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

Optimal. Leaf size=115 \[ -\frac {2615 \sqrt {1-2 x}}{28 \sqrt {5 x+3}}+\frac {173 \sqrt {1-2 x}}{28 (3 x+2) \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}+\frac {17951 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 115, 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 = {99, 151, 152, 12, 93, 204} \begin {gather*} -\frac {2615 \sqrt {1-2 x}}{28 \sqrt {5 x+3}}+\frac {173 \sqrt {1-2 x}}{28 (3 x+2) \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{2 (3 x+2)^2 \sqrt {5 x+3}}+\frac {17951 \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*(3 + 5*x)^(3/2)),x]

[Out]

(-2615*Sqrt[1 - 2*x])/(28*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (173*Sqrt[1 - 2*x])/(
28*(2 + 3*x)*Sqrt[3 + 5*x]) + (17951*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

Rule 12

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

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 99

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(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[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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 (3+5 x)^{3/2}} \, dx &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}-\frac {1}{2} \int \frac {-\frac {31}{2}+20 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 \sqrt {1-2 x}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {1}{14} \int \frac {-\frac {3677}{4}+865 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {2615 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 \sqrt {1-2 x}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {1}{77} \int -\frac {197461}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {2615 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 \sqrt {1-2 x}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {17951}{56} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {2615 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 \sqrt {1-2 x}}{28 (2+3 x) \sqrt {3+5 x}}-\frac {17951}{28} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {2615 \sqrt {1-2 x}}{28 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{2 (2+3 x)^2 \sqrt {3+5 x}}+\frac {173 \sqrt {1-2 x}}{28 (2+3 x) \sqrt {3+5 x}}+\frac {17951 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 74, normalized size = 0.64 \begin {gather*} \frac {1}{196} \left (17951 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {7 \sqrt {1-2 x} \left (23535 x^2+30861 x+10100\right )}{(3 x+2)^2 \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*Sqrt[1 - 2*x]*(10100 + 30861*x + 23535*x^2))/((2 + 3*x)^2*Sqrt[3 + 5*x]) + 17951*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/196

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.20, size = 106, normalized size = 0.92 \begin {gather*} \frac {17951 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{28 \sqrt {7}}-\frac {\sqrt {1-2 x} \left (\frac {1400 (1-2 x)^2}{(5 x+3)^2}+\frac {29929 (1-2 x)}{5 x+3}+125657\right )}{28 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/28*(Sqrt[1 - 2*x]*(125657 + (1400*(1 - 2*x)^2)/(3 + 5*x)^2 + (29929*(1 - 2*x))/(3 + 5*x)))/(Sqrt[3 + 5*x]*(
7 + (1 - 2*x)/(3 + 5*x))^2) + (17951*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

________________________________________________________________________________________

fricas [A]  time = 1.33, size = 101, normalized size = 0.88 \begin {gather*} \frac {17951 \, \sqrt {7} {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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 (23535 \, x^{2} + 30861 \, x + 10100\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\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)^(3/2),x, algorithm="fricas")

[Out]

1/392*(17951*sqrt(7)*(45*x^3 + 87*x^2 + 56*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1
)/(10*x^2 + x - 3)) - 14*(23535*x^2 + 30861*x + 10100)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(45*x^3 + 87*x^2 + 56*x +
12)

________________________________________________________________________________________

giac [B]  time = 2.05, size = 316, normalized size = 2.75 \begin {gather*} -\frac {1}{3920} \, \sqrt {5} {\left (17951 \, \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 )} + 9800 \, \sqrt {2} {\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 )} + \frac {9240 \, \sqrt {2} {\left (313 \, {\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 {69160 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {276640 \, \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)^(3/2),x, algorithm="giac")

[Out]

-1/3920*sqrt(5)*(17951*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)))) + 9800*sqrt(2)*((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))) + 9240*sqrt(2)*(313*((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 + 69160*
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 276640*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)^2)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 202, normalized size = 1.76 \begin {gather*} -\frac {\left (807795 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1561737 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+329490 \sqrt {-10 x^{2}-x +3}\, x^{2}+1005256 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+432054 \sqrt {-10 x^{2}-x +3}\, x +215412 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+141400 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{392 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \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)^(3/2),x)

[Out]

-1/392*(807795*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1561737*7^(1/2)*x^2*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1005256*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+32
9490*(-10*x^2-x+3)^(1/2)*x^2+215412*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+432054*(-10*x^2
-x+3)^(1/2)*x+141400*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(3*x+2)^2/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.15, size = 143, normalized size = 1.24 \begin {gather*} -\frac {17951}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2615 \, x}{14 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {8191}{84 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {7}{6 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {169}{12 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\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)^(3/2),x, algorithm="maxima")

[Out]

-17951/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2615/14*x/sqrt(-10*x^2 - x + 3) - 8191/
84/sqrt(-10*x^2 - x + 3) + 7/6/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x
+ 3)) + 169/12/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________