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]

177255/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-6205/7546/(1-2*x)^(1/2)/(3+5*x)^(1/2)+3/14
/(2+3*x)^2/(1-2*x)^(1/2)/(3+5*x)^(1/2)+555/196/(2+3*x)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-3125575/166012*(1-2*x)^(1/2
)/(3+5*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 137, normalized size of antiderivative = 1.00, 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 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 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 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*}

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Mathematica [A]  time = 0.08, 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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fricas [A]  time = 1.24, size = 116, normalized size = 0.85 \[ \frac {21447855 \, \sqrt {7} {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (56260350 \, x^{3} + 45655035 \, x^{2} - 12730165 \, x - 12072596\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2324168 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \]

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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giac [B]  time = 2.34, size = 342, normalized size = 2.50 \[ -\frac {35451}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {125}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{207515 \, {\left (2 \, x - 1\right )}} - \frac {297 \, {\left (47 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 10520 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]

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

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maple [B]  time = 0.02, size = 257, normalized size = 1.88 \[ -\frac {\sqrt {-2 x +1}\, \left (1930306950 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-686331360 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-178222310 \sqrt {-10 x^{2}-x +3}\, x -257374260 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-169016344 \sqrt {-10 x^{2}-x +3}\right )}{2324168 \left (3 x +2\right )^{2} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2324168*(-2*x+1)^(1/2)*(1930306950*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]
5*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3-686331360*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^2-257374260*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*
x^2-x+3)^(1/2))-178222310*(-10*x^2-x+3)^(1/2)*x-169016344*(-10*x^2-x+3)^(1/2))/(3*x+2)^2/(2*x-1)/(-10*x^2-x+3)
^(1/2)/(5*x+3)^(1/2)

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maxima [A]  time = 1.44, size = 143, normalized size = 1.04 \[ -\frac {177255}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3125575 \, x}{83006 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3262085}{166012 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3}{14 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {555}{196 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]

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]

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

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