3.2358 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{22 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}+\frac{4}{15} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{14}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-22*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/15 + (14*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3

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Rubi [A]  time = 0.169635, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{22 \sqrt{1-2 x}}{5 \sqrt{5 x+3}}+\frac{4}{15} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{14}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-22*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) + (4*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/15 + (14*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3

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Rubi in Sympy [A]  time = 15.861, size = 78, normalized size = 0.91 \[ - \frac{22 \sqrt{- 2 x + 1}}{5 \sqrt{5 x + 3}} + \frac{4 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{75} + \frac{14 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22*sqrt(-2*x + 1)/(5*sqrt(5*x + 3)) + 4*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11
)/75 + 14*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/3

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Mathematica [A]  time = 0.225714, size = 97, normalized size = 1.13 \[ \frac{7}{3} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+\frac{2}{75} \left (\sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )-\frac{165 \sqrt{1-2 x}}{\sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/3 + (2*((-165*
Sqrt[1 - 2*x])/Sqrt[3 + 5*x] + Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[
30 + 50*x])]))/75

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Maple [B]  time = 0.02, size = 124, normalized size = 1.4 \[ -{\frac{1}{75} \left ( 875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-10\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+525\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -6\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +330\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/75*(875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-10*10^(1
/2)*arcsin(20/11*x+1/11)*x+525*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))-6*10^(1/2)*arcsin(20/11*x+1/11)+330*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)
/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49824, size = 93, normalized size = 1.08 \[ \frac{2}{75} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{7}{3} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{44 \, x}{5 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{22}{5 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/75*sqrt(10)*arcsin(20/11*x + 1/11) - 7/3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) +
20/11/abs(3*x + 2)) + 44/5*x/sqrt(-10*x^2 - x + 3) - 22/5/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.227015, size = 146, normalized size = 1.7 \[ -\frac{\sqrt{5}{\left (35 \, \sqrt{7} \sqrt{5}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 2 \, \sqrt{2}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 66 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}\right )}}{75 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/75*sqrt(5)*(35*sqrt(7)*sqrt(5)*(5*x + 3)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqr
t(5*x + 3)*sqrt(-2*x + 1))) - 2*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(2
0*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 66*sqrt(5)*sqrt(5*x + 3)*sqrt(-2*x +
1))/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273833, size = 270, normalized size = 3.14 \[ -\frac{7}{30} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{2}{75} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11}{50} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/30*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
)) + 2/75*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)))) - 11/50*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)))