3.2605 \(\int \frac{2+3 x}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{164 \sqrt{5 x+3}}{3993 \sqrt{1-2 x}}+\frac{82 \sqrt{5 x+3}}{1815 (1-2 x)^{3/2}}-\frac{2}{55 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

[Out]

-2/(55*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (82*Sqrt[3 + 5*x])/(1815*(1 - 2*x)^(3/2)
) + (164*Sqrt[3 + 5*x])/(3993*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0675651, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{164 \sqrt{5 x+3}}{3993 \sqrt{1-2 x}}+\frac{82 \sqrt{5 x+3}}{1815 (1-2 x)^{3/2}}-\frac{2}{55 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(55*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (82*Sqrt[3 + 5*x])/(1815*(1 - 2*x)^(3/2)
) + (164*Sqrt[3 + 5*x])/(3993*Sqrt[1 - 2*x])

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Rubi in Sympy [A]  time = 6.70042, size = 60, normalized size = 0.9 \[ \frac{164 \sqrt{5 x + 3}}{3993 \sqrt{- 2 x + 1}} + \frac{82 \sqrt{5 x + 3}}{1815 \left (- 2 x + 1\right )^{\frac{3}{2}}} - \frac{2}{55 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

164*sqrt(5*x + 3)/(3993*sqrt(-2*x + 1)) + 82*sqrt(5*x + 3)/(1815*(-2*x + 1)**(3/
2)) - 2/(55*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.0464369, size = 32, normalized size = 0.48 \[ \frac{-1640 x^2+738 x+888}{3993 (1-2 x)^{3/2} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(888 + 738*x - 1640*x^2)/(3993*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])

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Maple [A]  time = 0.004, size = 27, normalized size = 0.4 \[ -{\frac{1640\,{x}^{2}-738\,x-888}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(820*x^2-369*x-444)/(3+5*x)^(1/2)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.359, size = 86, normalized size = 1.28 \[ \frac{820 \, x}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{41}{3993 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{7}{33 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

820/3993*x/sqrt(-10*x^2 - x + 3) + 41/3993/sqrt(-10*x^2 - x + 3) - 7/33/(2*sqrt(
-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.21695, size = 58, normalized size = 0.87 \[ -\frac{2 \,{\left (820 \, x^{2} - 369 \, x - 444\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(820*x^2 - 369*x - 444)*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. \[ \int \frac{3 x + 2}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.25364, size = 135, normalized size = 2.01 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2662 \, \sqrt{5 \, x + 3}} - \frac{2 \,{\left (152 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1221 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{99825 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{1331 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2662*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 2/99825*(1
52*sqrt(5)*(5*x + 3) - 1221*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 +
2/1331*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))