3.2403 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\)

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]

(-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])

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Rubi [A]  time = 0.466291, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{2543 \sqrt{1-2 x} (5 x+3)^{5/2}}{1296 (3 x+2)^3}+\frac{37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{72 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}-\frac{32453 \sqrt{1-2 x} (5 x+3)^{3/2}}{36288 (3 x+2)^2}-\frac{3248687 \sqrt{1-2 x} \sqrt{5 x+3}}{1524096 (3 x+2)}-\frac{200}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{109715471 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4572288 \sqrt{7}} \]

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])

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Rubi in Sympy [A]  time = 44.2636, size = 189, normalized size = 0.91 \[ - \frac{4783 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{63504 \left (3 x + 2\right )^{3}} - \frac{37 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{504 \left (3 x + 2\right )^{4}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{15 \left (3 x + 2\right )^{5}} + \frac{14557 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{28224 \left (3 x + 2\right )^{2}} + \frac{1994287 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1524096 \left (3 x + 2\right )} - \frac{200 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{729} - \frac{109715471 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{32006016} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**6,x)

[Out]

-4783*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(63504*(3*x + 2)**3) - 37*(-2*x + 1)**(5/2
)*(5*x + 3)**(3/2)/(504*(3*x + 2)**4) - (-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/(15*(
3*x + 2)**5) + 14557*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(28224*(3*x + 2)**2) + 1994
287*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1524096*(3*x + 2)) - 200*sqrt(10)*asin(sqrt(22
)*sqrt(5*x + 3)/11)/729 - 109715471*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(
5*x + 3)))/32006016

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Mathematica [A]  time = 0.290551, size = 122, normalized size = 0.59 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (490413015 x^4+1809469170 x^3+2146957188 x^2+1044006792 x+180761312\right )}{(3 x+2)^5}-548577355 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-43904000 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{320060160} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(180761312 + 1044006792*x + 2146957188*x^2 + 18
09469170*x^3 + 490413015*x^4))/(2 + 3*x)^5 - 548577355*Sqrt[7]*ArcTan[(-20 - 37*
x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 43904000*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sq
rt[1 - 2*x]*Sqrt[30 + 50*x])])/320060160

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Maple [B]  time = 0.02, size = 377, normalized size = 1.8 \[{\frac{1}{320060160\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 133304297265\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}-10668672000\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-35562240000\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-47416320000\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-31610880000\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-10536960000\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+90172201896\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1404928000\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^6,x)

[Out]

1/320060160*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(133304297265*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5-10668672000*10^(1/2)*arcsin(20/11*x+1/11)*x
^5+444347657550*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-3
5562240000*arcsin(20/11*x+1/11)*10^(1/2)*x^4+592463543400*7^(1/2)*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-47416320000*10^(1/2)*arcsin(20/11*x+1/1
1)*x^3+20597346630*x^4*(-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^2-31610880000*10^(1/2)*arcsin(20/11*x+1/11)
*x^2+75997705140*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-10536960000*10^(1/2)*arcsin(20/11*x+1/11)*x+9
0172201896*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))-1404928000*10^(1/2)*arcsin(20/11*x+1/11)[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 = 1.52531, size = 360, normalized size = 1.74 \[ \frac{44881}{691488} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{333 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{1960 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{6347 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{27440 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{44881 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{768320 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{3156205}{1382976} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{52017151}{24893568} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{9235489 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{13829760 \,{\left (3 \, x + 2\right )}} + \frac{17832215}{1778112} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{100}{729} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{109715471}{64012032} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{49508071}{10668672} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^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 + 8
10*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
) - 3156205/1382976*(-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) + 17832215/177811
2*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)) - 49508071/10
668672*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.24142, size = 246, normalized size = 1.19 \[ -\frac{\sqrt{7}{\left (6272000 \, \sqrt{10} \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{7}{\left (490413015 \, x^{4} + 1809469170 \, x^{3} + 2146957188 \, x^{2} + 1044006792 \, x + 180761312\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 548577355 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{320060160 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/320060160*sqrt(7)*(6272000*sqrt(10)*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 7
20*x^2 + 240*x + 32)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x +
1))) - 6*sqrt(7)*(490413015*x^4 + 1809469170*x^3 + 2146957188*x^2 + 1044006792*x
+ 180761312)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 548577355*(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))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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

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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GIAC/XCAS [A]  time = 0.620478, size = 684, normalized size = 3.3 \[ \frac{109715471}{640120320} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{100}{729} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (3248687 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 4238260880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 2165236899840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 364930179712000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 12258004702720000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{762048 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^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)
))) - 11/762048*(3248687*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)))^9 + 4238260880*sqr
t(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 2165236899840*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)))^5 - 364930179712000*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 - 1225800470
2720000*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))))/(((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 + 2
80)^5