3.2413 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=115 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}+\frac{74}{45} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{346}{135} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{175}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(74*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/45 + (7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 +
3*x)) + (346*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/135 - (175*Sqrt[7]*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi [A]  time = 0.237379, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{7 \sqrt{5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}+\frac{74}{45} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{346}{135} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{175}{27} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(74*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/45 + (7*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 +
3*x)) + (346*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/135 - (175*Sqrt[7]*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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Rubi in Sympy [A]  time = 23.4036, size = 102, normalized size = 0.89 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{3 \left (3 x + 2\right )} + \frac{74 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{45} + \frac{346 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{675} - \frac{175 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(3*(3*x + 2)) + 74*sqrt(-2*x + 1)*sqrt(5*x + 3
)/45 + 346*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/675 - 175*sqrt(7)*atan(sqrt(
7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/27

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Mathematica [A]  time = 0.182567, size = 107, normalized size = 0.93 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} (12 x+253)}{3 x+2}-4375 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+346 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{1350} \]

Antiderivative was successfully verified.

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

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(253 + 12*x))/(2 + 3*x) - 4375*Sqrt[7]*ArcTan[(
-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 346*Sqrt[10]*ArcTan[(1 + 20*x)/(
2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/1350

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Maple [A]  time = 0.02, size = 146, normalized size = 1.3 \[{\frac{1}{2700+4050\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1038\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+13125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+692\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +8750\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +360\,x\sqrt{-10\,{x}^{2}-x+3}+7590\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1350*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1038*10^(1/2)*arcsin(20/11*x+1/11)*x+13125*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+692*10^(1/2)*arcsin(
20/11*x+1/11)+8750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+36
0*x*(-10*x^2-x+3)^(1/2)+7590*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.49639, size = 101, normalized size = 0.88 \[ \frac{173}{675} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{175}{54} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{4}{45} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

173/675*sqrt(10)*arcsin(20/11*x + 1/11) + 175/54*sqrt(7)*arcsin(37/11*x/abs(3*x
+ 2) + 20/11/abs(3*x + 2)) + 4/45*sqrt(-10*x^2 - x + 3) + 49/9*sqrt(-10*x^2 - x
+ 3)/(3*x + 2)

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Fricas [A]  time = 0.230922, size = 153, normalized size = 1.33 \[ \frac{\sqrt{5}{\left (875 \, \sqrt{7} \sqrt{5}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 6 \, \sqrt{5}{\left (12 \, x + 253\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 346 \, \sqrt{2}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1350 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1350*sqrt(5)*(875*sqrt(7)*sqrt(5)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)/(s
qrt(5*x + 3)*sqrt(-2*x + 1))) + 6*sqrt(5)*(12*x + 253)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 346*sqrt(2)*(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)**(5/2)/(2+3*x)**2/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.33766, size = 377, normalized size = 3.28 \[ \frac{35}{108} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{173}{675} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{4}{225} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{9 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35/108*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)
))) + 173/675*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)))) + 4/225*s
qrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 1078/9*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)