3.27.36 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\) [2636]

Optimal. Leaf size=181 \[ -\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]

[Out]

-3715/3234/(1-2*x)^(3/2)/(3+5*x)^(3/2)+3/14/(1-2*x)^(3/2)/(2+3*x)^2/(3+5*x)^(3/2)+111/28/(1-2*x)^(3/2)/(2+3*x)
/(3+5*x)^(3/2)-538245/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-40765/83006/(3+5*x)^(3/2)/(
1-2*x)^(1/2)-34551425/5478396*(1-2*x)^(1/2)/(3+5*x)^(3/2)[phone]/60262356*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {105, 156, 157, 12, 95, 210} \begin {gather*} -\frac {538245 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {5 x+3}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac {3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-3715/(3234*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 40765/(83006*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (34551425*Sqrt[1
- 2*x])/(5478396*(3 + 5*x)^(3/2)) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 111/(28*(1 - 2*x)^(3/
2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + (3443814775*Sqrt[1 - 2*x])/(60262356*Sqrt[3 + 5*x]) - (538245*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 95

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

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] && ILtQ[m, -1]

Rule 157

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 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 (3+5 x)^{5/2}} \, dx &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{14} \int \frac {\frac {59}{2}-150 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {1}{98} \int \frac {\frac {5075}{4}-15540 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {1484385}{8}+\frac {1170225 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx}{11319}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {2 \int \frac {\frac {83479305}{16}-\frac {12840975 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{871563}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {4 \int \frac {\frac {9239846595}{32}-\frac {2176739775 x}{8}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{28761579}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}+\frac {8 \int \frac {496468037835}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{316377369}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}+\frac {538245 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}+\frac {538245 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}-\frac {538245 \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 = 0.22, size = 89, normalized size = 0.49 \begin {gather*} \frac {39900939556+28838387211 x-297937101390 x^2-276089438305 x^3+564878517900 x^4+619886659500 x^5}{60262356 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(39900939556 + 28838387211*x - 297937101390*x^2 - 276089438305*x^3 + 564878517900*x^4 + 619886659500*x^5)/(602
62356*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (538245*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1
372*Sqrt[7])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(352\) vs. \(2(136)=272\).
time = 0.09, size = 353, normalized size = 1.95

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \left (21277201621500 \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}-16123390562070 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}-5366583075645 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-3865252136270 x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -4171119419460 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{843672984 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/843672984*(1-2*x)^(1/2)*(21277201621500*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^6+32625
042486300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+2576905529715*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+8678413233000*x^5*(-10*x^2-x+3)^(1/2)-16123390562070*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+7908299250600*x^4*(-10*x^2-x+3)^(1/2)-5366583075645*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-3865252136270*x^3*(-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))*x-4171119419460*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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2
-x+3)^(1/2))/(2+3*x)^2/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]
time = 0.51, size = 172, normalized size = 0.95 \begin {gather*} \frac {538245}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {3443814775 \, x}{30131178 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3595841045}{60262356 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1022125 \, x}{35574 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {3}{14 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {111}{28 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1103855}{71148 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

538245/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 3443814775/30131178*x/sqrt(-10*x^2 -
x + 3) + 3595841045/60262356/sqrt(-10*x^2 - x + 3) + 1022125/35574*x/(-10*x^2 - x + 3)^(3/2) + 3/14/(9*(-10*x^
2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 111/28/(3*(-10*x^2 - x + 3)
^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1103855/71148/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 0.45, size = 146, normalized size = 0.81 \begin {gather*} -\frac {23641335135 \, \sqrt {7} {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\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 (619886659500 \, x^{5} + 564878517900 \, x^{4} - 276089438305 \, x^{3} - 297937101390 \, x^{2} + 28838387211 \, x + 39900939556\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{843672984 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/843672984*(23641335135*sqrt(7)*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*arctan(1/14*s
qrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(619886659500*x^5 + 564878517900*x^4 -
276089438305*x^3 - 297937101390*x^2 + 28838387211*x + 39900939556)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(900*x^6 + 13
80*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (136) = 272\).
time = 2.01, size = 412, normalized size = 2.28 \begin {gather*} \frac {107649}{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 {625}{702768} \, \sqrt {10} {\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 )}^{3} - \frac {2232 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {8928 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {128 \, {\left (577 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3366 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2636478075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {8019 \, {\left (159 \, \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} + 38360 \, \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 )}}{4802 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

107649/38416*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)))) - 625/702768*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 - 2232*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) + 8928*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 128/2636478075*(57
7*sqrt(5)*(5*x + 3) - 3366*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 8019/4802*(159*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 + 3836
0*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) - sq
rt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))^2 + 280)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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