3.2617 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{3471145 \sqrt{5 x+3}}{3486252 \sqrt{1-2 x}}+\frac{423 \sqrt{5 x+3}}{56 (1-2 x)^{3/2} (3 x+2)}-\frac{101485 \sqrt{5 x+3}}{45276 (1-2 x)^{3/2}}+\frac{193 \sqrt{5 x+3}}{196 (1-2 x)^{3/2} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}-\frac{330255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(-101485*Sqrt[3 + 5*x])/(45276*(1 - 2*x)^(3/2)) - (3471145*Sqrt[3 + 5*x])/(3486252*Sqrt[1 - 2*x]) + Sqrt[3 + 5
*x]/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (193*Sqrt[3 + 5*x])/(196*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (423*Sqrt[3 + 5*
x])/(56*(1 - 2*x)^(3/2)*(2 + 3*x)) - (330255*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0598795, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac{3471145 \sqrt{5 x+3}}{3486252 \sqrt{1-2 x}}+\frac{423 \sqrt{5 x+3}}{56 (1-2 x)^{3/2} (3 x+2)}-\frac{101485 \sqrt{5 x+3}}{45276 (1-2 x)^{3/2}}+\frac{193 \sqrt{5 x+3}}{196 (1-2 x)^{3/2} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)^3}-\frac{330255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-101485*Sqrt[3 + 5*x])/(45276*(1 - 2*x)^(3/2)) - (3471145*Sqrt[3 + 5*x])/(3486252*Sqrt[1 - 2*x]) + Sqrt[3 + 5
*x]/(7*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + (193*Sqrt[3 + 5*x])/(196*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + (423*Sqrt[3 + 5*
x])/(56*(1 - 2*x)^(3/2)*(2 + 3*x)) - (330255*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

Rule 103

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[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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{1}{(1-2 x)^{5/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx &=\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{1}{21} \int \frac{\frac{33}{2}-120 x}{(1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{1}{294} \int \frac{-\frac{2433}{4}-8685 x}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{423 \sqrt{3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}+\frac{\int \frac{-\frac{887565}{8}-310905 x}{(1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x}} \, dx}{2058}\\ &=-\frac{101485 \sqrt{3+5 x}}{45276 (1-2 x)^{3/2}}+\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{423 \sqrt{3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}-\frac{\int \frac{\frac{8958495}{16}+\frac{31967775 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{237699}\\ &=-\frac{101485 \sqrt{3+5 x}}{45276 (1-2 x)^{3/2}}-\frac{3471145 \sqrt{3+5 x}}{3486252 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{423 \sqrt{3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}+\frac{2 \int \frac{2517533865}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{18302823}\\ &=-\frac{101485 \sqrt{3+5 x}}{45276 (1-2 x)^{3/2}}-\frac{3471145 \sqrt{3+5 x}}{3486252 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{423 \sqrt{3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}+\frac{330255 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{38416}\\ &=-\frac{101485 \sqrt{3+5 x}}{45276 (1-2 x)^{3/2}}-\frac{3471145 \sqrt{3+5 x}}{3486252 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{423 \sqrt{3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}+\frac{330255 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{19208}\\ &=-\frac{101485 \sqrt{3+5 x}}{45276 (1-2 x)^{3/2}}-\frac{3471145 \sqrt{3+5 x}}{3486252 \sqrt{1-2 x}}+\frac{\sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)^3}+\frac{193 \sqrt{3+5 x}}{196 (1-2 x)^{3/2} (2+3 x)^2}+\frac{423 \sqrt{3+5 x}}{56 (1-2 x)^{3/2} (2+3 x)}-\frac{330255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{19208 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0740136, size = 100, normalized size = 0.6 \[ -\frac{-7 \sqrt{5 x+3} \left (374883660 x^4+140350860 x^3-244982277 x^2-48873610 x+44829024\right )-119882565 \sqrt{7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{48807528 (1-2 x)^{3/2} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-7*Sqrt[3 + 5*x]*(44829024 - 48873610*x - 244982277*x^2 + 140350860*x^3 + 374883660*x^4) - 119882565*Sqrt[7
- 14*x]*(-1 + 2*x)*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(48807528*(1 - 2*x)^(3/2)*(2 + 3
*x)^3)

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 305, normalized size = 1.8 \begin{align*}{\frac{1}{97615056\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) ^{2}} \left ( 12947317020\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-5394715425\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6953188770\,\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-3429751878\,{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 ) -684230540\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/97615056*(12947317020*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+12947317020*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-5394715425*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^3+5248371240*x^4*(-10*x^2-x+3)^(1/2)-6953188770*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))*x^2+1964912040*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-3429751878*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))-684230540*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2+3*x)^3/(2*x
-1)^2/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{4}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.5764, size = 424, normalized size = 2.55 \begin{align*} -\frac{119882565 \, \sqrt{7}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, 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 (374883660 \, x^{4} + 140350860 \, x^{3} - 244982277 \, x^{2} - 48873610 \, x + 44829024\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{97615056 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/97615056*(119882565*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)
*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(374883660*x^4 + 140350860*x^3 - 244982277*x^2 - 48873610
*x + 44829024)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [B]  time = 3.97885, size = 482, normalized size = 2.9 \begin{align*} \frac{66051}{537824} \, \sqrt{70} \sqrt{10}{\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{32 \,{\left (932 \, \sqrt{5}{\left (5 \, x + 3\right )} - 5511 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{152523525 \,{\left (2 \, x - 1\right )}^{2}} + \frac{297 \,{\left (15599 \, \sqrt{10}{\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} + 5723200 \, \sqrt{10}{\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} + 607208000 \, \sqrt{10}{\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 )}\right )}}{67228 \,{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

66051/537824*sqrt(70)*sqrt(10)*(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)))) - 32/152523525*(932*sqrt(5)*(5*x + 3) - 5511*sqrt(
5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 297/67228*(15599*sqrt(10)*((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 + 5723200*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 607208000*sqrt(
10)*((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))
))/(((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