3.2429 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt{5 x+3}}+\frac{102293609 \sqrt{1-2 x}}{18816 (3 x+2) \sqrt{5 x+3}}+\frac{587477 \sqrt{1-2 x}}{1344 (3 x+2)^2 \sqrt{5 x+3}}+\frac{12023 \sqrt{1-2 x}}{240 (3 x+2)^3 \sqrt{5 x+3}}+\frac{2513 \sqrt{1-2 x}}{360 (3 x+2)^4 \sqrt{5 x+3}}-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{5 x+3}}+\frac{3538809681 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(-4639661185*Sqrt[1 - 2*x])/(56448*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(15*(2 +
3*x)^5*Sqrt[3 + 5*x]) + (2513*Sqrt[1 - 2*x])/(360*(2 + 3*x)^4*Sqrt[3 + 5*x]) +
(12023*Sqrt[1 - 2*x])/(240*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (587477*Sqrt[1 - 2*x])/(
1344*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (102293609*Sqrt[1 - 2*x])/(18816*(2 + 3*x)*Sqr
t[3 + 5*x]) + (3538809681*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*S
qrt[7])

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Rubi [A]  time = 0.473397, antiderivative size = 202, 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 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt{5 x+3}}+\frac{102293609 \sqrt{1-2 x}}{18816 (3 x+2) \sqrt{5 x+3}}+\frac{587477 \sqrt{1-2 x}}{1344 (3 x+2)^2 \sqrt{5 x+3}}+\frac{12023 \sqrt{1-2 x}}{240 (3 x+2)^3 \sqrt{5 x+3}}+\frac{2513 \sqrt{1-2 x}}{360 (3 x+2)^4 \sqrt{5 x+3}}-\frac{4639661185 \sqrt{1-2 x}}{56448 \sqrt{5 x+3}}+\frac{3538809681 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(3/2)),x]

[Out]

(-4639661185*Sqrt[1 - 2*x])/(56448*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(15*(2 +
3*x)^5*Sqrt[3 + 5*x]) + (2513*Sqrt[1 - 2*x])/(360*(2 + 3*x)^4*Sqrt[3 + 5*x]) +
(12023*Sqrt[1 - 2*x])/(240*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (587477*Sqrt[1 - 2*x])/(
1344*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (102293609*Sqrt[1 - 2*x])/(18816*(2 + 3*x)*Sqr
t[3 + 5*x]) + (3538809681*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(6272*S
qrt[7])

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Rubi in Sympy [A]  time = 44.214, size = 187, normalized size = 0.93 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{5} \sqrt{5 x + 3}} - \frac{4639661185 \sqrt{- 2 x + 1}}{56448 \sqrt{5 x + 3}} + \frac{102293609 \sqrt{- 2 x + 1}}{18816 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{587477 \sqrt{- 2 x + 1}}{1344 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{12023 \sqrt{- 2 x + 1}}{240 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{2513 \sqrt{- 2 x + 1}}{360 \left (3 x + 2\right )^{4} \sqrt{5 x + 3}} + \frac{3538809681 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{43904} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**6/(3+5*x)**(3/2),x)

[Out]

7*(-2*x + 1)**(3/2)/(15*(3*x + 2)**5*sqrt(5*x + 3)) - 4639661185*sqrt(-2*x + 1)/
(56448*sqrt(5*x + 3)) + 102293609*sqrt(-2*x + 1)/(18816*(3*x + 2)*sqrt(5*x + 3))
+ 587477*sqrt(-2*x + 1)/(1344*(3*x + 2)**2*sqrt(5*x + 3)) + 12023*sqrt(-2*x + 1
)/(240*(3*x + 2)**3*sqrt(5*x + 3)) + 2513*sqrt(-2*x + 1)/(360*(3*x + 2)**4*sqrt(
5*x + 3)) + 3538809681*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/43
904

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Mathematica [A]  time = 0.148788, size = 92, normalized size = 0.46 \[ \frac{17694048405 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (626354259975 x^5+2074037896035 x^4+2746600901250 x^3+1818284414692 x^2+601741553688 x+79638637088\right )}{(3 x+2)^5 \sqrt{5 x+3}}}{439040} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^6*(3 + 5*x)^(3/2)),x]

[Out]

((-14*Sqrt[1 - 2*x]*(79638637088 + 601741553688*x + 1818284414692*x^2 + 27466009
01250*x^3 + 2074037896035*x^4 + 626354259975*x^5))/((2 + 3*x)^5*Sqrt[3 + 5*x]) +
17694048405*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/4390
40

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Maple [B]  time = 0.023, size = 346, normalized size = 1.7 \[ -{\frac{1}{439040\, \left ( 2+3\,x \right ) ^{5}} \left ( 21498268812075\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+25455981805688\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^6/(3+5*x)^(3/2),x)

[Out]

-1/439040*(21498268812075*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1
/2))*x^6+84559857327495*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^5+138544399011150*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^4+8768959639650*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+29036530544490*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+
38452412617500*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+25455981805688*x^2*(-10*x^2-x+3)^(1/2)+169862
8646880*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)^
5/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.51907, size = 537, normalized size = 2.66 \[ -\frac{3538809681}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{4639661185 \, x}{28224 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{4844248403}{56448 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{135 \,{\left (243 \, \sqrt{-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt{-10 \, x^{2} - x + 3} x + 32 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{5341}{360 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{242879}{2160 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{315689}{320 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{33314567}{2688 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

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)^6),x, algorithm="maxima")

[Out]

-3538809681/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 46
39661185/28224*x/sqrt(-10*x^2 - x + 3) - 4844248403/56448/sqrt(-10*x^2 - x + 3)
+ 343/135/(243*sqrt(-10*x^2 - x + 3)*x^5 + 810*sqrt(-10*x^2 - x + 3)*x^4 + 1080*
sqrt(-10*x^2 - x + 3)*x^3 + 720*sqrt(-10*x^2 - x + 3)*x^2 + 240*sqrt(-10*x^2 - x
+ 3)*x + 32*sqrt(-10*x^2 - x + 3)) + 5341/360/(81*sqrt(-10*x^2 - x + 3)*x^4 + 2
16*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)) + 242879/2160/(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)) + 315689/320/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x
+ 4*sqrt(-10*x^2 - x + 3)) + 33314567/2688/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-
10*x^2 - x + 3))

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Fricas [A]  time = 0.225546, size = 188, normalized size = 0.93 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (626354259975 \, x^{5} + 2074037896035 \, x^{4} + 2746600901250 \, x^{3} + 1818284414692 \, x^{2} + 601741553688 \, x + 79638637088\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 17694048405 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{439040 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]

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)^6),x, algorithm="fricas")

[Out]

-1/439040*sqrt(7)*(2*sqrt(7)*(626354259975*x^5 + 2074037896035*x^4 + 27466009012
50*x^3 + 1818284414692*x^2 + 601741553688*x + 79638637088)*sqrt(5*x + 3)*sqrt(-2
*x + 1) + 17694048405*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 88
0*x + 96)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(1215
*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)

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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)**6/(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.545801, size = 674, normalized size = 3.34 \[ -\frac{3538809681}{878080} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{3025}{2} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{121 \,{\left (34728039 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 30879615760 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 10961021460480 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1791349451136000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 112299870108160000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{3136 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

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)^6),x, algorithm="giac")

[Out]

-3538809681/878080*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)))) - 3025/2*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))) - 121/3136*(347280
39*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)))^9 + 30879615760*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 + 10961021460480*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 1791349
451136000*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 + 112299870108160000*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))))/(((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)^5