3.2330 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{137752591 \sqrt{1-2 x} \sqrt{5 x+3}}{14224896 (3 x+2)}+\frac{1316353 \sqrt{1-2 x} \sqrt{5 x+3}}{1016064 (3 x+2)^2}+\frac{37333 \sqrt{1-2 x} \sqrt{5 x+3}}{181440 (3 x+2)^3}-\frac{7591 \sqrt{1-2 x} \sqrt{5 x+3}}{30240 (3 x+2)^4}-\frac{19457889 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(-7591*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(30240*(2 + 3*x)^4) + (37333*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(181440*(2 + 3*x)^3) + (1316353*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1016
064*(2 + 3*x)^2) + (137752591*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14224896*(2 + 3*x))
- ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*(3 + 5*
x)^(3/2))/(180*(2 + 3*x)^5) - (19457889*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5
*x])])/(175616*Sqrt[7])

_______________________________________________________________________________________

Rubi [A]  time = 0.445394, 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 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{180 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac{137752591 \sqrt{1-2 x} \sqrt{5 x+3}}{14224896 (3 x+2)}+\frac{1316353 \sqrt{1-2 x} \sqrt{5 x+3}}{1016064 (3 x+2)^2}+\frac{37333 \sqrt{1-2 x} \sqrt{5 x+3}}{181440 (3 x+2)^3}-\frac{7591 \sqrt{1-2 x} \sqrt{5 x+3}}{30240 (3 x+2)^4}-\frac{19457889 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-7591*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(30240*(2 + 3*x)^4) + (37333*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(181440*(2 + 3*x)^3) + (1316353*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1016
064*(2 + 3*x)^2) + (137752591*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14224896*(2 + 3*x))
- ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*(3 + 5*
x)^(3/2))/(180*(2 + 3*x)^5) - (19457889*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5
*x])])/(175616*Sqrt[7])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.9276, size = 190, normalized size = 0.91 \[ - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{1260 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{18 \left (3 x + 2\right )^{6}} + \frac{137752591 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14224896 \left (3 x + 2\right )} + \frac{1316353 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1016064 \left (3 x + 2\right )^{2}} + \frac{37333 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{181440 \left (3 x + 2\right )^{3}} + \frac{311 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4320 \left (3 x + 2\right )^{4}} - \frac{19457889 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-37*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(1260*(3*x + 2)**5) - (-2*x + 1)**(3/2)*(5*x
+ 3)**(3/2)/(18*(3*x + 2)**6) + 137752591*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1422489
6*(3*x + 2)) + 1316353*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1016064*(3*x + 2)**2) + 373
33*sqrt(-2*x + 1)*sqrt(5*x + 3)/(181440*(3*x + 2)**3) + 311*sqrt(-2*x + 1)*sqrt(
5*x + 3)/(4320*(3*x + 2)**4) - 19457889*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*s
qrt(5*x + 3)))/1229312

_______________________________________________________________________________________

Mathematica [A]  time = 0.137596, size = 92, normalized size = 0.44 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (2066288865 x^5+6979774260 x^4+9434103472 x^3+6379024416 x^2+2157325040 x+291805632\right )}{(3 x+2)^6}-97289445 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{12293120} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(291805632 + 2157325040*x + 6379024416*x^2 + 94
34103472*x^3 + 6979774260*x^4 + 2066288865*x^5))/(2 + 3*x)^6 - 97289445*Sqrt[7]*
ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/12293120

_______________________________________________________________________________________

Maple [B]  time = 0.02, size = 346, normalized size = 1.7 \[{\frac{1}{12293120\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 70924005405\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+89306341824\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12293120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(70924005405*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+283696021620*7^(1/2)*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))*x^5+472826702700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^4+28928044110*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+97716839640*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+132077448608*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+89306341824*x^2*(-10*x^2-x+3)^(1/2)+622
6524480*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))/(-10*x^2-x+3)^(1/2)/(2+3*x)
^6

_______________________________________________________________________________________

Maxima [A]  time = 1.51457, size = 369, normalized size = 1.77 \[ \frac{3652535}{921984} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{140 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{1329 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1568 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{49173 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{2191521 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{29749665}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19457889}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{26211867}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{8670839 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3687936 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3652535/921984*(-10*x^2 - x + 3)^(3/2) + 1/14*(-10*x^2 - x + 3)^(5/2)/(729*x^6 +
2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 37/140*(-10*x^2 - x +
3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 1329/1568*(-10
*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 49173/21952*(-10*
x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 2191521/614656*(-10*x^2 - x +
3)^(5/2)/(9*x^2 + 12*x + 4) + 29749665/614656*sqrt(-10*x^2 - x + 3)*x + 19457889
/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 26211867/12
29312*sqrt(-10*x^2 - x + 3) + 8670839/3687936*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.228916, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (2066288865 \, x^{5} + 6979774260 \, x^{4} + 9434103472 \, x^{3} + 6379024416 \, x^{2} + 2157325040 \, x + 291805632\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 97289445 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{12293120 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12293120*sqrt(7)*(2*sqrt(7)*(2066288865*x^5 + 6979774260*x^4 + 9434103472*x^3
+ 6379024416*x^2 + 2157325040*x + 291805632)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 9728
9445*(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 + 2916*x^5 + 4
860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.598949, size = 676, normalized size = 3.23 \[ \frac{19457889}{24586240} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (1329 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 2108680 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 1434500480 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 382530534400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 46289743360000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 2287257907200000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{87808 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19457889/24586240*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)))) - 14641/87808*(1329*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)))^11 + 210
8680*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 - 1434500480*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 - 382530534400*sqrt(10)*((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)))^5 - 46289743
360000*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 - 2287257907200000*sqrt(10)*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*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)^6