3.2420 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=207 \[ \frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{1296 (3 x+2)^3}+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{72 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{36288 (3 x+2)^2}-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{1524096 (3 x+2)}-\frac {200}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {109715471 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4572288 \sqrt {7}} \]

[Out]

-1/15*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^5+37/72*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^4-109715471/32006016*arc
tan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-200/729*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-3245
3/36288*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2+2543/1296*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3-3248687/1524096*
(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.08, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ \frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{1296 (3 x+2)^3}+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{72 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{36288 (3 x+2)^2}-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{1524096 (3 x+2)}-\frac {200}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {109715471 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4572288 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3248687*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1524096*(2 + 3*x)) - (32453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36288*(2 +
3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(15*(2 + 3*x)^5) + (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(72*(2 +
3*x)^4) + (2543*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1296*(2 + 3*x)^3) - (200*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/729 - (109715471*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4572288*Sqrt[7])

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

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 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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])

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}-\frac {1}{180} \int \frac {\left (-\frac {5305}{4}-400 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {\int \frac {\left (\frac {149465}{8}-2400 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1620}\\ &=-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {\int \frac {\left (\frac {14451435}{16}-168000 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{68040}\\ &=-\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {\int \frac {\frac {423137355}{32}-5880000 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1428840}\\ &=-\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac {1000}{729} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {109715471 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{9144576}\\ &=-\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {109715471 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{4572288}-\frac {1}{729} \left (400 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac {200}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {109715471 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4572288 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.38, size = 144, normalized size = 0.70 \[ \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (490413015 x^4+1809469170 x^3+2146957188 x^2+1044006792 x+180761312\right )-548577355 \sqrt {14 x-7} (3 x+2)^5 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )[phone] \sqrt {10-20 x} (3 x+2)^5 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{160030080 \sqrt {2 x-1} (3 x+2)^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(21*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(180761312 + 1044006792*x + 2146957188*x^2 + 1809469170*x^3 + 490413015*x
^4) + 43904000*Sqrt[10 - 20*x]*(2 + 3*x)^5*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 548577355*(2 + 3*x)^5*Sqrt[-7
+ 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(160030080*Sqrt[-1 + 2*x]*(2 + 3*x)^5)

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fricas [A]  time = 0.93, size = 196, normalized size = 0.95 \[ -\frac {548577355 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 43904000 \, \sqrt {10} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (490413015 \, x^{4} + 1809469170 \, x^{3} + 2146957188 \, x^{2} + 1044006792 \, x + 180761312\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{320060160 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

-1/320060160*(548577355*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)) - 43904000*sqrt(10)*(243*x^5 + 810*x^4 + 1080*x^3 + 720
*x^2 + 240*x + 32)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(490413
015*x^4 + 1809469170*x^3 + 2146957188*x^2 + 1044006792*x + 180761312)*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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giac [B]  time = 5.69, size = 493, normalized size = 2.38 \[ \frac {109715471}{640120320} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {100}{729} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (3248687 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 4238260880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 2165236899840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 364930179712000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {12258004702720000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {49032018810880000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{762048 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

109715471/640120320*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)))) - 100/729*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 11/7
62048*sqrt(10)*(3248687*((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)))^9 + 4238260880*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sq
rt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 2165236899840*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*s
qrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 364930179712000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 12258004702720000*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) + 49032018810880000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqr
t(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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maple [B]  time = 0.02, size = 377, normalized size = 1.82 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (10668672000 \sqrt {10}\, x^{5} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-133304297265 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {10}\, x^{4} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-444347657550 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-20597346630 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-592463543400 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-75997705140 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-394975695600 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-90172201896 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-131658565200 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-43848285264 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-17554475360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7591975104 \sqrt {-10 x^{2}-x +3}\right )}{320060160 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/320060160*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(10668672000*10^(1/2)*arcsin(20/11*x+1/11)*x^5-133304297265*7^(1/2)*
x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*10^(1/2)*x^4*arcsin(20/11*x+1/11)-444347657
550*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*10^(1/2)*x^3*arcsin(20/11*x+1/1
1)-592463543400*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-20597346630*(-10*x^2-x+3)^(1/2)
*x^4+31610880000*10^(1/2)*x^2*arcsin(20/11*x+1/11)-394975695600*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10
*x^2-x+3)^(1/2))-75997705140*(-10*x^2-x+3)^(1/2)*x^3+10536960000*10^(1/2)*x*arcsin(20/11*x+1/11)-131658565200*
7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-90172201896*(-10*x^2-x+3)^(1/2)*x^2+1404928000*10
^(1/2)*arcsin(20/11*x+1/11)-17554475360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-43848285264
*(-10*x^2-x+3)^(1/2)*x-7591975104*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^5

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maxima [A]  time = 1.36, size = 267, normalized size = 1.29 \[ \frac {44881}{691488} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {333 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{1960 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{27440 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {44881 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{768320 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {3156205}{1382976} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {52017151}{24893568} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {9235489 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{13829760 \, {\left (3 \, x + 2\right )}} + \frac {17832215}{1778112} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {100}{729} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {109715471}{64012032} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {49508071}{10668672} \, \sqrt {-10 \, x^{2} - x + 3} \]

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)^6,x, algorithm="maxima")

[Out]

44881/691488*(-10*x^2 - x + 3)^(5/2) + 3/35*(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 +
240*x + 32) + 333/1960*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 6347/27440*(-10*x^2
- x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 44881/768320*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 315620
5/1382976*(-10*x^2 - x + 3)^(3/2)*x + 52017151/24893568*(-10*x^2 - x + 3)^(3/2) - 9235489/13829760*(-10*x^2 -
x + 3)^(5/2)/(3*x + 2) + 17832215/1778112*sqrt(-10*x^2 - x + 3)*x - 100/729*sqrt(10)*arcsin(20/11*x + 1/11) +
109715471/64012032*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 49508071/10668672*sqrt(-10*x^2
- x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^6} \,d x \]

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)^6,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**6,x)

[Out]

Timed out

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