3.2308 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac{18083 \sqrt{1-2 x} \sqrt{5 x+3}}{1176 (3 x+2)}+\frac{173 \sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}-\frac{68959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (173*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^2) + (18083*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(1176*(2 + 3*x)) - (68959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[
7])

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Rubi [A]  time = 0.0365923, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {99, 151, 12, 93, 204} \[ \frac{18083 \sqrt{1-2 x} \sqrt{5 x+3}}{1176 (3 x+2)}+\frac{173 \sqrt{1-2 x} \sqrt{5 x+3}}{84 (3 x+2)^2}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}-\frac{68959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/((2 + 3*x)^4*Sqrt[3 + 5*x]),x]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) + (173*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84*(2 + 3*x)^2) + (18083*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(1176*(2 + 3*x)) - (68959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[
7])

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 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 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)^4 \sqrt{3+5 x}} \, dx &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{1}{3} \int \frac{-\frac{31}{2}+20 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{173 \sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^2}-\frac{1}{42} \int \frac{-\frac{3721}{4}+865 x}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{173 \sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^2}+\frac{18083 \sqrt{1-2 x} \sqrt{3+5 x}}{1176 (2+3 x)}-\frac{1}{294} \int -\frac{206877}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{173 \sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^2}+\frac{18083 \sqrt{1-2 x} \sqrt{3+5 x}}{1176 (2+3 x)}+\frac{68959}{784} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{173 \sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^2}+\frac{18083 \sqrt{1-2 x} \sqrt{3+5 x}}{1176 (2+3 x)}+\frac{68959}{392} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}+\frac{173 \sqrt{1-2 x} \sqrt{3+5 x}}{84 (2+3 x)^2}+\frac{18083 \sqrt{1-2 x} \sqrt{3+5 x}}{1176 (2+3 x)}-\frac{68959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{392 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.048418, size = 74, normalized size = 0.61 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (54249 x^2+74754 x+25856\right )}{(3 x+2)^3}-68959 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]/((2 + 3*x)^4*Sqrt[3 + 5*x]),x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25856 + 74754*x + 54249*x^2))/(2 + 3*x)^3 - 68959*Sqrt[7]*ArcTan[Sqrt[1 - 2*x
]/(Sqrt[7]*Sqrt[3 + 5*x])])/2744

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Maple [B]  time = 0.012, size = 202, normalized size = 1.7 \begin{align*}{\frac{1}{5488\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1861893\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+3723786\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+2482524\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+759486\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+551672\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1046556\,x\sqrt{-10\,{x}^{2}-x+3}+361984\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(1/2),x)

[Out]

1/5488*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1861893*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+372
3786*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+2482524*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x+759486*x^2*(-10*x^2-x+3)^(1/2)+551672*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))+1046556*x*(-10*x^2-x+3)^(1/2)+361984*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.51948, size = 144, normalized size = 1.18 \begin{align*} \frac{68959}{5488} \, \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}}{3 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{173 \, \sqrt{-10 \, x^{2} - x + 3}}{84 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{18083 \, \sqrt{-10 \, x^{2} - x + 3}}{1176 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

68959/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/3*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*
x^2 + 36*x + 8) + 173/84*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 18083/1176*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.33068, size = 305, normalized size = 2.5 \begin{align*} -\frac{68959 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 (54249 \, x^{2} + 74754 \, x + 25856\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{5488 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/5488*(68959*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 14*(54249*x^2 + 74754*x + 25856)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x
+ 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(1/2)/(2+3*x)**4/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 3.13485, size = 425, normalized size = 3.48 \begin{align*} \frac{11}{54880} \, \sqrt{5}{\left (6269 \, \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 (13331 \,{\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 )}^{5} + 4674880 \,{\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{491489600 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{1965958400 \, \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 )}^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(1/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

11/54880*sqrt(5)*(6269*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)*(13331*((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)))^5 + 4674880*((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 + 49148960
0*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1965958400*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(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)^3)