3.1820 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{337955 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{14555 \sqrt{1-2 x}}{1176 (3 x+2)^2}+\frac{139 \sqrt{1-2 x}}{84 (3 x+2)^3}+\frac{\sqrt{1-2 x}}{4 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

Sqrt[1 - 2*x]/(4*(2 + 3*x)^4) + (139*Sqrt[1 - 2*x])/(84*(2 + 3*x)^3) + (14555*Sq
rt[1 - 2*x])/(1176*(2 + 3*x)^2) + (337955*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (116
56955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[55]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.301435, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{337955 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{14555 \sqrt{1-2 x}}{1176 (3 x+2)^2}+\frac{139 \sqrt{1-2 x}}{84 (3 x+2)^3}+\frac{\sqrt{1-2 x}}{4 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - 2*x]/(4*(2 + 3*x)^4) + (139*Sqrt[1 - 2*x])/(84*(2 + 3*x)^3) + (14555*Sq
rt[1 - 2*x])/(1176*(2 + 3*x)^2) + (337955*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (116
56955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[55]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 34.9262, size = 117, normalized size = 0.88 \[ \frac{337955 \sqrt{- 2 x + 1}}{2744 \left (3 x + 2\right )} + \frac{14555 \sqrt{- 2 x + 1}}{1176 \left (3 x + 2\right )^{2}} + \frac{139 \sqrt{- 2 x + 1}}{84 \left (3 x + 2\right )^{3}} + \frac{\sqrt{- 2 x + 1}}{4 \left (3 x + 2\right )^{4}} + \frac{11656955 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{28812} - 250 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

337955*sqrt(-2*x + 1)/(2744*(3*x + 2)) + 14555*sqrt(-2*x + 1)/(1176*(3*x + 2)**2
) + 139*sqrt(-2*x + 1)/(84*(3*x + 2)**3) + sqrt(-2*x + 1)/(4*(3*x + 2)**4) + 116
56955*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/28812 - 250*sqrt(55)*atanh(sqrt(
55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.248668, size = 88, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (9124785 x^3+18555225 x^2+12587542 x+2849254\right )}{2744 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(2849254 + 12587542*x + 18555225*x^2 + 9124785*x^3))/(2744*(2 + 3
*x)^4) + (11656955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[
55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 84, normalized size = 0.6 \[ -162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{337955\, \left ( 1-2\,x \right ) ^{7/2}}{8232}}-{\frac{3070705\, \left ( 1-2\,x \right ) ^{5/2}}{10584}}+{\frac{3100927\, \left ( 1-2\,x \right ) ^{3/2}}{4536}}-{\frac{116015\,\sqrt{1-2\,x}}{216}} \right ) }+{\frac{11656955\,\sqrt{21}}{28812}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-250\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-162*(337955/8232*(1-2*x)^(7/2)-3070705/10584*(1-2*x)^(5/2)+3100927/4536*(1-2*x)
^(3/2)-116015/216*(1-2*x)^(1/2))/(-4-6*x)^4+11656955/28812*arctanh(1/7*21^(1/2)*
(1-2*x)^(1/2))*21^(1/2)-250*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.48267, size = 197, normalized size = 1.48 \[ 125 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11656955}{57624} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9124785 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 64484805 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 151945423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 119379435 \, \sqrt{-2 \, x + 1}}{1372 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) -
11656955/57624*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 1/1372*(9124785*(-2*x + 1)^(7/2) - 64484805*(-2*x + 1)^(5/2) + 1519
45423*(-2*x + 1)^(3/2) - 119379435*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x -
1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 0.220837, size = 212, normalized size = 1.59 \[ \frac{\sqrt{21}{\left (343000 \, \sqrt{55} \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (9124785 \, x^{3} + 18555225 \, x^{2} + 12587542 \, x + 2849254\right )} \sqrt{-2 \, x + 1} + 11656955 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{57624 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/57624*sqrt(21)*(343000*sqrt(55)*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x +
16)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*(9124785*x^3 +
18555225*x^2 + 12587542*x + 2849254)*sqrt(-2*x + 1) + 11656955*(81*x^4 + 216*x^
3 + 216*x^2 + 96*x + 16)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2))
)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 159.506, size = 794, normalized size = 5.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3300*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))) - 1320*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))) + 528*Piecewise((sqrt(21)*(-5*lo
g(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))) - 224*Piecewise((sqrt(21)*(35*l
og(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/256 - 35*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)/25
6 + 35/(256*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 15/(256*(sqrt(21)*sqrt(-2*x + 1)/
7 + 1)**2) + 5/(192*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)**3) + 1/(128*(sqrt(21)*sqrt(
-2*x + 1)/7 + 1)**4) + 35/(256*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 15/(256*(sqrt(
21)*sqrt(-2*x + 1)/7 - 1)**2) + 5/(192*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**3) - 1/(
128*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)**4))/50421, (x <= 1/2) & (x > -2/3))) - 8250
*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sq
rt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3)) + 13750*Piecewise((
-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 > 11/5), (-sqrt(55)*ata
nh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 < 11/5))

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GIAC/XCAS [A]  time = 0.23176, size = 188, normalized size = 1.41 \[ 125 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{11656955}{57624} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{9124785 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 64484805 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 151945423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 119379435 \, \sqrt{-2 \, x + 1}}{21952 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x
+ 1))) - 11656955/57624*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 1/21952*(9124785*(2*x - 1)^3*sqrt(-2*x + 1) + 6448
4805*(2*x - 1)^2*sqrt(-2*x + 1) - 151945423*(-2*x + 1)^(3/2) + 119379435*sqrt(-2
*x + 1))/(3*x + 2)^4