3.24.5 \(\int \frac {\sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\) [2305]

Optimal. Leaf size=64 \[ -\frac {2}{3} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {2}{3} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]

[Out]

-2/15*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/3*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 64, 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 = {132, 56, 222, 12, 95, 210} \begin {gather*} -\frac {2}{3} \sqrt {\frac {2}{5}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2}{3} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 132

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[b*d^(m
+ n)*f^p, Int[(a + b*x)^(m - 1)/(c + d*x)^m, x], x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandTo
Sum[(a + b*x)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n,
-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])

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*} \int \frac {\sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx &=-\left (\frac {2}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\right )+\frac {7}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {14}{3} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3 \sqrt {5}}\\ &=-\frac {2}{3} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {2}{3} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(64)=128\).
time = 0.94, size = 140, normalized size = 2.19 \begin {gather*} \frac {2}{15} \left (5 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+2 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )+5 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 2*Sqrt[10]*ArcT
an[Sqrt[6 + 10*x]/(Sqrt[11] - Sqrt[5 - 10*x])] + 5*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt
[11] + Sqrt[5 - 10*x]))]))/15

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Maple [A]
time = 0.10, size = 68, normalized size = 1.06

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (\sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-5 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )\right )}{15 \sqrt {-10 x^{2}-x +3}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10^(1/2)*arcsin(20/11*x+1/11)-5*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2)))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.54, size = 40, normalized size = 0.62 \begin {gather*} -\frac {1}{15} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1}{3} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*sqrt(10)*arcsin(20/11*x + 1/11) + 1/3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2))

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Fricas [A]
time = 0.70, size = 87, normalized size = 1.36 \begin {gather*} \frac {1}{15} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\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 {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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*sqrt(5)*sqrt(2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 1
/3*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (44) = 88\).
time = 0.97, size = 145, normalized size = 2.27 \begin {gather*} \frac {1}{30} \, \sqrt {5} {\left (\sqrt {70} \sqrt {2} {\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 )} - 2 \, \sqrt {2} {\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 )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*sqrt(5)*(sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 2*sqrt(2)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))))

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Mupad [B]
time = 3.56, size = 138, normalized size = 2.16 \begin {gather*} \frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {5580\,\sqrt {21}\,x+2699\,\sqrt {21}-5489\,\sqrt {35\,x+21}+649\,\sqrt {21-42\,x}+2141\,\sqrt {7}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}}{7400\,x-5489\,\sqrt {1-2\,x}-4543\,\sqrt {15\,x+9}+3063\,\sqrt {3}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}+9929}\right )}{3}+\frac {4\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}-\sqrt {10-20\,x}}{2\,\sqrt {3}-2\,\sqrt {5\,x+3}}\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*7^(1/2)*atan((5580*21^(1/2)*x + 2699*21^(1/2) - 5489*(35*x + 21)^(1/2) + 649*(21 - 42*x)^(1/2) + 2141*7^(1/
2)*(1 - 2*x)^(1/2)*(5*x + 3)^(1/2))/(7400*x - 5489*(1 - 2*x)^(1/2) - 4543*(15*x + 9)^(1/2) + 3063*3^(1/2)*(1 -
2*x)^(1/2)*(5*x + 3)^(1/2) + 9929)))/3 + (4*10^(1/2)*atan((10^(1/2) - (10 - 20*x)^(1/2))/(2*3^(1/2) - 2*(5*x
+ 3)^(1/2))))/15

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