3.1890 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=153 \[ \frac{2788127 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{120077 \sqrt{1-2 x}}{882 (3 x+2)^2}+\frac{5732 \sqrt{1-2 x}}{315 (3 x+2)^3}+\frac{41 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{7 \sqrt{1-2 x}}{15 (3 x+2)^5}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (5732
*Sqrt[1 - 2*x])/(315*(2 + 3*x)^3) + (120077*Sqrt[1 - 2*x])/(882*(2 + 3*x)^2) + (
2788127*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (96169877*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(1029*Sqrt[21]) - 2750*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.369469, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2788127 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{120077 \sqrt{1-2 x}}{882 (3 x+2)^2}+\frac{5732 \sqrt{1-2 x}}{315 (3 x+2)^3}+\frac{41 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{7 \sqrt{1-2 x}}{15 (3 x+2)^5}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(3/2)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(7*Sqrt[1 - 2*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (5732
*Sqrt[1 - 2*x])/(315*(2 + 3*x)^3) + (120077*Sqrt[1 - 2*x])/(882*(2 + 3*x)^2) + (
2788127*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (96169877*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(1029*Sqrt[21]) - 2750*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 42.4778, size = 138, normalized size = 0.9 \[ \frac{2788127 \sqrt{- 2 x + 1}}{2058 \left (3 x + 2\right )} + \frac{120077 \sqrt{- 2 x + 1}}{882 \left (3 x + 2\right )^{2}} + \frac{5732 \sqrt{- 2 x + 1}}{315 \left (3 x + 2\right )^{3}} + \frac{41 \sqrt{- 2 x + 1}}{15 \left (3 x + 2\right )^{4}} + \frac{7 \sqrt{- 2 x + 1}}{15 \left (3 x + 2\right )^{5}} + \frac{96169877 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21609} - 2750 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)/(2+3*x)**6/(3+5*x),x)

[Out]

2788127*sqrt(-2*x + 1)/(2058*(3*x + 2)) + 120077*sqrt(-2*x + 1)/(882*(3*x + 2)**
2) + 5732*sqrt(-2*x + 1)/(315*(3*x + 2)**3) + 41*sqrt(-2*x + 1)/(15*(3*x + 2)**4
) + 7*sqrt(-2*x + 1)/(15*(3*x + 2)**5) + 96169877*sqrt(21)*atanh(sqrt(21)*sqrt(-
2*x + 1)/7)/21609 - 2750*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.174382, size = 93, normalized size = 0.61 \[ \frac{\sqrt{1-2 x} \left (1129191435 x^4+3049001415 x^3+3088510878 x^2+1391064622 x+235067382\right )}{10290 (3 x+2)^5}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(235067382 + 1391064622*x + 3088510878*x^2 + 3049001415*x^3 + 112
9191435*x^4))/(10290*(2 + 3*x)^5) + (96169877*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
(1029*Sqrt[21]) - 2750*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.019, size = 93, normalized size = 0.6 \[ -486\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ({\frac{2788127\, \left ( 1-2\,x \right ) ^{9/2}}{6174}}-{\frac{2406977\, \left ( 1-2\,x \right ) ^{7/2}}{567}}+{\frac{127289798\, \left ( 1-2\,x \right ) ^{5/2}}{8505}}-{\frac{17098361\, \left ( 1-2\,x \right ) ^{3/2}}{729}}+{\frac{20099611\,\sqrt{1-2\,x}}{1458}} \right ) }+{\frac{96169877\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-2750\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x),x)

[Out]

-486*(2788127/6174*(1-2*x)^(9/2)-2406977/567*(1-2*x)^(7/2)[phone]/8505*(1-2*x
)^(5/2)-17098361/729*(1-2*x)^(3/2)[phone]/1458*(1-2*x)^(1/2))/(-4-6*x)^5+96169
877/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2750*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51715, size = 221, normalized size = 1.44 \[ 1375 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{96169877}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1129191435 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 10614768570 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 37423200612 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 58647378230 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 34470832865 \, \sqrt{-2 \, x + 1}}{5145 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^6),x, algorithm="maxima")

[Out]

1375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 96169877/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/5145*(1129191435*(-2*x + 1)^(9/2) - 10614768570*(-2*x + 1)^(7/2)
+ 37423200612*(-2*x + 1)^(5/2) - 58647378230*(-2*x + 1)^(3/2) + 34470832865*sqr
t(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*
x - 1)^2 + 72030*x - 19208)

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Fricas [A]  time = 0.219211, size = 239, normalized size = 1.56 \[ \frac{\sqrt{21}{\left (14148750 \, \sqrt{55} \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (1129191435 \, x^{4} + 3049001415 \, x^{3} + 3088510878 \, x^{2} + 1391064622 \, x + 235067382\right )} \sqrt{-2 \, x + 1} + 480849385 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{216090 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^6),x, algorithm="fricas")

[Out]

1/216090*sqrt(21)*(14148750*sqrt(55)*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 72
0*x^2 + 240*x + 32)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21
)*(1129191435*x^4 + 3049001415*x^3 + 3088510878*x^2 + 1391064622*x + 235067382)*
sqrt(-2*x + 1) + 480849385*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(243*x^5 + 810*x^4 + 1
080*x^3 + 720*x^2 + 240*x + 32)

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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)**6/(3+5*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217569, size = 209, normalized size = 1.37 \[ 1375 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{96169877}{43218} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1129191435 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 10614768570 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 37423200612 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 58647378230 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 34470832865 \, \sqrt{-2 \, x + 1}}{164640 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(3/2)/((5*x + 3)*(3*x + 2)^6),x, algorithm="giac")

[Out]

1375*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) - 96169877/43218*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) + 1/164640*(1129191435*(2*x - 1)^4*sqrt(-2*x + 1) +
10614768570*(2*x - 1)^3*sqrt(-2*x + 1) + 37423200612*(2*x - 1)^2*sqrt(-2*x + 1)
- 58647378230*(-2*x + 1)^(3/2) + 34470832865*sqrt(-2*x + 1))/(3*x + 2)^5