3.25.22 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx\) [2422]

Optimal. Leaf size=238 \[ -\frac {29794435 \sqrt {1-2 x} \sqrt {3+5 x}}{2458624 (2+3 x)}-\frac {2708585 \sqrt {1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac {49247 \sqrt {1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}-\frac {327738785 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2458624 \sqrt {7}} \]

[Out]

3/49*(1-2*x)^(7/2)*(3+5*x)^(7/2)/(2+3*x)^7+37/84*(1-2*x)^(5/2)*(3+5*x)^(7/2)/(2+3*x)^6+407/168*(1-2*x)^(3/2)*(
3+5*x)^(7/2)/(2+3*x)^5-327738785/17210368*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-2708585/5268
48*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-49247/18816*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3+4477/448*(3+5*x)^(7
/2)*(1-2*x)^(1/2)/(2+3*x)^4-29794435/2458624*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.06, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \begin {gather*} -\frac {327738785 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2458624 \sqrt {7}}+\frac {4477 \sqrt {1-2 x} (5 x+3)^{7/2}}{448 (3 x+2)^4}+\frac {407 (1-2 x)^{3/2} (5 x+3)^{7/2}}{168 (3 x+2)^5}+\frac {37 (1-2 x)^{5/2} (5 x+3)^{7/2}}{84 (3 x+2)^6}+\frac {3 (1-2 x)^{7/2} (5 x+3)^{7/2}}{49 (3 x+2)^7}-\frac {49247 \sqrt {1-2 x} (5 x+3)^{5/2}}{18816 (3 x+2)^3}-\frac {2708585 \sqrt {1-2 x} (5 x+3)^{3/2}}{526848 (3 x+2)^2}-\frac {29794435 \sqrt {1-2 x} \sqrt {5 x+3}}{2458624 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]

[Out]

(-29794435*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2458624*(2 + 3*x)) - (2708585*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(526848*
(2 + 3*x)^2) - (49247*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(18816*(2 + 3*x)^3) + (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(7/2))
/(49*(2 + 3*x)^7) + (37*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(84*(2 + 3*x)^6) + (407*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/
2))/(168*(2 + 3*x)^5) + (4477*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(448*(2 + 3*x)^4) - (327738785*ArcTan[Sqrt[1 - 2*
x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2458624*Sqrt[7])

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 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37}{14} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {2035}{168} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477}{112} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac {49247}{896} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {49247 \sqrt {1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac {2708585 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{37632}\\ &=-\frac {2708585 \sqrt {1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac {49247 \sqrt {1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac {29794435 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{351232}\\ &=-\frac {29794435 \sqrt {1-2 x} \sqrt {3+5 x}}{2458624 (2+3 x)}-\frac {2708585 \sqrt {1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac {49247 \sqrt {1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac {327738785 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4917248}\\ &=-\frac {29794435 \sqrt {1-2 x} \sqrt {3+5 x}}{2458624 (2+3 x)}-\frac {2708585 \sqrt {1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac {49247 \sqrt {1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}+\frac {327738785 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2458624}\\ &=-\frac {29794435 \sqrt {1-2 x} \sqrt {3+5 x}}{2458624 (2+3 x)}-\frac {2708585 \sqrt {1-2 x} (3+5 x)^{3/2}}{526848 (2+3 x)^2}-\frac {49247 \sqrt {1-2 x} (3+5 x)^{5/2}}{18816 (2+3 x)^3}+\frac {3 (1-2 x)^{7/2} (3+5 x)^{7/2}}{49 (2+3 x)^7}+\frac {37 (1-2 x)^{5/2} (3+5 x)^{7/2}}{84 (2+3 x)^6}+\frac {407 (1-2 x)^{3/2} (3+5 x)^{7/2}}{168 (2+3 x)^5}+\frac {4477 \sqrt {1-2 x} (3+5 x)^{7/2}}{448 (2+3 x)^4}-\frac {327738785 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2458624 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.46, size = 94, normalized size = 0.39 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (5897927808+52456780256 x+194338741616 x^2+384048502848 x^3+427105196104 x^4+253441751890 x^5+62659925205 x^6\right )}{(2+3 x)^7}-983216355 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{51631104} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^8,x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5897927808 + 52456780256*x + 194338741616*x^2 + 384048502848*x^3 + 4271051961
04*x^4 + 253441751890*x^5 + 62659925205*x^6))/(2 + 3*x)^7 - 983216355*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sq
rt[3 + 5*x])])/51631104

