3.1888 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=113 \[ \frac{1138 \sqrt{1-2 x}}{21 (3 x+2)}+\frac{49 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3) + (49*Sqrt[1 - 2*x])/(9*(2 + 3*x)^2) + (1138*S
qrt[1 - 2*x])/(21*(2 + 3*x)) + (78506*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt
[21]) - 110*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.243521, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1138 \sqrt{1-2 x}}{21 (3 x+2)}+\frac{49 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3) + (49*Sqrt[1 - 2*x])/(9*(2 + 3*x)^2) + (1138*S
qrt[1 - 2*x])/(21*(2 + 3*x)) + (78506*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt
[21]) - 110*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 28.1951, size = 100, normalized size = 0.88 \[ \frac{1138 \sqrt{- 2 x + 1}}{21 \left (3 x + 2\right )} + \frac{49 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{2}} + \frac{7 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{3}} + \frac{78506 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{441} - 110 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)/(2+3*x)**4/(3+5*x),x)

[Out]

1138*sqrt(-2*x + 1)/(21*(3*x + 2)) + 49*sqrt(-2*x + 1)/(9*(3*x + 2)**2) + 7*sqrt
(-2*x + 1)/(9*(3*x + 2)**3) + 78506*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/44
1 - 110*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.182168, size = 83, normalized size = 0.73 \[ \frac{\sqrt{1-2 x} \left (10242 x^2+13999 x+4797\right )}{21 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(4797 + 13999*x + 10242*x^2))/(21*(2 + 3*x)^3) + (78506*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21]) - 110*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Maple [A]  time = 0.017, size = 75, normalized size = 0.7 \[ -54\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1138\, \left ( 1-2\,x \right ) ^{5/2}}{63}}-{\frac{6926\, \left ( 1-2\,x \right ) ^{3/2}}{81}}+{\frac{8204\,\sqrt{1-2\,x}}{81}} \right ) }+{\frac{78506\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-110\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x),x)

[Out]

-54*(1138/63*(1-2*x)^(5/2)-6926/81*(1-2*x)^(3/2)+8204/81*(1-2*x)^(1/2))/(-4-6*x)
^3+78506/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-110*arctanh(1/11*55^(1
/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50485, size = 173, normalized size = 1.53 \[ 55 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{39253}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (5121 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 24241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 28714 \, \sqrt{-2 \, x + 1}\right )}}{21 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^4),x, algorithm="maxima")

[Out]

55*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
39253/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 4/21*(5121*(-2*x + 1)^(5/2) - 24241*(-2*x + 1)^(3/2) + 28714*sqrt(-2*x +
1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.235447, size = 185, normalized size = 1.64 \[ \frac{\sqrt{21}{\left (1155 \, \sqrt{55} \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (10242 \, x^{2} + 13999 \, x + 4797\right )} \sqrt{-2 \, x + 1} + 39253 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{441 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^4),x, algorithm="fricas")

[Out]

1/441*sqrt(21)*(1155*sqrt(55)*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + s
qrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(10242*x^2 + 13999*x + 4797)*s
qrt(-2*x + 1) + 39253*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) - 21*
sqrt(-2*x + 1))/(3*x + 2)))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**(3/2)/(2+3*x)**4/(3+5*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214848, size = 166, normalized size = 1.47 \[ 55 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{39253}{441} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{5121 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 24241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 28714 \, \sqrt{-2 \, x + 1}}{42 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^4),x, algorithm="giac")

[Out]

55*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 39253/441*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21)
+ 3*sqrt(-2*x + 1))) + 1/42*(5121*(2*x - 1)^2*sqrt(-2*x + 1) - 24241*(-2*x + 1)
^(3/2) + 28714*sqrt(-2*x + 1))/(3*x + 2)^3