3.2471 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac{122343637 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)}+\frac{958171 \sqrt{1-2 x} \sqrt{5 x+3}}{16595712 (3 x+2)^2}-\frac{71369 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^3}-\frac{149951 \sqrt{1-2 x} \sqrt{5 x+3}}{1481760 (3 x+2)^4}+\frac{503 \sqrt{1-2 x} \sqrt{5 x+3}}{26460 (3 x+2)^5}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8605184 \sqrt{7}} \]

[Out]

(503*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26460*(2 + 3*x)^5) - (149951*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/(1481760*(2 + 3*x)^4) - (71369*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(296352
0*(2 + 3*x)^3) + (958171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16595712*(2 + 3*x)^2) + (
122343637*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(232339968*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(126*(2 + 3*x)^6) - (52573169*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/(8605184*Sqrt[7])

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Rubi [A]  time = 0.454823, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac{122343637 \sqrt{1-2 x} \sqrt{5 x+3}}{232339968 (3 x+2)}+\frac{958171 \sqrt{1-2 x} \sqrt{5 x+3}}{16595712 (3 x+2)^2}-\frac{71369 \sqrt{1-2 x} \sqrt{5 x+3}}{2963520 (3 x+2)^3}-\frac{149951 \sqrt{1-2 x} \sqrt{5 x+3}}{1481760 (3 x+2)^4}+\frac{503 \sqrt{1-2 x} \sqrt{5 x+3}}{26460 (3 x+2)^5}-\frac{52573169 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8605184 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]

[Out]

(503*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26460*(2 + 3*x)^5) - (149951*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/(1481760*(2 + 3*x)^4) - (71369*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(296352
0*(2 + 3*x)^3) + (958171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(16595712*(2 + 3*x)^2) + (
122343637*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(232339968*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3
+ 5*x)^(3/2))/(126*(2 + 3*x)^6) - (52573169*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[
3 + 5*x])])/(8605184*Sqrt[7])

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Rubi in Sympy [A]  time = 44.0773, size = 190, normalized size = 0.91 \[ \frac{122343637 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{232339968 \left (3 x + 2\right )} + \frac{958171 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{16595712 \left (3 x + 2\right )^{2}} - \frac{71369 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2963520 \left (3 x + 2\right )^{3}} - \frac{149951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1481760 \left (3 x + 2\right )^{4}} + \frac{503 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{26460 \left (3 x + 2\right )^{5}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{126 \left (3 x + 2\right )^{6}} - \frac{52573169 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{60236288} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(5/2)/(2+3*x)**7/(1-2*x)**(1/2),x)

[Out]

122343637*sqrt(-2*x + 1)*sqrt(5*x + 3)/(232339968*(3*x + 2)) + 958171*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(16595712*(3*x + 2)**2) - 71369*sqrt(-2*x + 1)*sqrt(5*x + 3)/
(2963520*(3*x + 2)**3) - 149951*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1481760*(3*x + 2)*
*4) + 503*sqrt(-2*x + 1)*sqrt(5*x + 3)/(26460*(3*x + 2)**5) + sqrt(-2*x + 1)*(5*
x + 3)**(3/2)/(126*(3*x + 2)**6) - 52573169*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/
(7*sqrt(5*x + 3)))/60236288

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Mathematica [A]  time = 0.134551, size = 92, normalized size = 0.44 \[ \frac{\frac{378 \sqrt{1-2 x} \sqrt{5 x+3} \left (16516390995 x^5+55658284380 x^4+74931979536 x^3+50261760608 x^2+16771747280 x+2225100096\right )}{(3 x+2)^6}-21292133445 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{48791393280} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^7),x]

[Out]

((378*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2225100096 + 16771747280*x + 50261760608*x^2
+ 74931979536*x^3 + 55658284380*x^4 + 16516390995*x^5))/(2 + 3*x)^6 - 2129213344
5*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/48791393280

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Maple [B]  time = 0.023, size = 346, normalized size = 1.7 \[{\frac{1}{1807088640\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 574887603015\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+703664648512\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(5/2)/(2+3*x)^7/(1-2*x)^(1/2),x)

[Out]

1/1807088640*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(574887603015*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+2299550412060*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+3832584020100*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+231229473930*x^5*(-10*x^2-x+3)^(1/2)[phone]
00*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+779215981320*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+1049047713504*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+703664648512*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))+
234804461920*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3
)^(1/2)/(2+3*x)^6

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Maxima [A]  time = 1.51971, size = 311, normalized size = 1.49 \[ \frac{52573169}{120472576} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{378 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{853 \, \sqrt{-10 \, x^{2} - x + 3}}{26460 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{149951 \, \sqrt{-10 \, x^{2} - x + 3}}{1481760 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{71369 \, \sqrt{-10 \, x^{2} - x + 3}}{2963520 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{958171 \, \sqrt{-10 \, x^{2} - x + 3}}{16595712 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{122343637 \, \sqrt{-10 \, x^{2} - x + 3}}{232339968 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^7*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

52573169/120472576*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1
/378*sqrt(-10*x^2 - x + 3)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64) + 853/26460*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 +
720*x^2 + 240*x + 32) - 149951/1481760*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 +
216*x^2 + 96*x + 16) - 71369/2963520*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 3
6*x + 8) + 958171/16595712*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 122343637/
232339968*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.230771, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (16516390995 \, x^{5} + 55658284380 \, x^{4} + 74931979536 \, x^{3} + 50261760608 \, x^{2} + 16771747280 \, x + 2225100096\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 788597535 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1807088640 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^7*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

1/1807088640*sqrt(7)*(2*sqrt(7)*(16516390995*x^5 + 55658284380*x^4 + 74931979536
*x^3 + 50261760608*x^2 + 16771747280*x + 2225100096)*sqrt(5*x + 3)*sqrt(-2*x + 1
) + 788597535*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(729*x^6 + 291
6*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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((3+5*x)**(5/2)/(2+3*x)**7/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.644462, size = 676, normalized size = 3.23 \[ \frac{52573169}{1204725760} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (118497 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 188015240 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 122630175360 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 17238395059200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 3670540357120000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 197895383347200000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{12907776 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^7*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

52573169/1204725760*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)))) - 1331/12907776*(118497*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11
+ 188015240*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq
rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 122630175360*sqrt(10)*((sq
rt(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 - 17238395059200*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 - 3670540357120000*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 - 197895383347200000
*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))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^6