3.1958 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=95 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{49 \sqrt{1-2 x}}{2 (3 x+2)}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2) + (49*Sqrt[1 - 2*x])/(2*(2 + 3*x)) + 235*Sqr
t[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt
[1 - 2*x]]

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Rubi [A]  time = 0.188595, antiderivative size = 95, 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 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2}+\frac{49 \sqrt{1-2 x}}{2 (3 x+2)}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2) + (49*Sqrt[1 - 2*x])/(2*(2 + 3*x)) + 235*Sqr
t[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt
[1 - 2*x]]

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Rubi in Sympy [A]  time = 20.925, size = 83, normalized size = 0.87 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{6 \left (3 x + 2\right )^{2}} + \frac{49 \sqrt{- 2 x + 1}}{2 \left (3 x + 2\right )} + \frac{235 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{3} - \frac{242 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(6*(3*x + 2)**2) + 49*sqrt(-2*x + 1)/(2*(3*x + 2)) + 235*sqr
t(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/3 - 242*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x
+ 1)/11)/5

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Mathematica [A]  time = 0.14918, size = 80, normalized size = 0.84 \[ \frac{7 \sqrt{1-2 x} (61 x+43)}{6 (3 x+2)^2}+235 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-242 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[1 - 2*x]*(43 + 61*x))/(6*(2 + 3*x)^2) + 235*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]] - 242*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.018, size = 66, normalized size = 0.7 \[ -126\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{61\, \left ( 1-2\,x \right ) ^{3/2}}{54}}-{\frac{49\,\sqrt{1-2\,x}}{18}} \right ) }+{\frac{235\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{242\,\sqrt{55}}{5}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-126*(61/54*(1-2*x)^(3/2)-49/18*(1-2*x)^(1/2))/(-4-6*x)^2+235/3*arctanh(1/7*21^(
1/2)*(1-2*x)^(1/2))*21^(1/2)-242/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48093, size = 149, normalized size = 1.57 \[ \frac{121}{5} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{235}{6} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{7 \,{\left (61 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 147 \, \sqrt{-2 \, x + 1}\right )}}{3 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 235/6*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 7/3*(61*(-2*x + 1)^(3/2) - 147*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 0.219496, size = 188, normalized size = 1.98 \[ \frac{\sqrt{5} \sqrt{3}{\left (726 \, \sqrt{11} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 705 \, \sqrt{7} \sqrt{5}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + 7 \, \sqrt{5} \sqrt{3}{\left (61 \, x + 43\right )} \sqrt{-2 \, x + 1}\right )}}{90 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*sqrt(5)*sqrt(3)*(726*sqrt(11)*sqrt(3)*(9*x^2 + 12*x + 4)*log((sqrt(5)*(5*x
- 8) + 5*sqrt(11)*sqrt(-2*x + 1))/(5*x + 3)) + 705*sqrt(7)*sqrt(5)*(9*x^2 + 12*x
+ 4)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*x + 2)) + 7*sqrt(5)*
sqrt(3)*(61*x + 43)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216867, size = 144, normalized size = 1.52 \[ \frac{121}{5} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{235}{6} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{7 \,{\left (61 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 147 \, \sqrt{-2 \, x + 1}\right )}}{12 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/5*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) - 235/6*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) - 7/12*(61*(-2*x + 1)^(3/2) - 147*sqrt(-2*x + 1))/(3*x + 2)
^2