3.24.70 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx\) [2370]

Optimal. Leaf size=180 \[ \frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]

[Out]

-333216939/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+7/15*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+
3*x)^5+293/120*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+23909/1680*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+835409/9
408*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+87374783/131712*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.04, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 156, 12, 95, 210} \begin {gather*} -\frac {333216939 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}}+\frac {87374783 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {835409 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {23909 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {293 \sqrt {1-2 x} \sqrt {5 x+3}}{120 (3 x+2)^4}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (293*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(120*(2 + 3*x)^4) + (2390
9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3) + (835409*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) +
(87374783*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(131712*(2 + 3*x)) - (333216939*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(43904*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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 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 {(1-2 x)^{3/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\frac {337}{2}-260 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {1}{420} \int \frac {\frac {85323}{4}-30765 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {\int \frac {\frac {15850275}{8}-2510445 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{8820}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {\int \frac {\frac {1888544805}{16}-\frac {438589725 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{123480}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {\int \frac {104963335785}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{864360}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {333216939 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {333216939 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 84, normalized size = 0.47 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (2499608096+14720806136 x+32535654204 x^2+31981229550 x^3+11795595705 x^4\right )}{(2+3 x)^5}-1666084695 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1536640} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^6*Sqrt[3 + 5*x]),x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2499608096 + 14720806136*x + 32535654204*x^2 + 31981229550*x^3 + 11795595705*
x^4))/(2 + 3*x)^5 - 1666084695*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/1536640

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(141)=282\).
time = 0.12, size = 298, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (11795595705 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{219520 \left (2+3 x \right )^{5} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {333216939 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(134\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (404858580885 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(404858580885*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^5+1349528602950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1799371470600*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+165138339870*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+447737213700*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+455499158856*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))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]
time = 0.49, size = 184, normalized size = 1.02 \begin {gather*} \frac {333216939}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{15 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {293 \, \sqrt {-10 \, x^{2} - x + 3}}{120 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {23909 \, \sqrt {-10 \, x^{2} - x + 3}}{1680 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {835409 \, \sqrt {-10 \, x^{2} - x + 3}}{9408 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {87374783 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

333216939/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/15*sqrt(-10*x^2 - x + 3)/(243*x
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 293/120*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 +
96*x + 16) + 23909/1680*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 835409/9408*sqrt(-10*x^2 - x + 3)
/(9*x^2 + 12*x + 4) + 87374783/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]
time = 0.83, size = 131, normalized size = 0.73 \begin {gather*} -\frac {1666084695 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (11795595705 \, x^{4} + 31981229550 \, x^{3} + 32535654204 \, x^{2} + 14720806136 \, x + 2499608096\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/3073280*(1666084695*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(11795595705*x^4 + 31981229550*x^3 + 32535654204*x^
2 + 14720806136*x + 2499608096)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)/(2+3*x)**6/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).
time = 1.02, size = 426, normalized size = 2.37 \begin {gather*} \frac {333216939}{6146560} \, \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 {121 \, \sqrt {10} {\left (8222141 \, {\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 )}^{9} + 5797080240 \, {\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 )}^{7} + 1842336276480 \, {\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 )}^{5} + 282112659584000 \, {\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 {16926759575040000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {67707038300160000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\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 )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

333216939/6146560*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 121/21952*sqrt(10)*(8222141*((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)))^9 + 5797080240*((
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)))^7 +
1842336276480*((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)))^5 + 282112659584000*((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 + 16926759575040000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 67707
038300160000*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)^5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^6\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)/((3*x + 2)^6*(5*x + 3)^(1/2)),x)

[Out]

int((1 - 2*x)^(3/2)/((3*x + 2)^6*(5*x + 3)^(1/2)), x)

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