\(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx\) [2267]

Optimal result

Integrand size = 26, antiderivative size = 84 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9 \sqrt {10}}+\frac {2}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]

[Out]

2/9*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+37/90*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
+1/3*(1-2*x)^(1/2)*(3+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {103, 163, 56, 222, 95, 210} \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {37 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{9 \sqrt {10}}+\frac {2}{9} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{3} \sqrt {1-2 x} \sqrt {5 x+3} \]

[In]

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

[Out]

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

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 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 103

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

Rule 163

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 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])

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 {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{3} \int \frac {-10-\frac {37 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {7}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {37}{18} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {14}{9} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {37 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{9 \sqrt {5}} \\ & = \frac {1}{3} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{9 \sqrt {10}}+\frac {2}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {1}{90} \left (30 \sqrt {1-2 x} \sqrt {3+5 x}-37 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+20 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \]

[In]

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] - 37*Sqrt[10]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 20*Sqrt[7]*ArcTan[Sqrt[1
- 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/90

Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (20 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-60 \sqrt {-10 x^{2}-x +3}\right )}{180 \sqrt {-10 x^{2}-x +3}}\)	\(83\)
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{3 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{180}+\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{9}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(121\)

[In]

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

[Out]

-1/180*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(20*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(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 = 97, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {1}{9} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {37}{180} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + \frac {1}{3} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

1/9*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 37/180*sqrt(10)*a
rctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 1/3*sqrt(5*x + 3)*sqrt(-2*x +
1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {37}{180} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{9} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {1}{3} \, \sqrt {-10 \, x^{2} - x + 3} \]

[In]

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

[Out]

37/180*sqrt(10)*arcsin(20/11*x + 1/11) - 1/9*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/3*s
qrt(-10*x^2 - x + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (60) = 120\).

Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=-\frac {1}{90} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {37}{180} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{15} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} \]

[In]

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

[Out]

-1/90*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)))) + 37/180*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1/15*sqrt(5)*sqrt(5
*x + 3)*sqrt(-10*x + 5)

Mupad [B] (verification not implemented)

Time = 5.06 (sec) , antiderivative size = 566, normalized size of antiderivative = 6.74 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {37\,\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 )}{45}-\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {6645115904\,\sqrt {7}\,\left (\sqrt {1-2\,x}-1\right )}{3955078125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\frac {229677056\,{\left (\sqrt {1-2\,x}-1\right )}^2}{158203125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3168922624\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{1318359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {459354112}{791015625}\right )}-\frac {192432128\,\sqrt {3}\,\sqrt {7}}{439453125\,\left (\frac {229677056\,{\left (\sqrt {1-2\,x}-1\right )}^2}{158203125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3168922624\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{1318359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {459354112}{791015625}\right )}+\frac {96216064\,\sqrt {3}\,\sqrt {7}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{87890625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\frac {229677056\,{\left (\sqrt {1-2\,x}-1\right )}^2}{158203125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {3168922624\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{1318359375\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {459354112}{791015625}\right )}\right )}{9}+\frac {2\,{\left (\sqrt {1-2\,x}-1\right )}^3}{15\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3\,\left (\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}\right )}-\frac {4\,\left (\sqrt {1-2\,x}-1\right )}{75\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )\,\left (\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}\right )}+\frac {16\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2\,\left (\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}\right )} \]

[In]

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

[Out]

(37*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/45 - (2*7^(1/2)*atan((664
5115904*7^(1/2)*((1 - 2*x)^(1/2) - 1))/(3955078125*(3^(1/2) - (5*x + 3)^(1/2))*((229677056*((1 - 2*x)^(1/2) -
1)^2)/(158203125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (3168922624*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(1318359375*(3^(1
/2) - (5*x + 3)^(1/2))) - 459354112/791015625)) - (192432128*3^(1/2)*7^(1/2))/(439453125*((229677056*((1 - 2*x
)^(1/2) - 1)^2)/(158203125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (3168922624*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(131835
9375*(3^(1/2) - (5*x + 3)^(1/2))) - 459354112/791015625)) + (96216064*3^(1/2)*7^(1/2)*((1 - 2*x)^(1/2) - 1)^2)
/(87890625*(3^(1/2) - (5*x + 3)^(1/2))^2*((229677056*((1 - 2*x)^(1/2) - 1)^2)/(158203125*(3^(1/2) - (5*x + 3)^
(1/2))^2) + (3168922624*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(1318359375*(3^(1/2) - (5*x + 3)^(1/2))) - 459354112/79
1015625))))/9 + (2*((1 - 2*x)^(1/2) - 1)^3)/(15*(3^(1/2) - (5*x + 3)^(1/2))^3*((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)) - (4*((1 - 2*x
)^(1/2) - 1))/(75*(3^(1/2) - (5*x + 3)^(1/2))*((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)) + (16*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(15*(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))