3.2340 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=178 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{871 \sqrt{1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac{77269 \sqrt{1-2 x} \sqrt{5 x+3}}{254016 (3 x+2)}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{7}} \]

[Out]

(-77269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(254016*(2 + 3*x)) - (871*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(6048*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*
x)^4) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3) + (100*Sqrt[10]*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (1922677*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(762048*Sqrt[7])

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Rubi [A]  time = 0.386115, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{871 \sqrt{1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac{77269 \sqrt{1-2 x} \sqrt{5 x+3}}{254016 (3 x+2)}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-77269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(254016*(2 + 3*x)) - (871*Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(6048*(2 + 3*x)^2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*
x)^4) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3) + (100*Sqrt[10]*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (1922677*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqr
t[3 + 5*x])])/(762048*Sqrt[7])

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Rubi in Sympy [A]  time = 36.8071, size = 162, normalized size = 0.91 \[ - \frac{7093 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{42336 \left (3 x + 2\right )^{2}} - \frac{181 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1512 \left (3 x + 2\right )^{3}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{12 \left (3 x + 2\right )^{4}} + \frac{390869 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{254016 \left (3 x + 2\right )} + \frac{100 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{243} - \frac{1922677 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{5334336} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7093*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(42336*(3*x + 2)**2) - 181*(-2*x + 1)**(3/
2)*(5*x + 3)**(3/2)/(1512*(3*x + 2)**3) - (-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(12
*(3*x + 2)**4) + 390869*sqrt(-2*x + 1)*sqrt(5*x + 3)/(254016*(3*x + 2)) + 100*sq
rt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/243 - 1922677*sqrt(7)*atan(sqrt(7)*sqrt(-
2*x + 1)/(7*sqrt(5*x + 3)))/5334336

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Mathematica [A]  time = 0.245049, size = 117, normalized size = 0.66 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (13290147 x^3+23185560 x^2+13434180 x+2583760\right )}{(3 x+2)^4}-1922677 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+2195200 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{10668672} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2583760 + 13434180*x + 23185560*x^2 + 13290147
*x^3))/(2 + 3*x)^4 - 1922677*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[
3 + 5*x])] + 2195200*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x]
)])/10668672

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Maple [B]  time = 0.018, size = 315, normalized size = 1.8 \[{\frac{1}{10668672\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 155736837\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+210739200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+973793520\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10668672*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(155736837*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+177811200*arcsin(20/11*x+1/11)*10^(1/2)*x^4+415
298232*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+474163200*
10^(1/2)*arcsin(20/11*x+1/11)*x^3+415298232*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^2+474163200*10^(1/2)*arcsin(20/11*x+1/11)*x^2+558186174
*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+210739200*10^(1/2)*arcsin(20/11*x+1/11)*x+973793520*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))+
35123200*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)^4

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Maxima [A]  time = 1.51684, size = 266, normalized size = 1.49 \[ \frac{27065}{148176} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1176 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{5413 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32928 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{528205}{296352} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{50}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1922677}{10668672} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{802877}{1778112} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3667 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{197568 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27065/148176*(-10*x^2 - x + 3)^(3/2) - 1/28*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 21
6*x^3 + 216*x^2 + 96*x + 16) + 169/1176*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2
+ 36*x + 8) + 5413/32928*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 528205/29
6352*sqrt(-10*x^2 - x + 3)*x + 50/243*sqrt(10)*arcsin(20/11*x + 1/11) + 1922677/
10668672*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 802877/1778
112*sqrt(-10*x^2 - x + 3) + 3667/197568*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.235769, size = 219, normalized size = 1.23 \[ \frac{\sqrt{7}{\left (313600 \, \sqrt{10} \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (13290147 \, x^{3} + 23185560 \, x^{2} + 13434180 \, x + 2583760\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1922677 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{10668672 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10668672*sqrt(7)*(313600*sqrt(10)*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x +
16)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)
*(13290147*x^3 + 23185560*x^2 + 13434180*x + 2583760)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 1922677*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x
+ 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.495242, size = 602, normalized size = 3.38 \[ \frac{1922677}{106686720} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{50}{243} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (77269 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 81002040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 31057924800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 8580356288000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{127008 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1922677/106686720*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)))) + 50/243*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))
- 11/127008*(77269*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)))^7 + 81002040*sqrt(10)*(
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))^5 + 31057924800*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 - 8580356288000*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(2
2)))^2 + 280)^4