3.2585 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ \frac{(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{25 (5 x+3)^{3/2}}{6 \sqrt{1-2 x}}-\frac{125}{8} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{275}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-125*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8 - (25*(3 + 5*x)^(3/2))/(6*Sqrt[1 - 2*x]) +
(3 + 5*x)^(5/2)/(3*(1 - 2*x)^(3/2)) + (275*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/8

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Rubi [A]  time = 0.0814997, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{25 (5 x+3)^{3/2}}{6 \sqrt{1-2 x}}-\frac{125}{8} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{275}{8} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-125*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8 - (25*(3 + 5*x)^(3/2))/(6*Sqrt[1 - 2*x]) +
(3 + 5*x)^(5/2)/(3*(1 - 2*x)^(3/2)) + (275*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/8

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Rubi in Sympy [A]  time = 9.10538, size = 83, normalized size = 0.86 \[ - \frac{125 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8} + \frac{275 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16} - \frac{25 \left (5 x + 3\right )^{\frac{3}{2}}}{6 \sqrt{- 2 x + 1}} + \frac{\left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*sqrt(-2*x + 1)*sqrt(5*x + 3)/8 + 275*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/1
1)/16 - 25*(5*x + 3)**(3/2)/(6*sqrt(-2*x + 1)) + (5*x + 3)**(5/2)/(3*(-2*x + 1)*
*(3/2))

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Mathematica [A]  time = 0.110108, size = 69, normalized size = 0.72 \[ \frac{825 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (300 x^2-1840 x+603\right )}{48 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(603 - 1840*x + 300*x^2) + 825*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcS
in[Sqrt[5/11]*Sqrt[1 - 2*x]])/(48*(1 - 2*x)^(3/2))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{1 \left ( 3+5\,x \right ) ^{{\frac{5}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50349, size = 174, normalized size = 1.81 \[ \frac{275}{32} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{55 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{24 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{605 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1925 \, \sqrt{-10 \, x^{2} - x + 3}}{48 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

275/32*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/2*(-10*x^2 - x + 3)^(5/2)/(16*
x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 55/24*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2
+ 6*x - 1) + 605/48*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 1925/48*sqrt(-10*
x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.222662, size = 122, normalized size = 1.27 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (300 \, x^{2} - 1840 \, x + 603\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 825 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*sqrt(2)*(2*sqrt(2)*(300*x^2 - 1840*x + 603)*sqrt(5*x + 3)*sqrt(-2*x + 1) -
825*sqrt(5)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 30.8273, size = 729, normalized size = 7.59 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-16500*sqrt(10)*I*(x + 3/5)**(27/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sq
rt(x + 3/5)/11)/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*s
qrt(10*x - 5)) + 8250*sqrt(10)*pi*(x + 3/5)**(27/2)*sqrt(10*x - 5)/(960*(x + 3/5
)**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 18150*sqrt(1
0)*I*(x + 3/5)**(25/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(960*(x
+ 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) - 9075*sq
rt(10)*pi*(x + 3/5)**(25/2)*sqrt(10*x - 5)/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5)
- 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) - 30000*I*(x + 3/5)**15/(960*(x + 3/5)
**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10*x - 5)) + 220000*I*(x +
3/5)**14/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(x + 3/5)**(25/2)*sqrt(10
*x - 5)) - 181500*I*(x + 3/5)**13/(960*(x + 3/5)**(27/2)*sqrt(10*x - 5) - 1056*(
x + 3/5)**(25/2)*sqrt(10*x - 5)), 10*Abs(x + 3/5)/11 > 1), (8250*sqrt(10)*sqrt(-
10*x + 5)*(x + 3/5)**(27/2)*asin(sqrt(110)*sqrt(x + 3/5)/11)/(480*sqrt(-10*x + 5
)*(x + 3/5)**(27/2) - 528*sqrt(-10*x + 5)*(x + 3/5)**(25/2)) - 9075*sqrt(10)*sqr
t(-10*x + 5)*(x + 3/5)**(25/2)*asin(sqrt(110)*sqrt(x + 3/5)/11)/(480*sqrt(-10*x
+ 5)*(x + 3/5)**(27/2) - 528*sqrt(-10*x + 5)*(x + 3/5)**(25/2)) + 15000*(x + 3/5
)**15/(480*sqrt(-10*x + 5)*(x + 3/5)**(27/2) - 528*sqrt(-10*x + 5)*(x + 3/5)**(2
5/2)) - 110000*(x + 3/5)**14/(480*sqrt(-10*x + 5)*(x + 3/5)**(27/2) - 528*sqrt(-
10*x + 5)*(x + 3/5)**(25/2)) + 90750*(x + 3/5)**13/(480*sqrt(-10*x + 5)*(x + 3/5
)**(27/2) - 528*sqrt(-10*x + 5)*(x + 3/5)**(25/2)), True))

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GIAC/XCAS [A]  time = 0.233866, size = 96, normalized size = 1. \[ \frac{275}{16} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \, \sqrt{5}{\left (5 \, x + 3\right )} - 110 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1815 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{120 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

275/16*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/120*(4*(3*sqrt(5)*(5*x +
3) - 110*sqrt(5))*(5*x + 3) + 1815*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1)^2