3.2634 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=123 \[ \frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{5 x+3}}-\frac{19130 \sqrt{1-2 x}}{195657 (5 x+3)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 412/(5929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (19130*Sqrt[1 - 2*x])/(19
5657*(3 + 5*x)^(3/2)) + (1001590*Sqrt[1 - 2*x])/(2152227*Sqrt[3 + 5*x]) - (162*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(49*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0459104, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {104, 152, 12, 93, 204} \[ \frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{5 x+3}}-\frac{19130 \sqrt{1-2 x}}{195657 (5 x+3)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 412/(5929*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (19130*Sqrt[1 - 2*x])/(19
5657*(3 + 5*x)^(3/2)) + (1001590*Sqrt[1 - 2*x])/(2152227*Sqrt[3 + 5*x]) - (162*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(49*Sqrt[7])

Rule 104

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] &&
IntegersQ[2*m, 2*n, 2*p]

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) (3+5 x)^{5/2}} \, dx &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2}{231} \int \frac{-\frac{219}{2}-90 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{4 \int \frac{\frac{27987}{4}+9270 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{17787}\\ &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{19130 \sqrt{1-2 x}}{195657 (3+5 x)^{3/2}}-\frac{8 \int \frac{\frac{93873}{8}-\frac{86085 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{586971}\\ &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{19130 \sqrt{1-2 x}}{195657 (3+5 x)^{3/2}}+\frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{3+5 x}}+\frac{16 \int \frac{10673289}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6456681}\\ &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{19130 \sqrt{1-2 x}}{195657 (3+5 x)^{3/2}}+\frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{3+5 x}}+\frac{81}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{19130 \sqrt{1-2 x}}{195657 (3+5 x)^{3/2}}+\frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{3+5 x}}+\frac{162}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{4}{231 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{412}{5929 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{19130 \sqrt{1-2 x}}{195657 (3+5 x)^{3/2}}+\frac{1001590 \sqrt{1-2 x}}{2152227 \sqrt{3+5 x}}-\frac{162 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0871249, size = 93, normalized size = 0.76 \[ \frac{14 \left (10015900 x^3-4427220 x^2-3234261 x+1490582\right )+7115526 \sqrt{7-14 x} \sqrt{5 x+3} \left (10 x^2+x-3\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{15065589 (1-2 x)^{3/2} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1490582 - 3234261*x - 4427220*x^2 + 10015900*x^3) + 7115526*Sqrt[7 - 14*x]*Sqrt[3 + 5*x]*(-3 + x + 10*x^2
)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(15065589*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 250, normalized size = 2. \begin{align*}{\frac{1}{15065589\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 355776300\,\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}-209908017\,\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}-21346578\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-61981080\,{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 ) -45279654\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15065589*(1-2*x)^(1/2)*(355776300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+71155260*7^
(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-209908017*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^2+140222600*x^3*(-10*x^2-x+3)^(1/2)-21346578*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))*x-61981080*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))-45279654*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3
/2)

________________________________________________________________________________________

Maxima [A]  time = 3.1943, size = 117, normalized size = 0.95 \begin{align*} \frac{81}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2003180 \, x}{2152227 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1085762}{2152227 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{740 \, x}{2541 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{326}{2541 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2003180/2152227*x/sqrt(-10*x^2 - x + 3) + 1
085762/2152227/sqrt(-10*x^2 - x + 3) + 740/2541*x/(-10*x^2 - x + 3)^(3/2) - 326/2541/(-10*x^2 - x + 3)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.82389, size = 366, normalized size = 2.98 \begin{align*} -\frac{3557763 \, \sqrt{7}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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 (10015900 \, x^{3} - 4427220 \, x^{2} - 3234261 \, x + 1490582\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{15065589 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15065589*(3557763*sqrt(7)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(10015900*x^3 - 4427220*x^2 - 3234261*x + 1490582)*sqrt(5*x + 3)*sqrt
(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.43531, size = 315, normalized size = 2.56 \begin{align*} -\frac{25}{702768} \, \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} + \frac{81}{3430} \, \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{675}{29282} \, \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 )} - \frac{32 \,{\left (379 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2277 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{53805675 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/702768*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)))^3 + 81/3430*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 675/29282*sqrt(10)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 32/53805675*
(379*sqrt(5)*(5*x + 3) - 2277*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2