3.25.66 \(\int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\) [2466]

Optimal. Leaf size=122 \[ -\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]

[Out]

-15235/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/21*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^
3+25/588*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+3895/8232*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.03, 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 = {101, 156, 12, 95, 210} \begin {gather*} -\frac {15235 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}}+\frac {3895 \sqrt {1-2 x} \sqrt {5 x+3}}{8232 (3 x+2)}+\frac {25 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^2}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x)^3 + (25*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^2) + (3895*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (15235*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqr
t[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 95

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 101

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 156

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] && ILtQ[m, -1]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^4} \, dx &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {1}{21} \int \frac {\frac {35}{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}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {1}{294} \int \frac {\frac {965}{4}-125 x}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {\int \frac {45705}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2058}\\ &=-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {15235 \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}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}+\frac {15235 \text {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}}{21 (2+3 x)^3}+\frac {25 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^2}+\frac {3895 \sqrt {1-2 x} \sqrt {3+5 x}}{8232 (2+3 x)}-\frac {15235 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 1.89, size = 142, normalized size = 1.16 \begin {gather*} \frac {5 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (5296+15930 x+11685 x^2\right )}{5 (2+3 x)^3}+3047 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+3047 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{19208} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5296 + 15930*x + 11685*x^2))/(5*(2 + 3*x)^3) + 3047*Sqrt[7]*ArcTan[(Sqrt[2
*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 3047*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34
+ Sqrt[1155]]*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/19208

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(95)=190\).
time = 0.08, size = 202, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (11685 x^{2}+15930 x +5296\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2744 \left (2+3 x \right )^{3} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {15235 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{38416 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(124\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (411345 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+822690 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+548460 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +163590 x^{2} \sqrt {-10 x^{2}-x +3}+121880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+223020 x \sqrt {-10 x^{2}-x +3}+74144 \sqrt {-10 x^{2}-x +3}\right )}{38416 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/38416*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(411345*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+822
690*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+548460*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x+163590*x^2*(-10*x^2-x+3)^(1/2)+121880*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))+223020*x*(-10*x^2-x+3)^(1/2)+74144*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]
time = 0.62, size = 107, normalized size = 0.88 \begin {gather*} \frac {15235}{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}}{21 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {25 \, \sqrt {-10 \, x^{2} - x + 3}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {3895 \, \sqrt {-10 \, x^{2} - x + 3}}{8232 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15235/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/21*sqrt(-10*x^2 - x + 3)/(27*x^3 + 5
4*x^2 + 36*x + 8) + 25/588*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 3895/8232*sqrt(-10*x^2 - x + 3)/(3*x + 2
)

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Fricas [A]
time = 0.41, size = 101, normalized size = 0.83 \begin {gather*} -\frac {15235 \, \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 (11685 \, x^{2} + 15930 \, x + 5296\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((3+5*x)^(1/2)/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/38416*(15235*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*(11685*x^2 + 15930*x + 5296)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x
+ 8)

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Sympy [F(-1)] Timed out
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((3+5*x)**(1/2)/(2+3*x)**4/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (95) = 190\).
time = 2.23, size = 310, normalized size = 2.54 \begin {gather*} \frac {3047}{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 {55 \, \sqrt {10} {\left (277 \, {\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} - 159040 \, {\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 {20713280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {82853120 \, \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((3+5*x)^(1/2)/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

3047/76832*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)))) - 55/1372*sqrt(10)*(277*((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 - 159040*((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 - 20713280*(sqrt(2)
*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 82853120*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((s
qrt(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 + 2
80)^3

