3.2603 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{111311 x+66967}{39930 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{27 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}} \]

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (66967 + 111311*x)/(39930*S
qrt[1 - 2*x]*Sqrt[3 + 5*x]) + (27*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*Sqrt[10]
)

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Rubi [A]  time = 0.125762, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{111311 x+66967}{39930 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{27 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - (66967 + 111311*x)/(39930*S
qrt[1 - 2*x]*Sqrt[3 + 5*x]) + (27*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10*Sqrt[10]
)

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Rubi in Sympy [A]  time = 12.3079, size = 78, normalized size = 0.93 \[ \frac{27 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{100} - \frac{2 \left (\frac{111311 x}{4} + \frac{66967}{4}\right )}{19965 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{7 \left (3 x + 2\right )^{2}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/100 - 2*(111311*x/4 + 66967/4)/(1996
5*sqrt(-2*x + 1)*sqrt(5*x + 3)) + 7*(3*x + 2)**2/(33*(-2*x + 1)**(3/2)*sqrt(5*x
+ 3))

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Mathematica [A]  time = 0.13362, size = 79, normalized size = 0.94 \[ -\frac{-10 \sqrt{5 x+3} \left (298852 x^2+124263 x-33087\right )-107811 \sqrt{10-20 x} \left (10 x^2+x-3\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{399300 (1-2 x)^{3/2} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

-(-10*Sqrt[3 + 5*x]*(-33087 + 124263*x + 298852*x^2) - 107811*Sqrt[10 - 20*x]*(-
3 + x + 10*x^2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(399300*(1 - 2*x)^(3/2)*(3 + 5
*x))

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Maple [B]  time = 0.022, size = 134, normalized size = 1.6 \[{\frac{1}{798600\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 2156220\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-862488\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-754677\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5977040\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+323433\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2485260\,x\sqrt{-10\,{x}^{2}-x+3}-661740\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/798600*(1-2*x)^(1/2)*(2156220*10^(1/2)*arcsin(20/11*x+1/11)*x^3-862488*10^(1/2
)*arcsin(20/11*x+1/11)*x^2-754677*10^(1/2)*arcsin(20/11*x+1/11)*x+5977040*x^2*(-
10*x^2-x+3)^(1/2)+323433*10^(1/2)*arcsin(20/11*x+1/11)+2485260*x*(-10*x^2-x+3)^(
1/2)-661740*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49172, size = 105, normalized size = 1.25 \[ \frac{27}{200} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{74713 \, x}{19965 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{273689}{79860 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{343}{132 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/200*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 74713/19965*x/sqrt(-10*x^2 - x +
3) - 273689/79860/sqrt(-10*x^2 - x + 3) - 343/132/(2*sqrt(-10*x^2 - x + 3)*x -
sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.223332, size = 127, normalized size = 1.51 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (298852 \, x^{2} + 124263 \, x - 33087\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 107811 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{798600 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/798600*sqrt(10)*(2*sqrt(10)*(298852*x^2 + 124263*x - 33087)*sqrt(5*x + 3)*sqrt
(-2*x + 1) + 107811*(20*x^3 - 8*x^2 - 7*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(
sqrt(5*x + 3)*sqrt(-2*x + 1))))/(20*x^3 - 8*x^2 - 7*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.25484, size = 159, normalized size = 1.89 \[ \frac{27}{100} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{66550 \, \sqrt{5 \, x + 3}} + \frac{49 \,{\left (244 \, \sqrt{5}{\left (5 \, x + 3\right )} - 957 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{199650 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{33275 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/100*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/66550*sqrt(10)*(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 49/199650*(244*sqrt(5)*(5*x + 3) - 9
57*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/33275*sqrt(10)*sqrt(5*
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))