3.2493 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{404 \sqrt{1-2 x}}{9075 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{825 (5 x+3)^{3/2}}+\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-2*Sqrt[1 - 2*x])/(825*(3 + 5*x)^(3/2)) - (404*Sqrt[1 - 2*x])/(9075*Sqrt[3 + 5*
x]) + (9*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

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Rubi [A]  time = 0.102133, antiderivative size = 74, 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 \[ -\frac{404 \sqrt{1-2 x}}{9075 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{825 (5 x+3)^{3/2}}+\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x])/(825*(3 + 5*x)^(3/2)) - (404*Sqrt[1 - 2*x])/(9075*Sqrt[3 + 5*
x]) + (9*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

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Rubi in Sympy [A]  time = 8.5222, size = 65, normalized size = 0.88 \[ - \frac{404 \sqrt{- 2 x + 1}}{9075 \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1}}{825 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{9 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-404*sqrt(-2*x + 1)/(9075*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)/(825*(5*x + 3)**(3/2
)) + 9*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/125

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Mathematica [A]  time = 0.13387, size = 57, normalized size = 0.77 \[ -\frac{2 \sqrt{1-2 x} (1010 x+617)}{9075 (5 x+3)^{3/2}}-\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(617 + 1010*x))/(9075*(3 + 5*x)^(3/2)) - (9*Sqrt[2/5]*ArcSin[S
qrt[5/11]*Sqrt[1 - 2*x]])/25

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Maple [A]  time = 0.021, size = 96, normalized size = 1.3 \[{\frac{1}{90750} \left ( 81675\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+98010\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+29403\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -20200\,x\sqrt{-10\,{x}^{2}-x+3}-12340\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90750*(81675*10^(1/2)*arcsin(20/11*x+1/11)*x^2+98010*10^(1/2)*arcsin(20/11*x+1
/11)*x+29403*10^(1/2)*arcsin(20/11*x+1/11)-20200*x*(-10*x^2-x+3)^(1/2)-12340*(-1
0*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.48039, size = 84, normalized size = 1.14 \[ \frac{9}{250} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{825 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{404 \, \sqrt{-10 \, x^{2} - x + 3}}{9075 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/825*sqrt(-10*x^2 - x + 3)/(25*x
^2 + 30*x + 9) - 404/9075*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.228941, size = 115, normalized size = 1.55 \[ -\frac{\sqrt{5}{\left (4 \, \sqrt{5}{\left (1010 \, x + 617\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3267 \, \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{90750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90750*sqrt(5)*(4*sqrt(5)*(1010*x + 617)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3267*s
qrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)/(sqrt(5*x + 3)
*sqrt(-2*x + 1))))/(25*x^2 + 30*x + 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257793, size = 194, normalized size = 2.62 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{726000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{9}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{27 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{405 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{45375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/726000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 9/12
5*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 27/12100*sqrt(10)*(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/45375*(405*sqrt(10)*(sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))^3