3.22.68 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^3 \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=93 \[ \frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{14 (3 x+2)^2}+\frac {107 \sqrt {5 x+3} \sqrt {1-2 x}}{28 (3 x+2)}-\frac {1177 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {3 \sqrt {5 x+3} (1-2 x)^{3/2}}{14 (3 x+2)^2}+\frac {107 \sqrt {5 x+3} \sqrt {1-2 x}}{28 (3 x+2)}-\frac {1177 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(28*(2 + 3*x)) - (1177*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

Rule 93

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 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 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}}{(2+3 x)^3 \sqrt {3+5 x}} \, dx &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107}{28} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {1177}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {1177}{28} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {3 (1-2 x)^{3/2} \sqrt {3+5 x}}{14 (2+3 x)^2}+\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}-\frac {1177 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 69, normalized size = 0.74 \begin {gather*} \frac {1}{196} \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (309 x+220)}{(3 x+2)^2}-1177 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(220 + 309*x))/(2 + 3*x)^2 - 1177*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/196

________________________________________________________________________________________

IntegrateAlgebraic [C]  time = 0.47, size = 127, normalized size = 1.37 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (309 \sqrt {5} (5 x+3)^{3/2}+173 \sqrt {5} \sqrt {5 x+3}\right )}{28 (3 (5 x+3)+1)^2}-\frac {1177 i \tanh ^{-1}\left (3 \sqrt {\frac {2}{35}} (5 x+3)+\frac {3 i \sqrt {11-2 (5 x+3)} \sqrt {5 x+3}}{\sqrt {35}}+\sqrt {\frac {2}{35}}\right )}{28 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(173*Sqrt[5]*Sqrt[3 + 5*x] + 309*Sqrt[5]*(3 + 5*x)^(3/2)))/(28*(1 + 3*(3 + 5*x))^2) -
(((1177*I)/28)*ArcTanh[Sqrt[2/35] + 3*Sqrt[2/35]*(3 + 5*x) + ((3*I)*Sqrt[3 + 5*x]*Sqrt[11 - 2*(3 + 5*x)])/Sqrt
[35]])/Sqrt[7]

________________________________________________________________________________________

fricas [A]  time = 1.60, size = 86, normalized size = 0.92 \begin {gather*} -\frac {1177 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\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 (309 \, x + 220\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/392*(1177*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 14*(309*x + 220)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

________________________________________________________________________________________

giac [B]  time = 1.45, size = 257, normalized size = 2.76 \begin {gather*} \frac {11}{3920} \, \sqrt {5} {\left (107 \, \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 )} + \frac {280 \, \sqrt {2} {\left (173 \, {\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 {29960 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {119840 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\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 )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/3920*sqrt(5)*(107*sqrt(70)*sqrt(2)*(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)))) + 280*sqrt(2)*(173*((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 + 29960*(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 119840*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 154, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (10593 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+14124 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4326 \sqrt {-10 x^{2}-x +3}\, x +4708 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3080 \sqrt {-10 x^{2}-x +3}\right )}{392 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/392*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(10593*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+14124
*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4708*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))+4326*(-10*x^2-x+3)^(1/2)*x+3080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^2

________________________________________________________________________________________

maxima [A]  time = 1.18, size = 76, normalized size = 0.82 \begin {gather*} \frac {1177}{392} \, \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}}{2 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {103 \, \sqrt {-10 \, x^{2} - x + 3}}{28 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1177/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/2*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x +
4) + 103/28*sqrt(-10*x^2 - x + 3)/(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 12.75, size = 1037, normalized size = 11.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((18323*((1 - 2*x)^(1/2) - 1)^5)/(875*(3^(1/2) - (5*x + 3)^(1/2))^5) - (36646*((1 - 2*x)^(1/2) - 1)^3)/(4375*(
3^(1/2) - (5*x + 3)^(1/2))^3) - (2326*((1 - 2*x)^(1/2) - 1))/(4375*(3^(1/2) - (5*x + 3)^(1/2))) + (1163*((1 -
2*x)^(1/2) - 1)^7)/(140*(3^(1/2) - (5*x + 3)^(1/2))^7) + (10607*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(4375*(3^(1/2
) - (5*x + 3)^(1/2))^2) - (3646*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(875*(3^(1/2) - (5*x + 3)^(1/2))^4) + (10607*
3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(700*(3^(1/2) - (5*x + 3)^(1/2))^6))/((544*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(
1/2) - (5*x + 3)^(1/2))^2) - (1764*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (136*((1 - 2
*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - (5*x + 3)^(1/2))^8 -
(96*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(625*(3^(1/2) - (5*x + 3)^(1/2))^3) + (48*3^(1/2)*((1 - 2*x)^(1/2) - 1)^
5)/(125*(3^(1/2) - (5*x + 3)^(1/2))^5) + (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(5*(3^(1/2) - (5*x + 3)^(1/2))^7
) - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(625*(3^(1/2) - (5*x + 3)^(1/2))) + 16/625) - (1177*7^(1/2)*atan(((1177
*7^(1/2)*((7062*3^(1/2))/875 + (3531*((1 - 2*x)^(1/2) - 1))/(875*(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)*1177i)/392 - (3531*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175*(3^(1/2) - (5*x + 3
)^(1/2))^2)))/392 + (1177*7^(1/2)*((7062*3^(1/2))/875 + (3531*((1 - 2*x)^(1/2) - 1))/(875*(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)*1177i)/392 - (3531*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)
/(175*(3^(1/2) - (5*x + 3)^(1/2))^2)))/392)/((7^(1/2)*((7062*3^(1/2))/875 + (3531*((1 - 2*x)^(1/2) - 1))/(875*
(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) + (8
88*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*1177i)/392 - (3531*3^(1/2)*((1
- 2*x)^(1/2) - 1)^2)/(175*(3^(1/2) - (5*x + 3)^(1/2))^2))*1177i)/392 - (7^(1/2)*((7062*3^(1/2))/875 + (3531*((
1 - 2*x)^(1/2) - 1))/(875*(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)*1177i
)/392 - (3531*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175*(3^(1/2) - (5*x + 3)^(1/2))^2))*1177i)/392 + (1385329*((1
- 2*x)^(1/2) - 1)^2)/(9800*(3^(1/2) - (5*x + 3)^(1/2))^2) + 1385329/24500)))/196

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________