3.2532 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=209 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^5}+\frac{426781 \sqrt{1-2 x} \sqrt{5 x+3}}{6453888 (3 x+2)}-\frac{55277 \sqrt{1-2 x} \sqrt{5 x+3}}{460992 (3 x+2)^2}-\frac{29297 \sqrt{1-2 x} \sqrt{5 x+3}}{82320 (3 x+2)^3}-\frac{42863 \sqrt{1-2 x} \sqrt{5 x+3}}{41160 (3 x+2)^4}+\frac{164 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^5}-\frac{3474273 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2151296 \sqrt{7}} \]

[Out]

(164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^5) - (42863*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(41160*(2 + 3*x)^4) - (29297*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(82320*(2 +
3*x)^3) - (55277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(460992*(2 + 3*x)^2) + (426781*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(6453888*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 -
2*x]*(2 + 3*x)^5) - (3474273*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21
51296*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.446731, 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 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^5}+\frac{426781 \sqrt{1-2 x} \sqrt{5 x+3}}{6453888 (3 x+2)}-\frac{55277 \sqrt{1-2 x} \sqrt{5 x+3}}{460992 (3 x+2)^2}-\frac{29297 \sqrt{1-2 x} \sqrt{5 x+3}}{82320 (3 x+2)^3}-\frac{42863 \sqrt{1-2 x} \sqrt{5 x+3}}{41160 (3 x+2)^4}+\frac{164 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^5}-\frac{3474273 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2151296 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(164*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^5) - (42863*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(41160*(2 + 3*x)^4) - (29297*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(82320*(2 +
3*x)^3) - (55277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(460992*(2 + 3*x)^2) + (426781*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(6453888*(2 + 3*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 -
2*x]*(2 + 3*x)^5) - (3474273*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21
51296*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.1544, size = 192, normalized size = 0.92 \[ \frac{426781 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6453888 \left (3 x + 2\right )} - \frac{55277 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{460992 \left (3 x + 2\right )^{2}} - \frac{29297 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{82320 \left (3 x + 2\right )^{3}} - \frac{42863 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{41160 \left (3 x + 2\right )^{4}} + \frac{164 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{735 \left (3 x + 2\right )^{5}} - \frac{3474273 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{15059072} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

426781*sqrt(-2*x + 1)*sqrt(5*x + 3)/(6453888*(3*x + 2)) - 55277*sqrt(-2*x + 1)*s
qrt(5*x + 3)/(460992*(3*x + 2)**2) - 29297*sqrt(-2*x + 1)*sqrt(5*x + 3)/(82320*(
3*x + 2)**3) - 42863*sqrt(-2*x + 1)*sqrt(5*x + 3)/(41160*(3*x + 2)**4) + 164*sqr
t(-2*x + 1)*sqrt(5*x + 3)/(735*(3*x + 2)**5) - 3474273*sqrt(7)*atan(sqrt(7)*sqrt
(-2*x + 1)/(7*sqrt(5*x + 3)))/15059072 + 11*(5*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(
3*x + 2)**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.158765, size = 92, normalized size = 0.44 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-115230870 x^5-180017865 x^4+19738914 x^3+164918884 x^2+95331368 x+16456032\right )}{\sqrt{1-2 x} (3 x+2)^5}-17371365 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{150590720} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[3 + 5*x]*(16456032 + 95331368*x + 164918884*x^2 + 19738914*x^3 - 18001
7865*x^4 - 115230870*x^5))/(Sqrt[1 - 2*x]*(2 + 3*x)^5) - 17371365*Sqrt[7]*ArcTan
[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/150590720

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 353, normalized size = 1.7 \[{\frac{1}{150590720\, \left ( 2+3\,x \right ) ^{5} \left ( -1+2\,x \right ) } \left ( 8442483390\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4169127600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-276344796\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-3057360240\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-2308864376\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-555883680\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1334639152\,x\sqrt{-10\,{x}^{2}-x+3}-230384448\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/150590720*(8442483390*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^6+23920369605*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
5+23451342750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+161
3232180*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+2520250110*x^4*(-10*x^2-x+3)^(1/2)-4169127600*7^(1/2)*
arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-276344796*x^3*(-10*x^2-x+
3)^(1/2)-3057360240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x
-2308864376*x^2*(-10*x^2-x+3)^(1/2)-555883680*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))-1334639152*x*(-10*x^2-x+3)^(1/2)-230384448*(-10*x^2-x+3
)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5/(-1+2*x)/(-10*x^2-x+3)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.56404, size = 537, normalized size = 2.57 \[ \frac{3474273}{30118144} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{2133905 \, x}{9680832 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4998019}{19361664 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{945 \,{\left (243 \, \sqrt{-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt{-10 \, x^{2} - x + 3} x + 32 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{331}{17640 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{83537}{740880 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{23353}{109760 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{137335}{921984 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^6*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

3474273/30118144*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 213
3905/9680832*x/sqrt(-10*x^2 - x + 3) + 4998019/19361664/sqrt(-10*x^2 - x + 3) +
1/945/(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)) - 331/17640/(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)) + 83537/740880/(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)) - 23353/109760/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x
+ 4*sqrt(-10*x^2 - x + 3)) - 137335/921984/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-
10*x^2 - x + 3))

_______________________________________________________________________________________

Fricas [A]  time = 0.237626, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (115230870 \, x^{5} + 180017865 \, x^{4} - 19738914 \, x^{3} - 164918884 \, x^{2} - 95331368 \, x - 16456032\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 17371365 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{150590720 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^6*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/150590720*sqrt(7)*(2*sqrt(7)*(115230870*x^5 + 180017865*x^4 - 19738914*x^3 - 1
64918884*x^2 - 95331368*x - 16456032)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 17371365*(4
86*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*arctan(1/14*sqrt(
7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(486*x^6 + 1377*x^5 + 1350*x^4 +
360*x^3 - 240*x^2 - 176*x - 32)

_______________________________________________________________________________________

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)/(1-2*x)**(3/2)/(2+3*x)**6,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.819509, size = 629, normalized size = 3.01 \[ \frac{3474273}{301181440} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1936 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{588245 \,{\left (2 \, x - 1\right )}} - \frac{121 \,{\left (203039 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 265495440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 136071290880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 774949504000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 650054039040000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{7529536 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^6*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

3474273/301181440*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)))) - 1936/588245*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 1
21/7529536*(203039*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 + 265495440*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)))^7 + 136071290880*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 - 774949504000*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 - 650054039040000*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))/sqrt(5*x
+ 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5