3.2569 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{5 x+3}}-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{555}{196 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{177255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-6205/(7546*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (3125575*Sqrt[1 - 2*x])/(166012*Sqrt[3 + 5*x]) + 3/(14*Sqrt[1 - 2*x
]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 555/(196*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (177255*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.0496286, antiderivative size = 137, 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 = {103, 151, 152, 12, 93, 204} \[ -\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{5 x+3}}-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{555}{196 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{177255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6205/(7546*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (3125575*Sqrt[1 - 2*x])/(166012*Sqrt[3 + 5*x]) + 3/(14*Sqrt[1 - 2*x
]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 555/(196*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x]) + (177255*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*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)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{1}{14} \int \frac{\frac{65}{2}-90 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{555}{196 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{1}{98} \int \frac{\frac{4895}{4}-5550 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{555}{196 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{-\frac{401735}{8}+\frac{93075 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{3773}\\ &=-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{3+5 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{555}{196 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{2 \int -\frac{21447855}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{41503}\\ &=-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{3+5 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{555}{196 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{177255 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{3+5 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{555}{196 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{177255 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=-\frac{6205}{7546 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{3125575 \sqrt{1-2 x}}{166012 \sqrt{3+5 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{555}{196 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{177255 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0705644, size = 79, normalized size = 0.58 \[ \frac{\frac{7 \left (56260350 x^3+45655035 x^2-12730165 x-12072596\right )}{\sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}[phone] \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1162084} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(-12072596 - 12730165*x + 45655035*x^2 + 56260350*x^3))/(Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 214478
55*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1162084

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Maple [B]  time = 0.015, size = 257, normalized size = 1.9 \begin{align*} -{\frac{1}{2324168\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) }\sqrt{1-2\,x} \left ( 1930306950\,\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}-686331360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+639170490\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-257374260\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -178222310\,x\sqrt{-10\,{x}^{2}-x+3}-169016344\,\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)^3/(3+5*x)^(3/2),x)

[Out]

-1/2324168*(1-2*x)^(1/2)*(1930306950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2766773295
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+536196375*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^2+787644900*x^3*(-10*x^2-x+3)^(1/2)-686331360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x+639170490*x^2*(-10*x^2-x+3)^(1/2)-257374260*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))-178222310*x*(-10*x^2-x+3)^(1/2)-169016344*(-10*x^2-x+3)^(1/2))/(2+3*x)^2/(2*x-1)/(-10*x^2-x+3)^
(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.85938, size = 193, normalized size = 1.41 \begin{align*} -\frac{177255}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{3125575 \, x}{83006 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3262085}{166012 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3}{14 \,{\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{555}{196 \,{\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)^3/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-177255/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 3125575/83006*x/sqrt(-10*x^2 - x + 3
) - 3262085/166012/sqrt(-10*x^2 - x + 3) + 3/14/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*
sqrt(-10*x^2 - x + 3)) + 555/196/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.53089, size = 374, normalized size = 2.73 \begin{align*} \frac{21447855 \, \sqrt{7}{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\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 (56260350 \, x^{3} + 45655035 \, x^{2} - 12730165 \, x - 12072596\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2324168 \,{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2324168*(21447855*sqrt(7)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(56260350*x^3 + 45655035*x^2 - 12730165*x - 12072596)*sqrt(5*x + 3)*
sqrt(-2*x + 1))/(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.37239, size = 462, normalized size = 3.37 \begin{align*} -\frac{35451}{38416} \, \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{125}{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{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{207515 \,{\left (2 \, x - 1\right )}} - \frac{297 \,{\left (47 \, \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} + 10520 \, \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 )}}{98 \,{\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35451/38416*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)))) - 125/242*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 32/207515*sqrt(5)*sqrt(5*x + 3)
*sqrt(-10*x + 5)/(2*x - 1) - 297/98*(47*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 + 10520*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)^2