3.2558 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=115 \[ -\frac {3895 \sqrt {5 x+3}}{7546 \sqrt {1-2 x}}+\frac {345 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}-\frac {12465 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]

[Out]

-12465/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-3895/7546*(3+5*x)^(1/2)/(1-2*x)^(1/2)+3/14
*(3+5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)+345/196*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {3895 \sqrt {5 x+3}}{7546 \sqrt {1-2 x}}+\frac {345 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}-\frac {12465 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3895*Sqrt[3 + 5*x])/(7546*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (345*Sqrt[3 +
5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)) - (12465*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 \sqrt {3+5 x}} \, dx &=\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {1}{14} \int \frac {\frac {35}{2}-60 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {1}{98} \int \frac {-\frac {445}{4}-1725 x}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {\int -\frac {137115}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{3773}\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {12465 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {12465 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {12465 \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 = 0.05, size = 85, normalized size = 0.74 \[ \frac {-7 \sqrt {5 x+3} \left (70110 x^2+13785 x-25204\right )-137115 \sqrt {7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{105644 \sqrt {1-2 x} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[3 + 5*x]*(-25204 + 13785*x + 70110*x^2) - 137115*Sqrt[7 - 14*x]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqr
t[7]*Sqrt[3 + 5*x])])/(105644*Sqrt[1 - 2*x]*(2 + 3*x)^2)

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fricas [A]  time = 0.91, size = 101, normalized size = 0.88 \[ -\frac {137115 \, \sqrt {7} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (70110 \, x^{2} + 13785 \, x - 25204\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{211288 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \]

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)^(1/2),x, algorithm="fricas")

[Out]

-1/211288*(137115*sqrt(7)*(18*x^3 + 15*x^2 - 4*x - 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 14*(70110*x^2 + 13785*x - 25204)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x
- 4)

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giac [B]  time = 2.60, size = 278, normalized size = 2.42 \[ \frac {2493}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{18865 \, {\left (2 \, x - 1\right )}} + \frac {297 \, \sqrt {10} {\left (9 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1640 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {6560 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]

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)^(1/2),x, algorithm="giac")

[Out]

2493/38416*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)))) - 16/18865*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*
x - 1) + 297/98*sqrt(10)*(9*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^3 + 1640*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 6560*sqrt(5*x + 3)/(sq
rt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2

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maple [B]  time = 0.02, size = 209, normalized size = 1.82 \[ \frac {\left (2468070 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2056725 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+981540 \sqrt {-10 x^{2}-x +3}\, x^{2}-548460 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+192990 \sqrt {-10 x^{2}-x +3}\, x -548460 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-352856 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{211288 \left (3 x +2\right )^{2} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]

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)^(1/2),x)

[Out]

1/211288*(2468070*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+2056725*7^(1/2)*x^2*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-548460*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+
981540*(-10*x^2-x+3)^(1/2)*x^2-548460*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+192990*(-10*x
^2-x+3)^(1/2)*x-352856*(-10*x^2-x+3)^(1/2))*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(3*x+2)^2/(2*x-1)/(-10*x^2-x+3)^(1/2)

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

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)^(1/2),x, algorithm="maxima")

[Out]

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

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

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)^(1/2)),x)

[Out]

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

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

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)**(1/2),x)

[Out]

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

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