3.2326 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=142 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{12 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}-\frac{205}{36} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{37}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1649 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108 \sqrt{7}} \]

[Out]

(-205*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/36 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(6*(2
+ 3*x)^2) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12*(2 + 3*x)) - (37*Sqrt[10]*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (1649*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(108*Sqrt[7])

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Rubi [A]  time = 0.303902, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{12 (3 x+2)}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{6 (3 x+2)^2}-\frac{205}{36} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{37}{27} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1649 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-205*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/36 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(6*(2
+ 3*x)^2) + (37*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12*(2 + 3*x)) - (37*Sqrt[10]*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (1649*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(108*Sqrt[7])

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Rubi in Sympy [A]  time = 29.6318, size = 129, normalized size = 0.91 \[ - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{84 \left (3 x + 2\right )} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{6 \left (3 x + 2\right )^{2}} - \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{126} - \frac{37 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{27} - \frac{1649 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{756} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(84*(3*x + 2)) - (-2*x + 1)**(3/2)*(5*x + 3)
**(3/2)/(6*(3*x + 2)**2) - 107*sqrt(-2*x + 1)*sqrt(5*x + 3)/126 - 37*sqrt(10)*as
in(sqrt(22)*sqrt(5*x + 3)/11)/27 - 1649*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*s
qrt(5*x + 3)))/756

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Mathematica [A]  time = 0.175083, size = 112, normalized size = 0.79 \[ \frac{-\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (120 x^2+345 x+172\right )}{(3 x+2)^2}-1649 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-1036 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1512} \]

Antiderivative was successfully verified.

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

[Out]

((-42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(172 + 345*x + 120*x^2))/(2 + 3*x)^2 - 1649*Sq
rt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 1036*Sqrt[10]*ArcT
an[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1512

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Maple [A]  time = 0.017, size = 208, normalized size = 1.5 \[{\frac{1}{1512\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 14841\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-9324\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+19788\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-12432\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-5040\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6596\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -4144\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -14490\,x\sqrt{-10\,{x}^{2}-x+3}-7224\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1512*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(14841*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x^2-9324*10^(1/2)*arcsin(20/11*x+1/11)*x^2+19788*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-12432*10^(1/2)*arcsin(20/11*x
+1/11)*x-5040*x^2*(-10*x^2-x+3)^(1/2)+6596*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))-4144*10^(1/2)*arcsin(20/11*x+1/11)-14490*x*(-10*x^2-x+3)^(
1/2)-7224*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 1.52703, size = 176, normalized size = 1.24 \[ \frac{5}{21} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{185}{42} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{37}{54} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1649}{1512} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{769}{252} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{84 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/21*(-10*x^2 - x + 3)^(3/2) + 3/14*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) +
185/42*sqrt(-10*x^2 - x + 3)*x - 37/54*sqrt(10)*arcsin(20/11*x + 1/11) + 1649/1
512*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 769/252*sqrt(-10
*x^2 - x + 3) + 37/84*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.232256, size = 171, normalized size = 1.2 \[ -\frac{\sqrt{7}{\left (148 \, \sqrt{10} \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{7}{\left (120 \, x^{2} + 345 \, x + 172\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1649 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1512 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1512*sqrt(7)*(148*sqrt(10)*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(10)*(2
0*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(7)*(120*x^2 + 345*x + 172)*sqr
t(5*x + 3)*sqrt(-2*x + 1) - 1649*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x +
20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(9*x^2 + 12*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{\left (3 x + 2\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/(3*x + 2)**3, x)

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GIAC/XCAS [A]  time = 0.393753, size = 463, normalized size = 3.26 \[ \frac{1649}{15120} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{37}{54} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{2}{27} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{55 \,{\left (23 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 10136 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{54 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1649/15120*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)))) - 37/54*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)))) - 2/27*
sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 55/54*(23*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 + 10136*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 + 280)^2