3.2317 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{5 x+3}}+\frac{14101 \sqrt{1-2 x}}{392 (3 x+2) \sqrt{5 x+3}}+\frac{81 \sqrt{1-2 x}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 \sqrt{5 x+3}}+\frac{1463447 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-639565*Sqrt[1 - 2*x])/(1176*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (81*Sqrt[1 - 2*x]
)/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (14101*Sqrt[1 - 2*x])/(392*(2 + 3*x)*Sqrt[3 + 5*x]) + (1463447*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[7])

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Rubi [A]  time = 0.0461948, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{5 x+3}}+\frac{14101 \sqrt{1-2 x}}{392 (3 x+2) \sqrt{5 x+3}}+\frac{81 \sqrt{1-2 x}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 \sqrt{5 x+3}}+\frac{1463447 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-639565*Sqrt[1 - 2*x])/(1176*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (81*Sqrt[1 - 2*x]
)/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (14101*Sqrt[1 - 2*x])/(392*(2 + 3*x)*Sqrt[3 + 5*x]) + (1463447*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 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 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 (3+5 x)^{3/2}} \, dx &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}-\frac{1}{3} \int \frac{-\frac{41}{2}+30 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 \sqrt{1-2 x}}{28 (2+3 x)^2 \sqrt{3+5 x}}-\frac{1}{42} \int \frac{-\frac{7621}{4}+2430 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 \sqrt{1-2 x}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{14101 \sqrt{1-2 x}}{392 (2+3 x) \sqrt{3+5 x}}-\frac{1}{294} \int \frac{-\frac{899407}{8}+\frac{211515 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 \sqrt{1-2 x}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{14101 \sqrt{1-2 x}}{392 (2+3 x) \sqrt{3+5 x}}+\frac{\int -\frac{48293751}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1617}\\ &=-\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 \sqrt{1-2 x}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{14101 \sqrt{1-2 x}}{392 (2+3 x) \sqrt{3+5 x}}-\frac{1463447}{784} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 \sqrt{1-2 x}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{14101 \sqrt{1-2 x}}{392 (2+3 x) \sqrt{3+5 x}}-\frac{1463447}{392} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{639565 \sqrt{1-2 x}}{1176 \sqrt{3+5 x}}+\frac{\sqrt{1-2 x}}{3 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 \sqrt{1-2 x}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{14101 \sqrt{1-2 x}}{392 (2+3 x) \sqrt{3+5 x}}+\frac{1463447 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{392 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.059001, size = 79, normalized size = 0.55 \[ \frac{1463447 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{7 \sqrt{1-2 x} \left (5756085 x^3+11385261 x^2+7502166 x+1646704\right )}{(3 x+2)^3 \sqrt{5 x+3}}}{2744} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*Sqrt[1 - 2*x]*(1646704 + 7502166*x + 11385261*x^2 + 5756085*x^3))/((2 + 3*x)^3*Sqrt[3 + 5*x]) + 1463447*S
qrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2744

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Maple [B]  time = 0.013, size = 250, normalized size = 1.7 \begin{align*} -{\frac{1}{5488\, \left ( 2+3\,x \right ) ^{3}} \left ( 197565345\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+159393654\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \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)^(3/2),x)

[Out]

-1/5488*(197565345*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+513669897*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+500498874*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^2+80585190*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x+159393654*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+105030
324*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1
/2)

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Maxima [A]  time = 3.23573, size = 285, normalized size = 1.98 \begin{align*} -\frac{1463447}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{639565 \, x}{588 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{222589}{392 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7}{9 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{235}{36 \,{\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{13777}{168 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 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)^(3/2),x, algorithm="maxima")

[Out]

-1463447/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 639565/588*x/sqrt(-10*x^2 - x + 3) -
222589/392/sqrt(-10*x^2 - x + 3) + 7/9/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt
(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 235/36/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3
)*x + 4*sqrt(-10*x^2 - x + 3)) + 13777/168/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.20253, size = 373, normalized size = 2.59 \begin{align*} \frac{1463447 \, \sqrt{7}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (5756085 \, x^{3} + 11385261 \, x^{2} + 7502166 \, x + 1646704\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{5488 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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)^(3/2),x, algorithm="fricas")

[Out]

1/5488*(1463447*sqrt(7)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(5756085*x^3 + 11385261*x^2 + 7502166*x + 1646704)*sqrt(5*x + 3)*sqrt
(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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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)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 3.06959, size = 505, normalized size = 3.51 \begin{align*} -\frac{1}{54880} \, \sqrt{5}{\left (1463447 \, \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 )} + 686000 \, \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{27720 \, \sqrt{2}{\left (11747 \,{\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} + 5216960 \,{\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{615675200 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{2462700800 \, \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)^(3/2),x, algorithm="giac")

[Out]

-1/54880*sqrt(5)*(1463447*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)))) + 686000*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))) + 27720*sqrt(2)*(11747*
((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
+ 5216960*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))^3 + 615675200*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 2462700800*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280)^3)