3.1968 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=92 \[ -\frac{11 (1-2 x)^{3/2}}{5 (5 x+3)}-\frac{58}{75} \sqrt{1-2 x}-\frac{98}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{836}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-58*Sqrt[1 - 2*x])/75 - (11*(1 - 2*x)^(3/2))/(5*(3 + 5*x)) - (98*Sqrt[7/3]*ArcT
anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + (836*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/25

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Rubi [A]  time = 0.191144, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{11 (1-2 x)^{3/2}}{5 (5 x+3)}-\frac{58}{75} \sqrt{1-2 x}-\frac{98}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{836}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-58*Sqrt[1 - 2*x])/75 - (11*(1 - 2*x)^(3/2))/(5*(3 + 5*x)) - (98*Sqrt[7/3]*ArcT
anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + (836*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/25

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Rubi in Sympy [A]  time = 21.491, size = 76, normalized size = 0.83 \[ - \frac{11 \left (- 2 x + 1\right )^{\frac{3}{2}}}{5 \left (5 x + 3\right )} - \frac{58 \sqrt{- 2 x + 1}}{75} - \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9} + \frac{836 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*(-2*x + 1)**(3/2)/(5*(5*x + 3)) - 58*sqrt(-2*x + 1)/75 - 98*sqrt(21)*atanh(s
qrt(21)*sqrt(-2*x + 1)/7)/9 + 836*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/125

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Mathematica [A]  time = 0.220928, size = 83, normalized size = 0.9 \[ \frac{1}{375} \left (\frac{5 \sqrt{1-2 x} (40 x-339)}{5 x+3}+2508 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )-\frac{98}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/3 + ((5*Sqrt[1 - 2*x]*(-339 + 4
0*x))/(3 + 5*x) + 2508*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/375

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Maple [A]  time = 0.018, size = 63, normalized size = 0.7 \[{\frac{8}{75}\sqrt{1-2\,x}}-{\frac{98\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{125}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}+{\frac{836\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/75*(1-2*x)^(1/2)-98/9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+242/125*(1-
2*x)^(1/2)/(-6/5-2*x)+836/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50505, size = 132, normalized size = 1.43 \[ -\frac{418}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{49}{9} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8}{75} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-418/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) + 49/9*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 8/75*sqrt(-2*x + 1) - 121/25*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.22265, size = 166, normalized size = 1.8 \[ \frac{\sqrt{5} \sqrt{3}{\left (1254 \, \sqrt{11} \sqrt{3}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} - 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 1225 \, \sqrt{7} \sqrt{5}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{5} \sqrt{3}{\left (40 \, x - 339\right )} \sqrt{-2 \, x + 1}\right )}}{1125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1125*sqrt(5)*sqrt(3)*(1254*sqrt(11)*sqrt(3)*(5*x + 3)*log((sqrt(5)*(5*x - 8) -
5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 1225*sqrt(7)*sqrt(5)*(5*x + 3)*log((sqr
t(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(5)*sqrt(3)*(40*x -
339)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [A]  time = 99.6752, size = 240, normalized size = 2.61 \[ \frac{8 \sqrt{- 2 x + 1}}{75} - \frac{5324 \left (\begin{cases} \frac{\sqrt{55} \left (- \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} - 1\right )}\right )}{605} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{25} + \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{3} - \frac{9438 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*sqrt(-2*x + 1)/75 - 5324*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(-2*x + 1)/11
- 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/1
1 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/605, (x <= 1/2) & (x > -3/5)))
/25 + 686*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7
/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/3 - 9438*P
iecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sq
rt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))/25

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GIAC/XCAS [A]  time = 0.21577, size = 140, normalized size = 1.52 \[ -\frac{418}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{49}{9} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{8}{75} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{25 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-418/125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) + 49/9*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) + 8/75*sqrt(-2*x + 1) - 121/25*sqrt(-2*x + 1)/(5*x + 3)