3.1956 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)} \, dx\)

Optimal. Leaf size=85 \[ -\frac{4}{45} (1-2 x)^{3/2}-\frac{272}{225} \sqrt{1-2 x}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-272*Sqrt[1 - 2*x])/225 - (4*(1 - 2*x)^(3/2))/45 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/9 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.187728, antiderivative size = 85, 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 \[ -\frac{4}{45} (1-2 x)^{3/2}-\frac{272}{225} \sqrt{1-2 x}+\frac{98}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{242}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-272*Sqrt[1 - 2*x])/225 - (4*(1 - 2*x)^(3/2))/45 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/9 - (242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi in Sympy [A]  time = 20.724, size = 73, normalized size = 0.86 \[ - \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}}}{45} - \frac{272 \sqrt{- 2 x + 1}}{225} + \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{27} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(-2*x + 1)**(3/2)/45 - 272*sqrt(-2*x + 1)/225 + 98*sqrt(21)*atanh(sqrt(21)*sq
rt(-2*x + 1)/7)/27 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/125

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Mathematica [A]  time = 0.113769, size = 71, normalized size = 0.84 \[ \frac{2 \left (30 \sqrt{1-2 x} (10 x-73)+6125 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-3267 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right )}{3375} \]

Antiderivative was successfully verified.

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

[Out]

(2*(30*Sqrt[1 - 2*x]*(-73 + 10*x) + 6125*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]] - 3267*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]))/3375

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Maple [A]  time = 0.014, size = 56, normalized size = 0.7 \[ -{\frac{4}{45} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{242\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{98\,\sqrt{21}}{27}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{272}{225}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/45*(1-2*x)^(3/2)-242/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+98/27*
arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-272/225*(1-2*x)^(1/2)

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Maxima [A]  time = 1.47872, size = 123, normalized size = 1.45 \[ -\frac{4}{45} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{272}{225} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/45*(-2*x + 1)^(3/2) + 121/125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sq
rt(55) + 5*sqrt(-2*x + 1))) - 49/27*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/
(sqrt(21) + 3*sqrt(-2*x + 1))) - 272/225*sqrt(-2*x + 1)

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Fricas [A]  time = 0.217301, size = 144, normalized size = 1.69 \[ \frac{1}{3375} \, \sqrt{5} \sqrt{3}{\left (4 \, \sqrt{5} \sqrt{3}{\left (10 \, x - 73\right )} \sqrt{-2 \, x + 1} + 1089 \, \sqrt{11} \sqrt{3} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 1225 \, \sqrt{7} \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3375*sqrt(5)*sqrt(3)*(4*sqrt(5)*sqrt(3)*(10*x - 73)*sqrt(-2*x + 1) + 1089*sqrt
(11)*sqrt(3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 12
25*sqrt(7)*sqrt(5)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2))
)

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Sympy [A]  time = 10.0773, size = 151, normalized size = 1.78 \[ - \frac{4 \left (- 2 x + 1\right )^{\frac{3}{2}}}{45} - \frac{272 \sqrt{- 2 x + 1}}{225} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{9} + \frac{2662 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(-2*x + 1)**(3/2)/45 - 272*sqrt(-2*x + 1)/225 - 686*Piecewise((-sqrt(21)*acot
h(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))/9 + 2662*Piecewise((-sqrt(55)*acoth(sqrt(55)*
sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1
)/11)/55, -2*x + 1 < 11/5))/25

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GIAC/XCAS [A]  time = 0.215343, size = 131, normalized size = 1.54 \[ -\frac{4}{45} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{121}{125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{49}{27} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{272}{225} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/45*(-2*x + 1)^(3/2) + 121/125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x
+ 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 49/27*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6
*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 272/225*sqrt(-2*x + 1)