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Mupad [B]
time = 13.69, size = 1273, normalized size = 10.43 \begin {gather*} \frac {\frac {8498458\,{\left (\sqrt {1-2\,x}-1\right )}^5}{5359375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {3429372\,{\left (\sqrt {1-2\,x}-1\right )}^3}{5359375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {52708\,\left (\sqrt {1-2\,x}-1\right )}{5359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {4249229\,{\left (\sqrt {1-2\,x}-1\right )}^7}{1071875\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {857343\,{\left (\sqrt {1-2\,x}-1\right )}^9}{85750\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {13177\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{13720\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {418634\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{5359375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {399977\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}-\frac {12159864\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{5359375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {399977\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{122500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {209317\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{68600\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {5856\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {4224\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}-\frac {14776\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}-\frac {1056\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {366\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}-\frac {7776\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {34704\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^5}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {17352\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^7}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {972\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^9}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {18\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}-\frac {576\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{15625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {64}{15625}}-\frac {15235\,\sqrt {7}\,\mathrm {atan}\left (\frac {\frac {15235\,\sqrt {7}\,\left (\frac {9141\,\sqrt {3}}{8575}+\frac {9141\,\left (\sqrt {1-2\,x}-1\right )}{17150\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {9141\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3430\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,15235{}\mathrm {i}}{38416}\right )}{38416}+\frac {15235\,\sqrt {7}\,\left (\frac {9141\,\sqrt {3}}{8575}+\frac {9141\,\left (\sqrt {1-2\,x}-1\right )}{17150\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {9141\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3430\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,15235{}\mathrm {i}}{38416}\right )}{38416}}{\frac {9284209\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3764768\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {9284209}{9411920}+\frac {\sqrt {7}\,\left (\frac {9141\,\sqrt {3}}{8575}+\frac {9141\,\left (\sqrt {1-2\,x}-1\right )}{17150\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {9141\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3430\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,15235{}\mathrm {i}}{38416}\right )\,15235{}\mathrm {i}}{38416}-\frac {\sqrt {7}\,\left (\frac {9141\,\sqrt {3}}{8575}+\frac {9141\,\left (\sqrt {1-2\,x}-1\right )}{17150\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {9141\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3430\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,15235{}\mathrm {i}}{38416}\right )\,15235{}\mathrm {i}}{38416}}\right )}{19208} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((8498458*((1 - 2*x)^(1/2) - 1)^5)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^5) - (3429372*((1 - 2*x)^(1/2) - 1)^3)
/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^3) - (52708*((1 - 2*x)^(1/2) - 1))/(5359375*(3^(1/2) - (5*x + 3)^(1/2)))
- (4249229*((1 - 2*x)^(1/2) - 1)^7)/(1071875*(3^(1/2) - (5*x + 3)^(1/2))^7) + (857343*((1 - 2*x)^(1/2) - 1)^9
)/(85750*(3^(1/2) - (5*x + 3)^(1/2))^9) + (13177*((1 - 2*x)^(1/2) - 1)^11)/(13720*(3^(1/2) - (5*x + 3)^(1/2))^
11) + (418634*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5359375*(3^(1/2) - (5*x + 3)^(1/2))^2) + (399977*3^(1/2)*((1 -
2*x)^(1/2) - 1)^4)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^4) - (12159864*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(53593
75*(3^(1/2) - (5*x + 3)^(1/2))^6) + (399977*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(122500*(3^(1/2) - (5*x + 3)^(1/2
))^8) + (209317*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) - (15235*7^(1/2)
*atan(((15235*7^(1/2)*((9141*3^(1/2))/8575 + (9141*((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)*15235i)/38416 - (9141*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430
*(3^(1/2) - (5*x + 3)^(1/2))^2)))/38416 + (15235*7^(1/2)*((9141*3^(1/2))/8575 + (9141*((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)*15235i)/38416 - (9141*3^(
1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430*(3^(1/2) - (5*x + 3)^(1/2))^2)))/38416)/((7^(1/2)*((9141*3^(1/2))/8575 + (
9141*((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))) - 536/1
25)*15235i)/38416 - (9141*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430*(3^(1/2) - (5*x + 3)^(1/2))^2))*15235i)/38416
- (7^(1/2)*((9141*3^(1/2))/8575 + (9141*((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)*15235i)/38416 - (9141*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(3430*(3^(1/2)
- (5*x + 3)^(1/2))^2))*15235i)/38416 + (9284209*((1 - 2*x)^(1/2) - 1)^2)/(3764768*(3^(1/2) - (5*x + 3)^(1/2))^
2) + 9284209/9411920)))/19208

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