-2\,x}-1\right )}{15\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )}-\frac {37\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{75\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {14\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {6\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {12\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{25\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {4}{25}\right )} \end {gather*}

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)^2,x)

[Out]

(37*7^(1/2)*atan((31139252096*7^(1/2)*((1 - 2*x)^(1/2) - 1))/(38759765625*(3^(1/2) - (5*x + 3)^(1/2))*((107627
4944*((1 - 2*x)^(1/2) - 1)^2)/(1550390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (14849685376*3^(1/2)*((1 - 2*x)^(1/
2) - 1))/(12919921875*(3^(1/2) - (5*x + 3)^(1/2))) - 2152549888/7751953125)) - (901743872*3^(1/2)*7^(1/2))/(43
06640625*((1076274944*((1 - 2*x)^(1/2) - 1)^2)/(1550390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (14849685376*3^(1/
2)*((1 - 2*x)^(1/2) - 1))/(12919921875*(3^(1/2) - (5*x + 3)^(1/2))) - 2152549888/7751953125)) + (450871936*3^(
1/2)*7^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(861328125*(3^(1/2) - (5*x + 3)^(1/2))^2*((1076274944*((1 - 2*x)^(1/2) -
1)^2)/(1550390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (14849685376*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(12919921875*(
3^(1/2) - (5*x + 3)^(1/2))) - 2152549888/7751953125))))/63 - (4*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))
/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/9 - (2*((1 - 2*x)^(1/2) - 1)^3)/(3*(3^(1/2) - (5*x + 3)^(1/2))^3*((14*((1 -
2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^(1/2) - (5*x + 3)^(1/2))^4
+ (6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(5*(3^(1/2) - (5*x + 3)^(1/2))^3) - (12*3^(1/2)*((1 - 2*x)^(1/2) - 1))/
(25*(3^(1/2) - (5*x + 3)^(1/2))) + 4/25)) + (4*((1 - 2*x)^(1/2) - 1))/(15*(3^(1/2) - (5*x + 3)^(1/2))*((14*((1
- 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^(1/2) - (5*x + 3)^(1/2))
^4 + (6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(5*(3^(1/2) - (5*x + 3)^(1/2))^3) - (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)
)/(25*(3^(1/2) - (5*x + 3)^(1/2))) + 4/25)) - (37*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(75*(3^(1/2) - (5*x + 3)^(1
/2))^2*((14*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^(1/2) - (
5*x + 3)^(1/2))^4 + (6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(5*(3^(1/2) - (5*x + 3)^(1/2))^3) - (12*3^(1/2)*((1 -
2*x)^(1/2) - 1))/(25*(3^(1/2) - (5*x + 3)^(1/2))) + 4/25))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{2}}\, dx \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)**2,x)

[Out]

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

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