3.2610 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=171 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{511 (3 x+2)^4}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7591 \sqrt{1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac{261331 \sqrt{1-2 x} (3 x+2)^2}{2196150 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (78981180 x+190406711)}{117128000}+\frac{753543 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

[Out]

(7591*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(39930*(3 + 5*x)^(3/2)) - (511*(2 + 3*x)^4)/(24
2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)
^(3/2)) + (261331*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(2196150*Sqrt[3 + 5*x]) - (7*Sqrt[1
- 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x))/117128000 + (753543*ArcSin[Sqrt[
2/11]*Sqrt[3 + 5*x]])/(8000*Sqrt[10])

_______________________________________________________________________________________

Rubi [A]  time = 0.358956, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{511 (3 x+2)^4}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{7591 \sqrt{1-2 x} (3 x+2)^3}{39930 (5 x+3)^{3/2}}+\frac{261331 \sqrt{1-2 x} (3 x+2)^2}{2196150 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (78981180 x+190406711)}{117128000}+\frac{753543 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7591*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(39930*(3 + 5*x)^(3/2)) - (511*(2 + 3*x)^4)/(24
2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^5)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)
^(3/2)) + (261331*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(2196150*Sqrt[3 + 5*x]) - (7*Sqrt[1
- 2*x]*Sqrt[3 + 5*x]*(190406711 + 78981180*x))/117128000 + (753543*ArcSin[Sqrt[
2/11]*Sqrt[3 + 5*x]])/(8000*Sqrt[10])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.2191, size = 160, normalized size = 0.94 \[ \frac{7591 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{39930 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{261331 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{2196150 \sqrt{5 x + 3}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{6219767925 x}{8} + \frac{59978113965}{32}\right )}{164711250} + \frac{753543 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{80000} - \frac{511 \left (3 x + 2\right )^{4}}{242 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{5}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7591*sqrt(-2*x + 1)*(3*x + 2)**3/(39930*(5*x + 3)**(3/2)) + 261331*sqrt(-2*x + 1
)*(3*x + 2)**2/(2196150*sqrt(5*x + 3)) - sqrt(-2*x + 1)*sqrt(5*x + 3)*(621976792
5*x/8 + 59978113965/32)/164711250 + 753543*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/
11)/80000 - 511*(3*x + 2)**4/(242*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 7*(3*x + 2)
**5/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.27552, size = 75, normalized size = 0.44 \[ -\frac{12807946800 x^5+97980793020 x^4-252342435560 x^3-274128335769 x^2+19932058554 x+44437106459}{351384000 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{753543 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{8000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(44437106459 + 19932058554*x - 274128335769*x^2 - 252342435560*x^3 + 9798079302
0*x^4 + 12807946800*x^5)/(351384000*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (753543*A
rcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(8000*Sqrt[10])

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 199, normalized size = 1.2 \[{\frac{1}{7027680000\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 3309786918900\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}-256158936000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-1959615860400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1952774282151\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-198587215134\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5482566715380\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -398641171080\,x\sqrt{-10\,{x}^{2}-x+3}-888742129180\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7027680000*(1-2*x)^(1/2)*(3309786918900*arcsin(20/11*x+1/11)*10^(1/2)*x^4-2561
58936000*x^5*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^3-
1959615860400*x^4*(-10*x^2-x+3)^(1/2)-1952774282151*10^(1/2)*arcsin(20/11*x+1/11
)*x^2+5046848711200*x^3*(-10*x^2-x+3)^(1/2)-198587215134*10^(1/2)*arcsin(20/11*x
+1/11)*x+5482566715380*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/1
1*x+1/11)-398641171080*x*(-10*x^2-x+3)^(1/2)-888742129180*(-10*x^2-x+3)^(1/2))/(
-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51096, size = 289, normalized size = 1.69 \[ -\frac{729 \, x^{5}}{20 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{111537 \, x^{4}}{400 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{251181}{234256000} \, x{\left (\frac{7220 \, x}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{361}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} - \frac{753543}{160000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{90676341}{117128000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{170985889 \, x}{7027680 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{766611 \, x^{2}}{1000 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1005653687}{878460000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{416356591 \, x}{3630000 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{496819753}{3630000 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-729/20*x^5/(-10*x^2 - x + 3)^(3/2) - 111537/400*x^4/(-10*x^2 - x + 3)^(3/2) + 2
51181/234256000*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x + 3)^(
3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^(3/2) - 87483/(-10*
x^2 - x + 3)^(3/2)) - 753543/160000*sqrt(10)*arcsin(-20/11*x - 1/11) + 90676341/
117128000*sqrt(-10*x^2 - x + 3) - 170985889/7027680*x/sqrt(-10*x^2 - x + 3) + 76
6611/1000*x^2/(-10*x^2 - x + 3)^(3/2) + 1005653687/878460000/sqrt(-10*x^2 - x +
3) + 416356591/3630000*x/(-10*x^2 - x + 3)^(3/2) - 496819753/3630000/(-10*x^2 -
x + 3)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.229629, size = 161, normalized size = 0.94 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (12807946800 \, x^{5} + 97980793020 \, x^{4} - 252342435560 \, x^{3} - 274128335769 \, x^{2} + 19932058554 \, x + 44437106459\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 33097869189 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7027680000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7027680000*sqrt(10)*(2*sqrt(10)*(12807946800*x^5 + 97980793020*x^4 - 25234243
5560*x^3 - 274128335769*x^2 + 19932058554*x + 44437106459)*sqrt(5*x + 3)*sqrt(-2
*x + 1) - 33097869189*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/20*sqrt(10)
*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x +
9)

_______________________________________________________________________________________

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((2+3*x)**6/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286803, size = 282, normalized size = 1.65 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{2196150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{753543}{80000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{37 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{16637500 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (32019867 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 93 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 110347010662 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1820310410259 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{219615000000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{1221 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{137259375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2196150000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) +
753543/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 37/16637500*sqrt(10)
*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/219615000000*(4*(3201986
7*(4*sqrt(5)*(5*x + 3) + 93*sqrt(5))*(5*x + 3) - 110347010662*sqrt(5))*(5*x + 3)
+ 1820310410259*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/13725937
5*(1221*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*
(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3