3.2135 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=54 \[ -\frac{407}{98 \sqrt{1-2 x}}+\frac{121}{42 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

[Out]

121/(42*(1 - 2*x)^(3/2)) - 407/(98*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(49*Sqrt[21])

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Rubi [A]  time = 0.0882123, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{407}{98 \sqrt{1-2 x}}+\frac{121}{42 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

121/(42*(1 - 2*x)^(3/2)) - 407/(98*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(49*Sqrt[21])

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Rubi in Sympy [A]  time = 11.8814, size = 48, normalized size = 0.89 \[ - \frac{2 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1029} - \frac{407}{98 \sqrt{- 2 x + 1}} + \frac{121}{42 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1029 - 407/(98*sqrt(-2*x + 1)) + 12
1/(42*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.130435, size = 46, normalized size = 0.85 \[ \frac{\frac{77 (111 x-17)}{(1-2 x)^{3/2}}-2 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029} \]

Antiderivative was successfully verified.

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

[Out]

((77*(-17 + 111*x))/(1 - 2*x)^(3/2) - 2*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/1029

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Maple [A]  time = 0.014, size = 38, normalized size = 0.7 \[{\frac{121}{42} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{407}{98}{\frac{1}{\sqrt{1-2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/42/(1-2*x)^(3/2)-2/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-407/98/
(1-2*x)^(1/2)

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Maxima [A]  time = 1.4747, size = 69, normalized size = 1.28 \[ \frac{1}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{11 \,{\left (111 \, x - 17\right )}}{147 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) + 11/147*(111*x - 17)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.226288, size = 97, normalized size = 1.8 \[ \frac{\sqrt{21}{\left (3 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - 11 \, \sqrt{21}{\left (111 \, x - 17\right )}\right )}}{3087 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3087*sqrt(21)*(3*(2*x - 1)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2
*x + 1))/(3*x + 2)) - 11*sqrt(21)*(111*x - 17))/((2*x - 1)*sqrt(-2*x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213598, size = 82, normalized size = 1.52 \[ \frac{1}{1029} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{11 \,{\left (111 \, x - 17\right )}}{147 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 11/147*(111*x - 17)/((2*x - 1)*sqrt(-2*x + 1))