3.2337 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=159 \[ -\frac{8}{27} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{25}{12} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{3065 \sqrt{1-2 x} \sqrt{5 x+3}}{1296}-\frac{43 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3888}-\frac{181}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-3065*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1296 + (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12
- (8*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/27 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(
2 + 3*x)) - (43*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3888 - (181*Sqrt[7]*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi [A]  time = 0.378629, antiderivative size = 159, 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 \[ -\frac{8}{27} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{3 (3 x+2)}+\frac{25}{12} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{3065 \sqrt{1-2 x} \sqrt{5 x+3}}{1296}-\frac{43 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3888}-\frac{181}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3065*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1296 + (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/12
- (8*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/27 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(3*(
2 + 3*x)) - (43*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3888 - (181*Sqrt[7]*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi in Sympy [A]  time = 37.6565, size = 141, normalized size = 0.89 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (3 x + 2\right )} - \frac{8 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{27} + \frac{25 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{12} - \frac{3065 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1296} - \frac{43 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{7776} - \frac{181 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{243} \]

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)**2,x)

[Out]

-(-2*x + 1)**(3/2)*(5*x + 3)**(5/2)/(3*(3*x + 2)) - 8*sqrt(-2*x + 1)*(5*x + 3)**
(5/2)/27 + 25*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/12 - 3065*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/1296 - 43*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/7776 - 181*sqrt(7)*atan(
sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/243

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Mathematica [A]  time = 0.209881, size = 117, normalized size = 0.74 \[ \frac{\frac{12 \sqrt{1-2 x} \sqrt{5 x+3} \left (-7200 x^3+1860 x^2+3513 x-730\right )}{3 x+2}-5792 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-43 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{15552} \]

Antiderivative was successfully verified.

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

[Out]

((12*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-730 + 3513*x + 1860*x^2 - 7200*x^3))/(2 + 3*x
) - 5792*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 43*Sqrt
[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/15552

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Maple [A]  time = 0.018, size = 180, normalized size = 1.1 \[{\frac{1}{31104+46656\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -86400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+17376\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-129\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+22320\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+11584\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -86\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +42156\,x\sqrt{-10\,{x}^{2}-x+3}-8760\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^2,x)

[Out]

1/15552*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-86400*x^3*(-10*x^2-x+3)^(1/2)+17376*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-129*10^(1/2)*arcsin(20/11
*x+1/11)*x+22320*x^2*(-10*x^2-x+3)^(1/2)+11584*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))-86*10^(1/2)*arcsin(20/11*x+1/11)+42156*x*(-10*x^2-x+3)
^(1/2)-8760*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.50101, size = 140, normalized size = 0.88 \[ \frac{5}{27} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{245}{108} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{43}{15552} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{181}{486} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1301}{1296} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{9 \,{\left (3 \, x + 2\right )}} \]

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)^2,x, algorithm="maxima")

[Out]

5/27*(-10*x^2 - x + 3)^(3/2) + 245/108*sqrt(-10*x^2 - x + 3)*x - 43/15552*sqrt(1
0)*arcsin(20/11*x + 1/11) + 181/486*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/
abs(3*x + 2)) - 1301/1296*sqrt(-10*x^2 - x + 3) + 1/9*(-10*x^2 - x + 3)^(3/2)/(3
*x + 2)

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Fricas [A]  time = 0.233743, size = 166, normalized size = 1.04 \[ \frac{\sqrt{2}{\left (2896 \, \sqrt{7} \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{2}{\left (7200 \, x^{3} - 1860 \, x^{2} - 3513 \, x + 730\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 43 \, \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{15552 \,{\left (3 \, x + 2\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/15552*sqrt(2)*(2896*sqrt(7)*sqrt(2)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(2)*(7200*x^3 - 1860*x^2 - 3513*x + 730)
*sqrt(5*x + 3)*sqrt(-2*x + 1) - 43*sqrt(5)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)
*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(3*x + 2)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.396441, size = 412, normalized size = 2.59 \[ \frac{181}{4860} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{2160} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 85 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 835 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{43}{15552} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{154 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{81 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

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)^2,x, algorithm="giac")

[Out]

181/4860*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(2
2)))) - 1/2160*(4*(8*sqrt(5)*(5*x + 3) - 85*sqrt(5))*(5*x + 3) + 835*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5) - 43/15552*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)))) - 154/81*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(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^2 + 280)