3.2439 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{5 x+3}}+\frac{2992825 \sqrt{1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{25024175 \sqrt{1-2 x}}{1344 (5 x+3)^{3/2}}-\frac{519421265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

[Out]

(-25024175*Sqrt[1 - 2*x])/(1344*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(12*(2 +
3*x)^4*(3 + 5*x)^(3/2)) + (847*Sqrt[1 - 2*x])/(72*(2 + 3*x)^3*(3 + 5*x)^(3/2)) +
(36817*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (2992825*Sqrt[1 - 2*x
])/(1344*(2 + 3*x)*(3 + 5*x)^(3/2)) + (227000875*Sqrt[1 - 2*x])/(1344*Sqrt[3 + 5
*x]) - (519421265*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

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Rubi [A]  time = 0.479682, antiderivative size = 195, 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 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac{227000875 \sqrt{1-2 x}}{1344 \sqrt{5 x+3}}+\frac{2992825 \sqrt{1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac{36817 \sqrt{1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{847 \sqrt{1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{25024175 \sqrt{1-2 x}}{1344 (5 x+3)^{3/2}}-\frac{519421265 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-25024175*Sqrt[1 - 2*x])/(1344*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(12*(2 +
3*x)^4*(3 + 5*x)^(3/2)) + (847*Sqrt[1 - 2*x])/(72*(2 + 3*x)^3*(3 + 5*x)^(3/2)) +
(36817*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (2992825*Sqrt[1 - 2*x
])/(1344*(2 + 3*x)*(3 + 5*x)^(3/2)) + (227000875*Sqrt[1 - 2*x])/(1344*Sqrt[3 + 5
*x]) - (519421265*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(448*Sqrt[7])

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Rubi in Sympy [A]  time = 43.4552, size = 180, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12 \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{227000875 \sqrt{- 2 x + 1}}{1344 \sqrt{5 x + 3}} - \frac{25024175 \sqrt{- 2 x + 1}}{1344 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2992825 \sqrt{- 2 x + 1}}{1344 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{36817 \sqrt{- 2 x + 1}}{288 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{847 \sqrt{- 2 x + 1}}{72 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{519421265 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3136} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(12*(3*x + 2)**4*(5*x + 3)**(3/2)) + 227000875*sqrt(-2*x + 1
)/(1344*sqrt(5*x + 3)) - 25024175*sqrt(-2*x + 1)/(1344*(5*x + 3)**(3/2)) + 29928
25*sqrt(-2*x + 1)/(1344*(3*x + 2)*(5*x + 3)**(3/2)) + 36817*sqrt(-2*x + 1)/(288*
(3*x + 2)**2*(5*x + 3)**(3/2)) + 847*sqrt(-2*x + 1)/(72*(3*x + 2)**3*(5*x + 3)**
(3/2)) - 519421265*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/3136

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Mathematica [A]  time = 0.124333, size = 92, normalized size = 0.47 \[ \frac{\sqrt{1-2 x} \left (91935354375 x^5+298295199450 x^4+386933096475 x^3+250814924064 x^2+81243850516 x+10520317456\right )}{1344 (3 x+2)^4 (5 x+3)^{3/2}}-\frac{519421265 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{896 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(10520317456 + 81243850516*x + 250814924064*x^2 + 386933096475*x^
3 + 298295199450*x^4 + 91935354375*x^5))/(1344*(2 + 3*x)^4*(3 + 5*x)^(3/2)) - (5
19421265*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(896*Sqrt[7])

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Maple [B]  time = 0.023, size = 346, normalized size = 1.8 \[{\frac{1}{18816\, \left ( 2+3\,x \right ) ^{4}} \left ( 3155484184875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3511408936896\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18816*(3155484184875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^6+12201205514850*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^5+19648148191155*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+1287094961250*x^5*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+4176132792300*x^4*(-10*x^2-x+3)^(1/2)+8140
370065080*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+5417063
350650*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x+3511408936896*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(
1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2
-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^4/(-10*x^2-x
+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.5182, size = 439, normalized size = 2.25 \[ \frac{519421265}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{227000875 \, x}{672 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{79003515}{448 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{24449315 \, x}{288 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{324 \,{\left (81 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 216 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 96 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 16 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{37387}{648 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{571291}{864 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{60813781}{5184 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{237706249}{5184 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

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)^5),x, algorithm="maxima")

[Out]

519421265/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 22700
0875/672*x/sqrt(-10*x^2 - x + 3) + 79003515/448/sqrt(-10*x^2 - x + 3) + 24449315
/288*x/(-10*x^2 - x + 3)^(3/2) + 2401/324/(81*(-10*x^2 - x + 3)^(3/2)*x^4 + 216*
(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^(3/2)*x^2 + 96*(-10*x^2 - x
+ 3)^(3/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 37387/648/(27*(-10*x^2 - x + 3)^(3/
2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*
x^2 - x + 3)^(3/2)) + 571291/864/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 -
x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 60813781/5184/(3*(-10*x^2 - x + 3)
^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 237706249/5184/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.227094, size = 188, normalized size = 0.96 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (91935354375 \, x^{5} + 298295199450 \, x^{4} + 386933096475 \, x^{3} + 250814924064 \, x^{2} + 81243850516 \, x + 10520317456\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1558263795 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{18816 \,{\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

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)^5),x, algorithm="fricas")

[Out]

1/18816*sqrt(7)*(2*sqrt(7)*(91935354375*x^5 + 298295199450*x^4 + 386933096475*x^
3 + 250814924064*x^2 + 81243850516*x + 10520317456)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 1558263795*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x +
144)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(2025*x^6
+ 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.513274, size = 674, normalized size = 3.46 \[ -\frac{55}{48} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{103884253}{12544} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{9295}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{55 \,{\left (6089929 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 4375094808 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1081495934400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 90973105216000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{224 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

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)^5),x, algorithm="giac")

[Out]

-55/48*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 + 103884253/12544*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)))) + 9295/2*sqrt(10)*(
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))) + 55/224*(6089929*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))
)^7 + 4375094808*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)))^5 + 1081495934400*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 + 90973105216000*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))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sq
rt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4