3.2568 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{17735 \sqrt{1-2 x}}{5929 \sqrt{5 x+3}}-\frac{58}{539 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{999 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

-58/(539*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (17735*Sqrt[1 - 2*x])/(5929*Sqrt[3 + 5*x]) + 3/(7*Sqrt[1 - 2*x]*(2 + 3
*x)*Sqrt[3 + 5*x]) + (999*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.0362919, antiderivative size = 108, 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 = {103, 152, 12, 93, 204} \[ -\frac{17735 \sqrt{1-2 x}}{5929 \sqrt{5 x+3}}-\frac{58}{539 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{999 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-58/(539*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (17735*Sqrt[1 - 2*x])/(5929*Sqrt[3 + 5*x]) + 3/(7*Sqrt[1 - 2*x]*(2 + 3
*x)*Sqrt[3 + 5*x]) + (999*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(49*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 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)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac{3}{7 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{1}{7} \int \frac{\frac{31}{2}-60 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{2}{539} \int \frac{-\frac{2503}{4}+435 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{17735 \sqrt{1-2 x}}{5929 \sqrt{3+5 x}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{4 \int -\frac{120879}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5929}\\ &=-\frac{58}{539 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{17735 \sqrt{1-2 x}}{5929 \sqrt{3+5 x}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{999}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{17735 \sqrt{1-2 x}}{5929 \sqrt{3+5 x}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{999}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{58}{539 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{17735 \sqrt{1-2 x}}{5929 \sqrt{3+5 x}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{999 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0574041, size = 74, normalized size = 0.69 \[ \frac{106410 x^2+15821 x-34205}{5929 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{999 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-34205 + 15821*x + 106410*x^2)/(5929*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (999*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Maple [B]  time = 0.016, size = 209, normalized size = 1.9 \begin{align*} -{\frac{1}{ \left ( 166012+249018\,x \right ) \left ( 2\,x-1 \right ) }\sqrt{1-2\,x} \left ( 3626370\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+2780217\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-846153\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1489740\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-725274\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +221494\,x\sqrt{-10\,{x}^{2}-x+3}-478870\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^(3/2),x)

[Out]

-1/83006*(1-2*x)^(1/2)*(3626370*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2780217*7^(1/2)
*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-846153*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x+1489740*x^2*(-10*x^2-x+3)^(1/2)-725274*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)+221494*x*(-10*x^2-x+3)^(1/2)-478870*(-10*x^2-x+3)^(1/2))/(2+3*x)/(2*x-1)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 3.34289, size = 124, normalized size = 1.15 \begin{align*} -\frac{999}{686} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{35470 \, x}{5929 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{18373}{5929 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3}{7 \,{\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/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-999/686*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 35470/5929*x/sqrt(-10*x^2 - x + 3) - 1837
3/5929/sqrt(-10*x^2 - x + 3) + 3/7/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.55407, size = 305, normalized size = 2.82 \begin{align*} \frac{120879 \, \sqrt{7}{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\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 (106410 \, x^{2} + 15821 \, x - 34205\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{83006 \,{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/83006*(120879*sqrt(7)*(30*x^3 + 23*x^2 - 7*x - 6)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 14*(106410*x^2 + 15821*x - 34205)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(30*x^3 + 23*x^2 - 7*x
- 6)

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 2.41473, size = 375, normalized size = 3.47 \begin{align*} -\frac{999}{6860} \, \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{25}{242} \, \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{16 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{29645 \,{\left (2 \, x - 1\right )}} - \frac{594 \, \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 )}}{49 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-999/6860*sqrt(70)*sqrt(10)*(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)))) - 25/242*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))) - 16/29645*sqrt(5)*sqrt(5*x + 3)*sqrt
(-10*x + 5)/(2*x - 1) - 594/49*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)