3.1819 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=113 \[ \frac{1207 \sqrt{1-2 x}}{49 (3 x+2)}+\frac{52 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

Sqrt[1 - 2*x]/(3*(2 + 3*x)^3) + (52*Sqrt[1 - 2*x])/(21*(2 + 3*x)^2) + (1207*Sqrt
[1 - 2*x])/(49*(2 + 3*x)) + (83264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21
]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.24113, 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 \[ \frac{1207 \sqrt{1-2 x}}{49 (3 x+2)}+\frac{52 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - 2*x]/(3*(2 + 3*x)^3) + (52*Sqrt[1 - 2*x])/(21*(2 + 3*x)^2) + (1207*Sqrt
[1 - 2*x])/(49*(2 + 3*x)) + (83264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21
]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 27.9971, size = 99, normalized size = 0.88 \[ \frac{1207 \sqrt{- 2 x + 1}}{49 \left (3 x + 2\right )} + \frac{52 \sqrt{- 2 x + 1}}{21 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{3}} + \frac{83264 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1029} - 50 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1207*sqrt(-2*x + 1)/(49*(3*x + 2)) + 52*sqrt(-2*x + 1)/(21*(3*x + 2)**2) + sqrt(
-2*x + 1)/(3*(3*x + 2)**3) + 83264*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/102
9 - 50*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.182588, size = 83, normalized size = 0.73 \[ \frac{\sqrt{1-2 x} \left (10863 x^2+14848 x+5087\right )}{49 (3 x+2)^3}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(5087 + 14848*x + 10863*x^2))/(49*(2 + 3*x)^3) + (83264*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]]

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Maple [A]  time = 0.019, size = 75, normalized size = 0.7 \[ -54\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{1207\, \left ( 1-2\,x \right ) ^{5/2}}{147}}-{\frac{7346\, \left ( 1-2\,x \right ) ^{3/2}}{189}}+{\frac{1243\,\sqrt{1-2\,x}}{27}} \right ) }+{\frac{83264\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-50\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-54*(1207/147*(1-2*x)^(5/2)-7346/189*(1-2*x)^(3/2)+1243/27*(1-2*x)^(1/2))/(-4-6*
x)^3+83264/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50*arctanh(1/11*55^
(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51494, size = 173, normalized size = 1.53 \[ 25 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{41632}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (10863 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 51422 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 60907 \, \sqrt{-2 \, x + 1}\right )}}{49 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
41632/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 2/49*(10863*(-2*x + 1)^(5/2) - 51422*(-2*x + 1)^(3/2) + 60907*sqrt(-2*x
+ 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.218538, size = 185, normalized size = 1.64 \[ \frac{\sqrt{21}{\left (1225 \, \sqrt{55} \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (10863 \, x^{2} + 14848 \, x + 5087\right )} \sqrt{-2 \, x + 1} + 41632 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1029 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1029*sqrt(21)*(1225*sqrt(55)*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x +
sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(10863*x^2 + 14848*x + 5087)*
sqrt(-2*x + 1) + 41632*(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 [A]  time = 99.8497, size = 559, normalized size = 4.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

660*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*sq
rt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*s
qrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 264*Piecewise((sqrt(21)*
(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/
16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 +
1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x + 1
)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3))) + 112*Piecewise((sqrt(21)*(-5*log(
sqrt(21)*sqrt(-2*x + 1)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/32 - 5/
(32*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**2)
- 1/(48*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**3) - 5/(32*(sqrt(21)*sqrt(-2*x + 1)/7
- 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(-2*x +
1)/7 - 1)**3))/7203, (x <= 1/2) & (x > -2/3))) - 1650*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)) + 2750*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqr
t(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/1
1)/55, -2*x + 1 < 11/5))

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GIAC/XCAS [A]  time = 0.216681, size = 166, normalized size = 1.47 \[ 25 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{41632}{1029} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{10863 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 51422 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 60907 \, \sqrt{-2 \, x + 1}}{196 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 41632/1029*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) + 1/196*(10863*(2*x - 1)^2*sqrt(-2*x + 1) - 51422*(-2*x +
1)^(3/2) + 60907*sqrt(-2*x + 1))/(3*x + 2)^3