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(187)=374\).
time = 0.16, size = 394, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (62659925205 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{7375872 \left (2+3 x \right )^{7} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {327738785 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{34420736 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(144\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (2150294168385 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}[phone] \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) \sqrt {7}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{103262208 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\)	\(394\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x,method=_RETURNVERBOSE)

[Out]

1/103262208*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2150294168385*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^7+10034706119130*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^6+20069412238260*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+877238952870*(-10*x^2-x+3)^(1/2)*x^6+22299346931400*7^(1
/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+3548184526460*x^5*(-10*x^2-x+3)^(1/2)[phone]
0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+5979472745456*x^4*(-10*x^2-x+3)^(1/2)+5946492
515040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+5376679039872*x^3*(-10*x^2-x+3)^(1/2)+13
21442781120*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+2720742382624*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)+82
570989312*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^7

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Maxima [A]
time = 0.58, size = 353, normalized size = 1.48 \begin {gather*} \frac {122277415}{271063296} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{49 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{196 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {1369 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{2744 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {162319 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{153664 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {3024121 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{2151296 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {24455483 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {2190708025}{180708864} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {4205402795}{361417728} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {4059472427 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1084253184 \, {\left (3 \, x + 2\right )}} + \frac {501088225}{8605184} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {327738785}{34420736} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {441499355}{17210368} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="maxima")

[Out]

122277415/271063296*(-10*x^2 - x + 3)^(5/2) + 3/49*(-10*x^2 - x + 3)^(7/2)/(2187*x^7 + 10206*x^6 + 20412*x^5 +
22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 37/196*(-10*x^2 - x + 3)^(7/2)/(729*x^6 + 2916*x^5 + 4860*
x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 1369/2744*(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 72
0*x^2 + 240*x + 32) + 162319/153664*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 3024121
/2151296*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 24455483/60236288*(-10*x^2 - x + 3)^(7/2)/(9*x
^2 + 12*x + 4) - 2190708025/180708864*(-10*x^2 - x + 3)^(3/2)*x + 4205402795/361417728*(-10*x^2 - x + 3)^(3/2)
- 4059472427/1084253184*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 501088225/8605184*sqrt(-10*x^2 - x + 3)*x + 32773
8785/34420736*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 441499355/17210368*sqrt(-10*x^2 - x
+ 3)

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Fricas [A]
time = 0.53, size = 161, normalized size = 0.68 \begin {gather*} -\frac {983216355 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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 (62659925205 \, x^{6} + 253441751890 \, x^{5} + 427105196104 \, x^{4} + 384048502848 \, x^{3} + 194338741616 \, x^{2} + 52456780256 \, x + 5897927808\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{103262208 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/103262208*(983216355*sqrt(7)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x
+ 128)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(62659925205*x^6 +
253441751890*x^5 + 427105196104*x^4 + 384048502848*x^3 + 194338741616*x^2 + 52456780256*x + 5897927808)*sqrt(5
*x + 3)*sqrt(-2*x + 1))/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**8,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).
time = 1.15, size = 542, normalized size = 2.28 \begin {gather*} \frac {65547757}{68841472} \, \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 {8857805 \, \sqrt {10} {\left (111 \, {\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 )}^{13} + 207200 \, {\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 )}^{11} + 164185280 \, {\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} - 63583027200 \, {\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} - 12872125952000 \, {\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} - 1273567232000000 \, {\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 {53489823744000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {213959294976000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3687936 \, {\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 )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^8,x, algorithm="giac")

[Out]

65547757/68841472*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)))) - 8857805/3687936*sqrt(10)*(111*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^13 + 207200*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 +
164185280*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22)))^9 - 63583027200*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^7 - 12872125952000*((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 - 1273567232000000*((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 - 53489823744000000*(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) + 213959294976000000*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)^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^8} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^8,x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^8, x)

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