3.2576 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{5 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{5}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-5*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (3 + 5*x)^(3/2)/(3*(1 - 2*x)^(3/2)) + (5*
Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/2

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Rubi [A]  time = 0.0612905, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{5 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{5}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (3 + 5*x)^(3/2)/(3*(1 - 2*x)^(3/2)) + (5*
Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/2

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Rubi in Sympy [A]  time = 7.18687, size = 63, normalized size = 0.85 \[ \frac{5 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{4} - \frac{5 \sqrt{5 x + 3}}{2 \sqrt{- 2 x + 1}} + \frac{\left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/4 - 5*sqrt(5*x + 3)/(2*sqrt(-2*x + 1)
) + (5*x + 3)**(3/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.105979, size = 64, normalized size = 0.86 \[ \frac{2 \sqrt{5 x+3} (40 x-9)+15 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x]*(-9 + 40*x) + 15*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*S
qrt[1 - 2*x]])/(12*(1 - 2*x)^(3/2))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{1 \left ( 3+5\,x \right ) ^{{\frac{3}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{5}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.50462, size = 126, normalized size = 1.7 \[ \frac{5}{8} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{35 \, \sqrt{-10 \, x^{2} - x + 3}}{12 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/8*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/6*(-10*x^2 - x + 3)^(3/2)/(8*x^3
- 12*x^2 + 6*x - 1) + 11/12*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 35/12*sqrt
(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 0.231894, size = 115, normalized size = 1.55 \[ \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (40 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 15 \, \sqrt{5}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{24 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(2)*(2*sqrt(2)*(40*x - 9)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 15*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 = 10.6786, size = 636, normalized size = 8.59 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-300*sqrt(10)*I*(x + 3/5)**(15/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt
(x + 3/5)/11)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt
(10*x - 5)) + 150*sqrt(10)*pi*(x + 3/5)**(15/2)*sqrt(10*x - 5)/(240*(x + 3/5)**(
15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) + 330*sqrt(10)*I*(x
+ 3/5)**(13/2)*sqrt(10*x - 5)*acosh(sqrt(110)*sqrt(x + 3/5)/11)/(240*(x + 3/5)*
*(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) - 165*sqrt(10)*pi
*(x + 3/5)**(13/2)*sqrt(10*x - 5)/(240*(x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x
+ 3/5)**(13/2)*sqrt(10*x - 5)) + 4000*I*(x + 3/5)**8/(240*(x + 3/5)**(15/2)*sqr
t(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)) - 3300*I*(x + 3/5)**7/(240*(
x + 3/5)**(15/2)*sqrt(10*x - 5) - 264*(x + 3/5)**(13/2)*sqrt(10*x - 5)), 10*Abs(
x + 3/5)/11 > 1), (150*sqrt(10)*sqrt(-10*x + 5)*(x + 3/5)**(15/2)*asin(sqrt(110)
*sqrt(x + 3/5)/11)/(120*sqrt(-10*x + 5)*(x + 3/5)**(15/2) - 132*sqrt(-10*x + 5)*
(x + 3/5)**(13/2)) - 165*sqrt(10)*sqrt(-10*x + 5)*(x + 3/5)**(13/2)*asin(sqrt(11
0)*sqrt(x + 3/5)/11)/(120*sqrt(-10*x + 5)*(x + 3/5)**(15/2) - 132*sqrt(-10*x + 5
)*(x + 3/5)**(13/2)) - 2000*(x + 3/5)**8/(120*sqrt(-10*x + 5)*(x + 3/5)**(15/2)
- 132*sqrt(-10*x + 5)*(x + 3/5)**(13/2)) + 1650*(x + 3/5)**7/(120*sqrt(-10*x + 5
)*(x + 3/5)**(15/2) - 132*sqrt(-10*x + 5)*(x + 3/5)**(13/2)), True))

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GIAC/XCAS [A]  time = 0.233358, size = 78, normalized size = 1.05 \[ \frac{5}{4} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{30 \,{\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/4*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/30*(8*sqrt(5)*(5*x + 3) - 3
3*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2