3.2526 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=93 \[ \frac{(5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{25}{8} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{825}{32} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{1815}{32} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(825*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/8 + (3
+ 5*x)^(5/2)/Sqrt[1 - 2*x] - (1815*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
32

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Rubi [A]  time = 0.0819144, antiderivative size = 93, 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 \[ \frac{(5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{25}{8} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{825}{32} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{1815}{32} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(825*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (25*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/8 + (3
+ 5*x)^(5/2)/Sqrt[1 - 2*x] - (1815*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/
32

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Rubi in Sympy [A]  time = 8.96746, size = 82, normalized size = 0.88 \[ \frac{25 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{8} + \frac{825 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{32} - \frac{1815 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{64} + \frac{\left (5 x + 3\right )^{\frac{5}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/8 + 825*sqrt(-2*x + 1)*sqrt(5*x + 3)/32 - 181
5*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/64 + (5*x + 3)**(5/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.0603539, size = 64, normalized size = 0.69 \[ \frac{1815 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (200 x^2+790 x-1413\right )}{64 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 5*x]*(-1413 + 790*x + 200*x^2) + 1815*Sqrt[10 - 20*x]*ArcSin[Sqrt[5
/11]*Sqrt[1 - 2*x]])/(64*Sqrt[1 - 2*x])

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{1 \left ( 3+5\,x \right ) ^{{\frac{5}{2}}} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.52487, size = 101, normalized size = 1.09 \[ -\frac{125 \, x^{3}}{4 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2275 \, x^{2}}{16 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1815}{128} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{4695 \, x}{32 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4239}{32 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/4*x^3/sqrt(-10*x^2 - x + 3) - 2275/16*x^2/sqrt(-10*x^2 - x + 3) + 1815/128*
sqrt(10)*arcsin(-20/11*x - 1/11) + 4695/32*x/sqrt(-10*x^2 - x + 3) + 4239/32/sqr
t(-10*x^2 - x + 3)

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Fricas [A]  time = 0.224578, size = 108, normalized size = 1.16 \[ \frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (200 \, x^{2} + 790 \, x - 1413\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1815 \, \sqrt{5}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{128 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*sqrt(2)*(2*sqrt(2)*(200*x^2 + 790*x - 1413)*sqrt(5*x + 3)*sqrt(-2*x + 1) -
1815*sqrt(5)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)*sq
rt(-2*x + 1))))/(2*x - 1)

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Sympy [A]  time = 31.7412, size = 187, normalized size = 2.01 \[ \begin{cases} \frac{125 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{4 \sqrt{10 x - 5}} + \frac{1375 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{16 \sqrt{10 x - 5}} - \frac{9075 i \sqrt{x + \frac{3}{5}}}{32 \sqrt{10 x - 5}} + \frac{1815 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{64} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\- \frac{1815 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{64} - \frac{125 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{4 \sqrt{- 10 x + 5}} - \frac{1375 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{16 \sqrt{- 10 x + 5}} + \frac{9075 \sqrt{x + \frac{3}{5}}}{32 \sqrt{- 10 x + 5}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((125*I*(x + 3/5)**(5/2)/(4*sqrt(10*x - 5)) + 1375*I*(x + 3/5)**(3/2)/(
16*sqrt(10*x - 5)) - 9075*I*sqrt(x + 3/5)/(32*sqrt(10*x - 5)) + 1815*sqrt(10)*I*
acosh(sqrt(110)*sqrt(x + 3/5)/11)/64, 10*Abs(x + 3/5)/11 > 1), (-1815*sqrt(10)*a
sin(sqrt(110)*sqrt(x + 3/5)/11)/64 - 125*(x + 3/5)**(5/2)/(4*sqrt(-10*x + 5)) -
1375*(x + 3/5)**(3/2)/(16*sqrt(-10*x + 5)) + 9075*sqrt(x + 3/5)/(32*sqrt(-10*x +
5)), True))

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GIAC/XCAS [A]  time = 0.22699, size = 96, normalized size = 1.03 \[ -\frac{1815}{64} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} + 55 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1815 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{160 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1815/64*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/160*(2*(4*sqrt(5)*(5*x
+ 3) + 55*sqrt(5))*(5*x + 3) - 1815*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1)