\(\int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\) [2492]

Optimal result

Integrand size = 24, antiderivative size = 50 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10 \sqrt {10}} \]

[Out]

37/100*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-3/10*(1-2*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {81, 56, 222} \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10 \sqrt {10}}-\frac {3}{10} \sqrt {1-2 x} \sqrt {5 x+3} \]

[In]

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

[Out]

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

Rule 56

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 81

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

Rule 222

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*} \text {integral}& = -\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37}{20} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10 \sqrt {5}} \\ & = -\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10 \sqrt {10}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.26 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-30 \sqrt {1-2 x} (3+5 x)-37 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{100 \sqrt {3+5 x}} \]

[In]

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

[Out]

(-30*Sqrt[1 - 2*x]*(3 + 5*x) - 37*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(100*Sqrt[3 + 5*x])

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-60 \sqrt {-10 x^{2}-x +3}\right )}{200 \sqrt {-10 x^{2}-x +3}}\)	\(55\)
risch	\(\frac {3 \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{200 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(88\)

[In]

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

[Out]

1/200*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(37*10^(1/2)*arcsin(20/11*x+1/11)-60*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2
)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.14 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {37}{200} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {3}{10} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-37/200*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 3/10*sqrt(5*
x + 3)*sqrt(-2*x + 1)

Sympy [F]

\[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {3 x + 2}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.52 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {37}{200} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {3}{10} \, \sqrt {-10 \, x^{2} - x + 3} \]

[In]

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

[Out]

-37/200*sqrt(10)*arcsin(-20/11*x - 1/11) - 3/10*sqrt(-10*x^2 - x + 3)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {1}{100} \, \sqrt {5} {\left (37 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - 6 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

[In]

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

[Out]

1/100*sqrt(5)*(37*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 6*sqrt(5*x + 3)*sqrt(-10*x + 5))

Mupad [B] (verification not implemented)

Time = 4.32 (sec) , antiderivative size = 220, normalized size of antiderivative = 4.40 \[ \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {\frac {3\,{\left (\sqrt {1-2\,x}-1\right )}^3}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {6\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {24\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}}{\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^2}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {4}{25}}-\frac {3\,\sqrt {10}\,\left (\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )+\frac {40\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}{5\,\left (\sqrt {1-2\,x}-1\right )}\right )}{3}\right )}{50} \]

[In]

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

[Out]

- ((3*((1 - 2*x)^(1/2) - 1)^3)/(25*(3^(1/2) - (5*x + 3)^(1/2))^3) - (6*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) -
(5*x + 3)^(1/2))) + (24*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2))/((4*((1 - 2*x)^(1
/2) - 1)^2)/(5*(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 + 4/25)
- (3*10^(1/2)*(atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))) + (40*atan((10^(1/2)*(3^
(1/2) - (5*x + 3)^(1/2)))/(5*((1 - 2*x)^(1/2) - 1))))/3))/50