3.23.70 \(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\) [2270]

Optimal. Leaf size=122 \[ -\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}-\frac {4477 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}} \]

[Out]

1/7*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^3-4477/2744*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+37
/28*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-407/392*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.02, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \begin {gather*} -\frac {4477 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}}+\frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{28 (3 x+2)^2}+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}-\frac {407 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-407*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) + ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(7*(2 + 3*x)^3) + (37*S
qrt[1 - 2*x]*(3 + 5*x)^(3/2))/(28*(2 + 3*x)^2) - (4477*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqr
t[7])

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 96

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[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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 {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx &=\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37}{14} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}+\frac {407}{56} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}+\frac {4477}{784} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}+\frac {4477}{392} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}-\frac {4477 \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 = 0.22, size = 74, normalized size = 0.61 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (1648+4902 x+3547 x^2\right )}{(2+3 x)^3}-4477 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1648 + 4902*x + 3547*x^2))/(2 + 3*x)^3 - 4477*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/2744

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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 = 0.12, size = 202, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (3547 x^{2}+4902 x +1648\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{392 \left (2+3 x \right )^{3} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {4477 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{5488 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(124\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (120879 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+241758 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+161172 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +49658 x^{2} \sqrt {-10 x^{2}-x +3}+35816 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+68628 x \sqrt {-10 x^{2}-x +3}+23072 \sqrt {-10 x^{2}-x +3}\right )}{5488 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{3}}\)	\(202\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5488*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(120879*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2417
58*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+161172*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x+49658*x^2*(-10*x^2-x+3)^(1/2)+35816*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))+68628*x*(-10*x^2-x+3)^(1/2)+23072*(-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 = 0.50, size = 121, normalized size = 0.99 \begin {gather*} \frac {4477}{5488} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {185}{294} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1369 \, \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)*(3+5*x)^(1/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

4477/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 185/294*sqrt(-10*x^2 - x + 3) + 1/7*(-10
*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 111/196*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1369/1
176*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]
time = 0.58, size = 101, normalized size = 0.83 \begin {gather*} -\frac {4477 \, \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 (3547 \, x^{2} + 4902 \, x + 1648\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)*(3+5*x)^(1/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/5488*(4477*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*(3547*x^2 + 4902*x + 1648)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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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 )^{4}}\, 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)**4,x)

[Out]

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

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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 = 0.62, size = 310, normalized size = 2.54 \begin {gather*} \frac {4477}{54880} \, \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 {121 \, \sqrt {10} {\left (37 \, {\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} - 24640 \, {\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 {2900800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {11603200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{196 \, {\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*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

4477/54880*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)))) - 121/196*sqrt(10)*(37*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 24640*((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 - 2900800*(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11603200*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

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Mupad [B]
time = 12.76, size = 1273, normalized size = 10.43 \begin {gather*} \frac {\frac {2630142\,{\left (\sqrt {1-2\,x}-1\right )}^5}{765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {1060932\,{\left (\sqrt {1-2\,x}-1\right )}^3}{765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {16732\,\left (\sqrt {1-2\,x}-1\right )}{765625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {1315071\,{\left (\sqrt {1-2\,x}-1\right )}^7}{153125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {265233\,{\left (\sqrt {1-2\,x}-1\right )}^9}{12250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {4183\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{1960\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {131174\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {121551\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{109375\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}-\frac {3688612\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{765625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {121551\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{17500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {65587\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{9800\,{\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 {4477\,\sqrt {7}\,\mathrm {atan}\left (\frac {\frac {4477\,\sqrt {7}\,\left (\frac {13431\,\sqrt {3}}{6125}+\frac {13431\,\left (\sqrt {1-2\,x}-1\right )}{12250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {13431\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{2450\,{\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 )\,4477{}\mathrm {i}}{5488}\right )}{5488}+\frac {4477\,\sqrt {7}\,\left (\frac {13431\,\sqrt {3}}{6125}+\frac {13431\,\left (\sqrt {1-2\,x}-1\right )}{12250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {13431\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{2450\,{\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 )\,4477{}\mathrm {i}}{5488}\right )}{5488}}{\frac {20043529\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1920800\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {20043529}{4802000}+\frac {\sqrt {7}\,\left (\frac {13431\,\sqrt {3}}{6125}+\frac {13431\,\left (\sqrt {1-2\,x}-1\right )}{12250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {13431\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{2450\,{\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 )\,4477{}\mathrm {i}}{5488}\right )\,4477{}\mathrm {i}}{5488}-\frac {\sqrt {7}\,\left (\frac {13431\,\sqrt {3}}{6125}+\frac {13431\,\left (\sqrt {1-2\,x}-1\right )}{12250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {13431\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{2450\,{\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 )\,4477{}\mathrm {i}}{5488}\right )\,4477{}\mathrm {i}}{5488}}\right )}{2744} \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)^4,x)

[Out]

((2630142*((1 - 2*x)^(1/2) - 1)^5)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^5) - (1060932*((1 - 2*x)^(1/2) - 1)^3)/
(765625*(3^(1/2) - (5*x + 3)^(1/2))^3) - (16732*((1 - 2*x)^(1/2) - 1))/(765625*(3^(1/2) - (5*x + 3)^(1/2))) -
(1315071*((1 - 2*x)^(1/2) - 1)^7)/(153125*(3^(1/2) - (5*x + 3)^(1/2))^7) + (265233*((1 - 2*x)^(1/2) - 1)^9)/(1
2250*(3^(1/2) - (5*x + 3)^(1/2))^9) + (4183*((1 - 2*x)^(1/2) - 1)^11)/(1960*(3^(1/2) - (5*x + 3)^(1/2))^11) +
(131174*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(765625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (121551*3^(1/2)*((1 - 2*x)^(
1/2) - 1)^4)/(109375*(3^(1/2) - (5*x + 3)^(1/2))^4) - (3688612*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(765625*(3^(1/
2) - (5*x + 3)^(1/2))^6) + (121551*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(17500*(3^(1/2) - (5*x + 3)^(1/2))^8) + (6
5587*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) - (5
76*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(15625*(3^(1/2) - (5*x + 3)^(1/2))) + 64/15625) - (4477*7^(1/2)*atan(((4477*
7^(1/2)*((13431*3^(1/2))/6125 + (13431*((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)*4477i)/5488 - (13431*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(2450*(3^(1/2) - (
5*x + 3)^(1/2))^2)))/5488 + (4477*7^(1/2)*((13431*3^(1/2))/6125 + (13431*((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)*4477i)/5488 - (13431*3^(1/2)*((1 - 2*x
)^(1/2) - 1)^2)/(2450*(3^(1/2) - (5*x + 3)^(1/2))^2)))/5488)/((7^(1/2)*((13431*3^(1/2))/6125 + (13431*((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)*4477i)/54
88 - (13431*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(2450*(3^(1/2) - (5*x + 3)^(1/2))^2))*4477i)/5488 - (7^(1/2)*((13
431*3^(1/2))/6125 + (13431*((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)*4477i)/5488 - (13431*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(2450*(3^(1/2) - (5*x + 3)^(1/
2))^2))*4477i)/5488 + (20043529*((1 - 2*x)^(1/2) - 1)^2)/(1920800*(3^(1/2) - (5*x + 3)^(1/2))^2) + 20043529/48
02000)))/2744

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