3.24.49 \(\int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)}+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{16595712 (3 x+2)^2}-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{2963520 (3 x+2)^3}-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{1481760 (3 x+2)^4}+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{26460 (3 x+2)^5}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8605184 \sqrt {7}} \]

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Rubi [A]  time = 0.08, antiderivative size = 209, 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 = {98, 149, 151, 12, 93, 204} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)}+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{16595712 (3 x+2)^2}-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{2963520 (3 x+2)^3}-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{1481760 (3 x+2)^4}+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{26460 (3 x+2)^5}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8605184 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]

[Out]

(503*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26460*(2 + 3*x)^5) - (149951*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1481760*(2 + 3*x
)^4) - (71369*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2963520*(2 + 3*x)^3) + (958171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16595
712*(2 + 3*x)^2) + (122343637*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(232339968*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^(3
/2))/(126*(2 + 3*x)^6) - (52573169*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8605184*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 98

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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 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 {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {1}{126} \int \frac {\left (-\frac {1179}{2}-1010 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {394523}{4}-166690 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{13230}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {5498457}{8}-\frac {2249265 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{370440}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {73502625}{16}-\frac {7493745 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{7779240}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {2940587895}{32}+\frac {503039775 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{108909360}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int -\frac {149044934115}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{762365520}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}+\frac {52573169 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{17210368}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}+\frac {52573169 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{8605184}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8605184 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 193, normalized size = 0.92 \begin {gather*} \frac {1}{42} \left (\frac {591 \sqrt {1-2 x} (5 x+3)^{7/2}}{70 (3 x+2)^5}+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{(3 x+2)^6}+\frac {352839984 \sqrt {1-2 x} (5 x+3)^{7/2}-39499 (3 x+2) \left (2744 \sqrt {1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt {1-2 x} \sqrt {5 x+3} (169 x+108)+363 \sqrt {7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )\right )}{21512960 (3 x+2)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]

[Out]

((3*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(2 + 3*x)^6 + (591*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(70*(2 + 3*x)^5) + (35283
9984*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2) - 39499*(2 + 3*x)*(2744*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2) + 55*(2 + 3*x)*(7*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x]*(108 + 169*x) + 363*Sqrt[7]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]
)))/(21512960*(2 + 3*x)^4))/42

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IntegrateAlgebraic [A]  time = 0.44, size = 154, normalized size = 0.74 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {592485 (1-2 x)^5}{(5 x+3)^5}+\frac {23501905 (1-2 x)^4}{(5 x+3)^4}+\frac {383219298 (1-2 x)^3}{(5 x+3)^3}-\frac {1346749614 (1-2 x)^2}{(5 x+3)^2}-\frac {7169024135 (1-2 x)}{5 x+3}-9662860515\right )}{129077760 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^6}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8605184 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]

[Out]

(-1331*Sqrt[1 - 2*x]*(-9662860515 + (592485*(1 - 2*x)^5)/(3 + 5*x)^5 + (23501905*(1 - 2*x)^4)/(3 + 5*x)^4 + (3
83219298*(1 - 2*x)^3)/(3 + 5*x)^3 - (1346749614*(1 - 2*x)^2)/(3 + 5*x)^2 - (7169024135*(1 - 2*x))/(3 + 5*x)))/
(129077760*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))^6) - (52573169*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])
])/(8605184*Sqrt[7])

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fricas [A]  time = 1.36, size = 146, normalized size = 0.70 \begin {gather*} -\frac {788597535 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (16516390995 \, x^{5} + 55658284380 \, x^{4} + 74931979536 \, x^{3} + 50261760608 \, x^{2} + 16771747280 \, x + 2225100096\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1807088640 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^7/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1807088640*(788597535*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/1
4*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(16516390995*x^5 + 55658284380*x^4 +
74931979536*x^3 + 50261760608*x^2 + 16771747280*x + 2225100096)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916
*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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giac [B]  time = 4.64, size = 484, normalized size = 2.32 \begin {gather*} \frac {52573169}{1204725760} \, \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 {1331 \, \sqrt {10} {\left (118497 \, {\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} + 188015240 \, {\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} + 122630175360 \, {\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} - 17238395059200 \, {\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} - 3670540357120000 \, {\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 {197895383347200000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {791581533388800000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{12907776 \, {\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 )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^7/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

52573169/1204725760*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)))) - 1331/12907776*sqrt(10)*(118497*((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)))^11 + 1880152
40*((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 + 122630175360*((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 - 17238395059200*((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 - 3670540357120000*((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 - 197895383347200000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
)/sqrt(5*x + 3) + 791581533388800000*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)^6

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maple [B]  time = 0.02, size = 346, normalized size = 1.66 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (574887603015 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{5} \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^{5}[phone] \sqrt {7}\, x^{4} \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^{4}[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}[phone] \sqrt {7}\, x \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 [phone] \sqrt {7}\, \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}\right )}{1807088640 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^(5/2)/(3*x+2)^7/(-2*x+1)^(1/2),x)

[Out]

1/1807088640*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(574887603015*7^(1/2)*x^6*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))[phone]*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^
4*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^5+3406741351200*7^(1/2
)*x^3*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^4+1703370675600*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+45423218016
0*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+5047024224
0*7^(1/2)*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+31151401344*(-
10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^6

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maxima [A]  time = 1.20, size = 230, normalized size = 1.10 \begin {gather*} \frac {52573169}{120472576} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{378 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {853 \, \sqrt {-10 \, x^{2} - x + 3}}{26460 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {149951 \, \sqrt {-10 \, x^{2} - x + 3}}{1481760 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {71369 \, \sqrt {-10 \, x^{2} - x + 3}}{2963520 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {958171 \, \sqrt {-10 \, x^{2} - x + 3}}{16595712 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {122343637 \, \sqrt {-10 \, x^{2} - x + 3}}{232339968 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)/(2+3*x)^7/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

52573169/120472576*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/378*sqrt(-10*x^2 - x + 3)/(72
9*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 853/26460*sqrt(-10*x^2 - x + 3)/(243*x^5 + 8
10*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) - 149951/1481760*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 +
96*x + 16) - 71369/2963520*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 958171/16595712*sqrt(-10*x^2
- x + 3)/(9*x^2 + 12*x + 4) + 122343637/232339968*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^(5/2)/((1 - 2*x)^(1/2)*(3*x + 2)^7),x)

[Out]

int((5*x + 3)^(5/2)/((1 - 2*x)^(1/2)*(3*x + 2)^7), x)

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sympy [F(-1)]  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((3+5*x)**(5/2)/(2+3*x)**7/(1-2*x)**(1/2),x)

[Out]

Timed out

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