3.2428 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=173 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{3997345 \sqrt{1-2 x}}{4032 (3 x+2) \sqrt{5 x+3}}+\frac{22957 \sqrt{1-2 x}}{288 (3 x+2)^2 \sqrt{5 x+3}}+\frac{2051 \sqrt{1-2 x}}{216 (3 x+2)^3 \sqrt{5 x+3}}-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{5 x+3}}+\frac{46095555 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

[Out]

(-181304825*Sqrt[1 - 2*x])/(12096*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(12*(2 +
3*x)^4*Sqrt[3 + 5*x]) + (2051*Sqrt[1 - 2*x])/(216*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (
22957*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (3997345*Sqrt[1 - 2*x])/(
4032*(2 + 3*x)*Sqrt[3 + 5*x]) + (46095555*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(448*Sqrt[7])

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Rubi [A]  time = 0.394061, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{3997345 \sqrt{1-2 x}}{4032 (3 x+2) \sqrt{5 x+3}}+\frac{22957 \sqrt{1-2 x}}{288 (3 x+2)^2 \sqrt{5 x+3}}+\frac{2051 \sqrt{1-2 x}}{216 (3 x+2)^3 \sqrt{5 x+3}}-\frac{181304825 \sqrt{1-2 x}}{12096 \sqrt{5 x+3}}+\frac{46095555 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{448 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-181304825*Sqrt[1 - 2*x])/(12096*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(12*(2 +
3*x)^4*Sqrt[3 + 5*x]) + (2051*Sqrt[1 - 2*x])/(216*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (
22957*Sqrt[1 - 2*x])/(288*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (3997345*Sqrt[1 - 2*x])/(
4032*(2 + 3*x)*Sqrt[3 + 5*x]) + (46095555*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 +
5*x])])/(448*Sqrt[7])

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Rubi in Sympy [A]  time = 36.1374, size = 160, normalized size = 0.92 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{12 \left (3 x + 2\right )^{4} \sqrt{5 x + 3}} - \frac{181304825 \sqrt{- 2 x + 1}}{12096 \sqrt{5 x + 3}} + \frac{3997345 \sqrt{- 2 x + 1}}{4032 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{22957 \sqrt{- 2 x + 1}}{288 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{2051 \sqrt{- 2 x + 1}}{216 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{46095555 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{3136} \]

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)**(3/2),x)

[Out]

7*(-2*x + 1)**(3/2)/(12*(3*x + 2)**4*sqrt(5*x + 3)) - 181304825*sqrt(-2*x + 1)/(
12096*sqrt(5*x + 3)) + 3997345*sqrt(-2*x + 1)/(4032*(3*x + 2)*sqrt(5*x + 3)) + 2
2957*sqrt(-2*x + 1)/(288*(3*x + 2)**2*sqrt(5*x + 3)) + 2051*sqrt(-2*x + 1)/(216*
(3*x + 2)**3*sqrt(5*x + 3)) + 46095555*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sq
rt(5*x + 3)))/3136

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Mathematica [A]  time = 0.116609, size = 87, normalized size = 0.5 \[ \frac{46095555 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (543914475 x^4+1438446565 x^3+1426133132 x^2+628209228 x+103735088\right )}{(3 x+2)^4 \sqrt{5 x+3}}}{6272} \]

Antiderivative was successfully verified.

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

[Out]

((-14*Sqrt[1 - 2*x]*(103735088 + 628209228*x + 1426133132*x^2 + 1438446565*x^3 +
543914475*x^4))/((2 + 3*x)^4*Sqrt[3 + 5*x]) + 46095555*Sqrt[7]*ArcTan[(-20 - 37
*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/6272

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Maple [B]  time = 0.022, size = 298, normalized size = 1.7 \[ -{\frac{1}{6272\, \left ( 2+3\,x \right ) ^{4}} \left ( 18668699775\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+19965863848\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6272*(18668699775*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^5+60984419265*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+7
9653119040*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+761480
2650*x^4*(-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^2+20138251910*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+19965863848*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))+8
794929192*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)^(1/2)

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Maxima [A]  time = 1.52273, size = 400, normalized size = 2.31 \[ -\frac{46095555}{6272} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{181304825 \, x}{6048 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{189299515}{12096 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{108 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{13181}{648 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{466361}{2592 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{1301839}{576 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-46095555/6272*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 18130
4825/6048*x/sqrt(-10*x^2 - x + 3) - 189299515/12096/sqrt(-10*x^2 - x + 3) + 343/
108/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3 + 216*sqrt(-10
*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 131
81/648/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-1
0*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 466361/2592/(9*sqrt(-10*x^2 - x +
3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 1301839/576/(3*
sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.22446, size = 167, normalized size = 0.97 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (543914475 \, x^{4} + 1438446565 \, x^{3} + 1426133132 \, x^{2} + 628209228 \, x + 103735088\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 46095555 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{6272 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6272*sqrt(7)*(2*sqrt(7)*(543914475*x^4 + 1438446565*x^3 + 1426133132*x^2 + 62
8209228*x + 103735088)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 46095555*(405*x^5 + 1323*x
^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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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)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.446509, size = 591, normalized size = 3.42 \[ -\frac{9219111}{12544} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{605}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{605 \,{\left (77025 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 51138136 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 12067876800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 984130112000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{224 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9219111/12544*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)))) - 605/2*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))) - 605/224*(77025*sqrt(1
0)*((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 + 51138136*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 + 12067876800*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 + 984130112000*sqrt(10)*
((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-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)^4