3.2557 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{4091 \sqrt{1-2 x}}{19965 \sqrt{5 x+3}}-\frac{3679 \sqrt{1-2 x}}{3630 (5 x+3)^{3/2}}+\frac{49}{22 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(3630*(3 + 5*x)^(3/
2)) - (4091*Sqrt[1 - 2*x])/(19965*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0891281, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{4091 \sqrt{1-2 x}}{19965 \sqrt{5 x+3}}-\frac{3679 \sqrt{1-2 x}}{3630 (5 x+3)^{3/2}}+\frac{49}{22 (5 x+3)^{3/2} \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

49/(22*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (3679*Sqrt[1 - 2*x])/(3630*(3 + 5*x)^(3/
2)) - (4091*Sqrt[1 - 2*x])/(19965*Sqrt[3 + 5*x])

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Rubi in Sympy [A]  time = 8.28504, size = 60, normalized size = 0.9 \[ \frac{8182 \sqrt{5 x + 3}}{99825 \sqrt{- 2 x + 1}} - \frac{412}{9075 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{2}{825 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8182*sqrt(5*x + 3)/(99825*sqrt(-2*x + 1)) - 412/(9075*sqrt(-2*x + 1)*sqrt(5*x +
3)) - 2/(825*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.0505029, size = 32, normalized size = 0.48 \[ \frac{2 \left (4091 x^2+4456 x+1196\right )}{3993 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(1196 + 4456*x + 4091*x^2))/(3993*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.005, size = 27, normalized size = 0.4 \[{\frac{8182\,{x}^{2}+8912\,x+2392}{3993} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3993*(4091*x^2+4456*x+1196)/(3+5*x)^(3/2)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.34495, size = 86, normalized size = 1.28 \[ \frac{8182 \, x}{19965 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{20014}{99825 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{825 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8182/19965*x/sqrt(-10*x^2 - x + 3) + 20014/99825/sqrt(-10*x^2 - x + 3) - 2/825/(
5*sqrt(-10*x^2 - x + 3)*x + 3*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.228541, size = 58, normalized size = 0.87 \[ -\frac{2 \,{\left (4091 \, x^{2} + 4456 \, x + 1196\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3993 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3993*(4091*x^2 + 4456*x + 1196)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(50*x^3 + 35*x^2
- 12*x - 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.255842, size = 205, normalized size = 3.06 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{1597200 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{139 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{133100 \, \sqrt{5 \, x + 3}} - \frac{98 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{6655 \,{\left (2 \, x - 1\right )}} + \frac{{\left (\frac{417 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{99825 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1597200*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 139
/133100*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 98/6655*sq
rt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) + 1/99825*(417*sqrt(10)*(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))^3