3.1829 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=131 \[ -\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)}+\frac{52 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)) +
(52*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) - (7209*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/7 + 1000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.245399, antiderivative size = 131, 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 \[ -\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)}+\frac{52 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)) +
(52*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) - (7209*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/7 + 1000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 28.5465, size = 109, normalized size = 0.83 \[ - \frac{1045 \sqrt{- 2 x + 1}}{14 \left (5 x + 3\right )} + \frac{52 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{\sqrt{- 2 x + 1}}{2 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} - \frac{7209 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} + \frac{1000 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1045*sqrt(-2*x + 1)/(14*(5*x + 3)) + 52*sqrt(-2*x + 1)/(7*(3*x + 2)*(5*x + 3))
+ sqrt(-2*x + 1)/(2*(3*x + 2)**2*(5*x + 3)) - 7209*sqrt(21)*atanh(sqrt(21)*sqrt(
-2*x + 1)/7)/49 + 1000*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/11

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Mathematica [A]  time = 0.192295, size = 94, normalized size = 0.72 \[ -\frac{\sqrt{1-2 x} \left (9405 x^2+12228 x+3965\right )}{14 (3 x+2)^2 (5 x+3)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(3965 + 12228*x + 9405*x^2))/(14*(2 + 3*x)^2*(3 + 5*x)) - (7209*
Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + 1000*Sqrt[5/11]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.02, size = 82, normalized size = 0.6 \[ 162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{139\, \left ( 1-2\,x \right ) ^{3/2}}{126}}-{\frac{47\,\sqrt{1-2\,x}}{18}} \right ) }-{\frac{7209\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+10\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+{\frac{1000\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

162*(139/126*(1-2*x)^(3/2)-47/18*(1-2*x)^(1/2))/(-4-6*x)^2-7209/49*arctanh(1/7*2
1^(1/2)*(1-2*x)^(1/2))*21^(1/2)+10*(1-2*x)^(1/2)/(-6/5-2*x)+1000/11*arctanh(1/11
*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50402, size = 173, normalized size = 1.32 \[ -\frac{500}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7209}{98} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9405 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 43266 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49721 \, \sqrt{-2 \, x + 1}}{7 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-500/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) + 7209/98*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/7*(9405*(-2*x + 1)^(5/2) - 43266*(-2*x + 1)^(3/2) + 49721*sqrt(-2*x
+ 1))/(45*(2*x - 1)^3 + 309*(2*x - 1)^2 + 1414*x - 168)

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Fricas [A]  time = 0.220664, size = 215, normalized size = 1.64 \[ \frac{\sqrt{11} \sqrt{7}{\left (7000 \, \sqrt{7} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} - 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 7209 \, \sqrt{11} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{11} \sqrt{7}{\left (9405 \, x^{2} + 12228 \, x + 3965\right )} \sqrt{-2 \, x + 1}\right )}}{1078 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1078*sqrt(11)*sqrt(7)*(7000*sqrt(7)*sqrt(5)*(45*x^3 + 87*x^2 + 56*x + 12)*log(
(sqrt(11)*(5*x - 8) - 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 7209*sqrt(11)*sqrt
(3)*(45*x^3 + 87*x^2 + 56*x + 12)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x +
1))/(3*x + 2)) - sqrt(11)*sqrt(7)*(9405*x^2 + 12228*x + 3965)*sqrt(-2*x + 1))/(
45*x^3 + 87*x^2 + 56*x + 12)

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Sympy [A]  time = 69.4744, size = 468, normalized size = 3.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-816*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*s
qrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*
sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) + 168*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))) - 1100*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)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/605
, (x <= 1/2) & (x > -3/5))) + 3030*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)) - 5050*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 < 1
1/5))

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GIAC/XCAS [A]  time = 0.246, size = 166, normalized size = 1.27 \[ -\frac{500}{11} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{7209}{98} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} + \frac{9 \,{\left (139 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 329 \, \sqrt{-2 \, x + 1}\right )}}{28 \,{\left (3 \, x + 2\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-500/11*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(
-2*x + 1))) + 7209/98*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) - 25*sqrt(-2*x + 1)/(5*x + 3) + 9/28*(139*(-2*x + 1)^(3
/2) - 329*sqrt(-2*x + 1))/(3*x + 2)^2