3.2525 \(\int \frac{(2+3 x) (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{7 (5 x+3)^{7/2}}{11 \sqrt{1-2 x}}+\frac{81}{44} \sqrt{1-2 x} (5 x+3)^{5/2}+\frac{405}{32} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{13365}{128} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{29403}{128} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(13365*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/128 + (405*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/32
+ (81*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/44 + (7*(3 + 5*x)^(7/2))/(11*Sqrt[1 - 2*x]
) - (29403*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/128

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Rubi [A]  time = 0.119116, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{7 (5 x+3)^{7/2}}{11 \sqrt{1-2 x}}+\frac{81}{44} \sqrt{1-2 x} (5 x+3)^{5/2}+\frac{405}{32} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{13365}{128} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{29403}{128} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(13365*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/128 + (405*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/32
+ (81*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/44 + (7*(3 + 5*x)^(7/2))/(11*Sqrt[1 - 2*x]
) - (29403*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/128

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Rubi in Sympy [A]  time = 11.1827, size = 105, normalized size = 0.89 \[ \frac{81 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{44} + \frac{405 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{32} + \frac{13365 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{128} - \frac{29403 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{256} + \frac{7 \left (5 x + 3\right )^{\frac{7}{2}}}{11 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/44 + 405*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/32 +
13365*sqrt(-2*x + 1)*sqrt(5*x + 3)/128 - 29403*sqrt(10)*asin(sqrt(22)*sqrt(5*x
+ 3)/11)/256 + 7*(5*x + 3)**(7/2)/(11*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.0796758, size = 69, normalized size = 0.58 \[ \frac{29403 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (1600 x^3+6120 x^2+14526 x-22545\right )}{256 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(-22545 + 14526*x + 6120*x^2 + 1600*x^3) + 29403*Sqrt[10 - 20*
x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(256*Sqrt[1 - 2*x])

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Maple [A]  time = 0.018, size = 123, normalized size = 1. \[ -{\frac{1}{-512+1024\,x} \left ( -6400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+58806\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-24480\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-29403\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -58104\,x\sqrt{-10\,{x}^{2}-x+3}+90180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/512*(-6400*x^3*(-10*x^2-x+3)^(1/2)+58806*10^(1/2)*arcsin(20/11*x+1/11)*x-2448
0*x^2*(-10*x^2-x+3)^(1/2)-29403*10^(1/2)*arcsin(20/11*x+1/11)-58104*x*(-10*x^2-x
+3)^(1/2)+90180*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)/(-10*x
^2-x+3)^(1/2)

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Maxima [A]  time = 1.5014, size = 124, normalized size = 1.05 \[ -\frac{125 \, x^{4}}{2 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{4425 \, x^{3}}{16 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{45495 \, x^{2}}{64 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{29403}{512} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{69147 \, x}{128 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{67635}{128 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/2*x^4/sqrt(-10*x^2 - x + 3) - 4425/16*x^3/sqrt(-10*x^2 - x + 3) - 45495/64*
x^2/sqrt(-10*x^2 - x + 3) + 29403/512*sqrt(10)*arcsin(-20/11*x - 1/11) + 69147/1
28*x/sqrt(-10*x^2 - x + 3) + 67635/128/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.228979, size = 115, normalized size = 0.97 \[ \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (1600 \, x^{3} + 6120 \, x^{2} + 14526 \, x - 22545\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 29403 \, \sqrt{5}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{512 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/512*sqrt(2)*(2*sqrt(2)*(1600*x^3 + 6120*x^2 + 14526*x - 22545)*sqrt(5*x + 3)*s
qrt(-2*x + 1) - 29403*sqrt(5)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(
sqrt(5*x + 3)*sqrt(-2*x + 1))))/(2*x - 1)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.232681, size = 113, normalized size = 0.96 \[ -\frac{29403}{256} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 81 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4455 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 147015 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3200 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-29403/256*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/3200*(2*(4*(8*sqrt(5
)*(5*x + 3) + 81*sqrt(5))*(5*x + 3) + 4455*sqrt(5))*(5*x + 3) - 147015*sqrt(5))*
sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)