3.2435 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{22 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{8}{75} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{98}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-22*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (814*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*
x]) - (8*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/75 - (98*Sqrt[7]*ArcTan[Sqr
t[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3

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Rubi [A]  time = 0.244266, antiderivative size = 108, 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 \[ -\frac{22 (1-2 x)^{3/2}}{15 (5 x+3)^{3/2}}+\frac{814 \sqrt{1-2 x}}{25 \sqrt{5 x+3}}-\frac{8}{75} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{98}{3} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-22*(1 - 2*x)^(3/2))/(15*(3 + 5*x)^(3/2)) + (814*Sqrt[1 - 2*x])/(25*Sqrt[3 + 5*
x]) - (8*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/75 - (98*Sqrt[7]*ArcTan[Sqr
t[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3

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Rubi in Sympy [A]  time = 22.9029, size = 99, normalized size = 0.92 \[ - \frac{22 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{814 \sqrt{- 2 x + 1}}{25 \sqrt{5 x + 3}} - \frac{8 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{375} - \frac{98 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22*(-2*x + 1)**(3/2)/(15*(5*x + 3)**(3/2)) + 814*sqrt(-2*x + 1)/(25*sqrt(5*x +
3)) - 8*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/375 - 98*sqrt(7)*atan(sqrt(7)*s
qrt(-2*x + 1)/(7*sqrt(5*x + 3)))/3

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Mathematica [A]  time = 0.400168, size = 103, normalized size = 0.95 \[ \frac{2}{375} \left (\frac{55 \sqrt{1-2 x} (565 x+328)}{(5 x+3)^{3/2}}-2 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )\right )-\frac{49}{3} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-49*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/3 + (2*((55*
Sqrt[1 - 2*x]*(328 + 565*x))/(3 + 5*x)^(3/2) - 2*Sqrt[10]*ArcTan[(1 + 20*x)/(2*S
qrt[1 - 2*x]*Sqrt[30 + 50*x])]))/375

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Maple [B]  time = 0.02, size = 184, normalized size = 1.7 \[{\frac{1}{375} \left ( 153125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-100\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+183750\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-120\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+55125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -36\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +62150\,x\sqrt{-10\,{x}^{2}-x+3}+36080\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/375*(153125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-100
*10^(1/2)*arcsin(20/11*x+1/11)*x^2+183750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x-120*10^(1/2)*arcsin(20/11*x+1/11)*x+55125*7^(1/2)*arctan(
1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-36*10^(1/2)*arcsin(20/11*x+1/11)+621
50*x*(-10*x^2-x+3)^(1/2)+36080*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^
(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51406, size = 220, normalized size = 2.04 \[ \frac{626336 \, x^{2}}{17788815 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{16 \, x^{3}}{15 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{4}{375} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{49}{3} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{313168}{88944075} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{5905573412 \, x}{88944075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3286544 \, x^{2}}{735075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{3102773174}{88944075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{11007824 \, x}{735075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{2075846}{245025 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

626336/17788815*x^2/sqrt(-10*x^2 - x + 3) - 16/15*x^3/(-10*x^2 - x + 3)^(3/2) -
4/375*sqrt(10)*arcsin(20/11*x + 1/11) + 49/3*sqrt(7)*arcsin(37/11*x/abs(3*x + 2)
+ 20/11/abs(3*x + 2)) + 313168/88944075*sqrt(-10*x^2 - x + 3) - 5905573412/8894
4075*x/sqrt(-10*x^2 - x + 3) + 3286544/735075*x^2/(-10*x^2 - x + 3)^(3/2) + 3102
773174/88944075/sqrt(-10*x^2 - x + 3) + 11007824/735075*x/(-10*x^2 - x + 3)^(3/2
) - 2075846/245025/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.23134, size = 173, normalized size = 1.6 \[ \frac{\sqrt{5}{\left (1225 \, \sqrt{7} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 22 \, \sqrt{5}{\left (565 \, x + 328\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 4 \, \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/375*sqrt(5)*(1225*sqrt(7)*sqrt(5)*(25*x^2 + 30*x + 9)*arctan(1/14*sqrt(7)*(37*
x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 22*sqrt(5)*(565*x + 328)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 4*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*
x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.302195, size = 352, normalized size = 3.26 \[ -\frac{11}{6000} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{49}{30} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4}{375} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{407}{250} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/6000*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 + 49/30*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)))) - 4/375*sqrt(10)*(pi + 2*ar
ctan(-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)))) + 407/250*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(2
2)))