3.2611 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{357 (3 x+2)^3}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{5281 \sqrt{1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac{\sqrt{1-2 x} (55300905 x+33035947)}{8784600 \sqrt{5 x+3}}+\frac{2997 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(5281*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*(3 + 5*x)^(3/2)) - (357*(2 + 3*x)^3)/(24
2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)
^(3/2)) - (Sqrt[1 - 2*x]*(33035947 + 55300905*x))/(8784600*Sqrt[3 + 5*x]) + (299
7*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi [A]  time = 0.283578, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{357 (3 x+2)^3}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{5281 \sqrt{1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac{\sqrt{1-2 x} (55300905 x+33035947)}{8784600 \sqrt{5 x+3}}+\frac{2997 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(5281*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*(3 + 5*x)^(3/2)) - (357*(2 + 3*x)^3)/(24
2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)
^(3/2)) - (Sqrt[1 - 2*x]*(33035947 + 55300905*x))/(8784600*Sqrt[3 + 5*x]) + (299
7*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi in Sympy [A]  time = 25.8499, size = 134, normalized size = 0.94 \[ \frac{5281 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{39930 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (\frac{829513575 x}{16} + \frac{495539205}{16}\right )}{16471125 \sqrt{5 x + 3}} + \frac{2997 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2000} - \frac{357 \left (3 x + 2\right )^{3}}{242 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{4}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5281*sqrt(-2*x + 1)*(3*x + 2)**2/(39930*(5*x + 3)**(3/2)) - 2*sqrt(-2*x + 1)*(82
9513575*x/16 + 495539205/16)/(16471125*sqrt(5*x + 3)) + 2997*sqrt(10)*asin(sqrt(
22)*sqrt(5*x + 3)/11)/2000 - 357*(3*x + 2)**3/(242*sqrt(-2*x + 1)*(5*x + 3)**(3/
2)) + 7*(3*x + 2)**4/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.263981, size = 70, normalized size = 0.49 \[ -\frac{213465780 x^4-1247811640 x^3-1260430251 x^2+19593966 x+168318961}{8784600 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{2997 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

-(168318961 + 19593966*x - 1260430251*x^2 - 1247811640*x^3 + 213465780*x^4)/(878
4600*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (2997*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/
(200*Sqrt[10])

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Maple [A]  time = 0.021, size = 182, normalized size = 1.3 \[{\frac{1}{175692000\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 13163723100\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-4269315600\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-7766596629\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-789823386\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+25208605020\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -391879320\,x\sqrt{-10\,{x}^{2}-x+3}-3366379220\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/175692000*(1-2*x)^(1/2)*(13163723100*arcsin(20/11*x+1/11)*10^(1/2)*x^4+2632744
620*10^(1/2)*arcsin(20/11*x+1/11)*x^3-4269315600*x^4*(-10*x^2-x+3)^(1/2)-7766596
629*10^(1/2)*arcsin(20/11*x+1/11)*x^2+24956232800*x^3*(-10*x^2-x+3)^(1/2)-789823
386*10^(1/2)*arcsin(20/11*x+1/11)*x+25208605020*x^2*(-10*x^2-x+3)^(1/2)[phone]
79*10^(1/2)*arcsin(20/11*x+1/11)-391879320*x*(-10*x^2-x+3)^(1/2)-3366379220*(-10
*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51178, size = 266, normalized size = 1.87 \[ -\frac{243 \, x^{4}}{10 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{999}{5856400} \, x{\left (\frac{7220 \, x}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{361}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} - \frac{2997}{4000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{360639}{2928200} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{5842159 \, x}{878460 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3429 \, x^{2}}{25 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{947293}{21961500 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3016649 \, x}{90750 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1851167}{90750 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/10*x^4/(-10*x^2 - x + 3)^(3/2) + 999/5856400*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/(-1
0*x^2 - x + 3)^(3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 2997/4000*sqrt(10)*arcsi
n(-20/11*x - 1/11) + 360639/2928200*sqrt(-10*x^2 - x + 3) - 5842159/878460*x/sqr
t(-10*x^2 - x + 3) + 3429/25*x^2/(-10*x^2 - x + 3)^(3/2) + 947293/21961500/sqrt(
-10*x^2 - x + 3) + 3016649/90750*x/(-10*x^2 - x + 3)^(3/2) - 1851167/90750/(-10*
x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.22649, size = 154, normalized size = 1.08 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (213465780 \, x^{4} - 1247811640 \, x^{3} - 1260430251 \, x^{2} + 19593966 \, x + 168318961\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 131637231 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{175692000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/175692000*sqrt(10)*(2*sqrt(10)*(213465780*x^4 - 1247811640*x^3 - 1260430251*x
^2 + 19593966*x + 168318961)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 131637231*(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)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282344, size = 265, normalized size = 1.87 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{439230000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{2997}{2000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{31 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{3327500 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (10673289 \, \sqrt{5}{\left (5 \, x + 3\right )} - 440040554 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 7233942969 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5490375000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{1023 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{27451875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/439230000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 2
997/2000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 31/3327500*sqrt(10)*(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/5490375000*(4*(10673289*sqrt(
5)*(5*x + 3) - 440040554*sqrt(5))*(5*x + 3) + 7233942969*sqrt(5))*sqrt(5*x + 3)*
sqrt(-10*x + 5)/(2*x - 1)^2 + 1/27451875*(1023*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