3.1961 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=153 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{584179 \sqrt{1-2 x}}{196 (3 x+2)}+\frac{25159 \sqrt{1-2 x}}{84 (3 x+2)^2}+\frac{1201 \sqrt{1-2 x}}{30 (3 x+2)^3}+\frac{63 \sqrt{1-2 x}}{10 (3 x+2)^4}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5) + (63*Sqrt[1 - 2*x])/(10*(2 + 3*x)^4) + (12
01*Sqrt[1 - 2*x])/(30*(2 + 3*x)^3) + (25159*Sqrt[1 - 2*x])/(84*(2 + 3*x)^2) + (5
84179*Sqrt[1 - 2*x])/(196*(2 + 3*x)) + (20149879*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/(98*Sqrt[21]) - 6050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.377968, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5}+\frac{584179 \sqrt{1-2 x}}{196 (3 x+2)}+\frac{25159 \sqrt{1-2 x}}{84 (3 x+2)^2}+\frac{1201 \sqrt{1-2 x}}{30 (3 x+2)^3}+\frac{63 \sqrt{1-2 x}}{10 (3 x+2)^4}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^5) + (63*Sqrt[1 - 2*x])/(10*(2 + 3*x)^4) + (12
01*Sqrt[1 - 2*x])/(30*(2 + 3*x)^3) + (25159*Sqrt[1 - 2*x])/(84*(2 + 3*x)^2) + (5
84179*Sqrt[1 - 2*x])/(196*(2 + 3*x)) + (20149879*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/(98*Sqrt[21]) - 6050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 41.4284, size = 138, normalized size = 0.9 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{5}} + \frac{584179 \sqrt{- 2 x + 1}}{196 \left (3 x + 2\right )} + \frac{25159 \sqrt{- 2 x + 1}}{84 \left (3 x + 2\right )^{2}} + \frac{1201 \sqrt{- 2 x + 1}}{30 \left (3 x + 2\right )^{3}} + \frac{63 \sqrt{- 2 x + 1}}{10 \left (3 x + 2\right )^{4}} + \frac{20149879 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2058} - 6050 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(15*(3*x + 2)**5) + 584179*sqrt(-2*x + 1)/(196*(3*x + 2)) +
25159*sqrt(-2*x + 1)/(84*(3*x + 2)**2) + 1201*sqrt(-2*x + 1)/(30*(3*x + 2)**3) +
63*sqrt(-2*x + 1)/(10*(3*x + 2)**4) + 20149879*sqrt(21)*atanh(sqrt(21)*sqrt(-2*
x + 1)/7)/2058 - 6050*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)

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Mathematica [A]  time = 0.192321, size = 93, normalized size = 0.61 \[ \frac{\sqrt{1-2 x} \left (709777485 x^4+1916515215 x^3+1941349752 x^2+874383298 x+147756688\right )}{2940 (3 x+2)^5}+\frac{20149879 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{98 \sqrt{21}}-6050 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(147756688 + 874383298*x + 1941349752*x^2 + 1916515215*x^3 + 7097
77485*x^4))/(2940*(2 + 3*x)^5) + (20149879*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(98
*Sqrt[21]) - 6050*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.017, size = 93, normalized size = 0.6 \[ -486\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ({\frac{584179\, \left ( 1-2\,x \right ) ^{9/2}}{588}}-{\frac{504319\, \left ( 1-2\,x \right ) ^{7/2}}{54}}+{\frac{13335122\, \left ( 1-2\,x \right ) ^{5/2}}{405}}-{\frac{75232787\, \left ( 1-2\,x \right ) ^{3/2}}{1458}}+{\frac{29479429\,\sqrt{1-2\,x}}{972}} \right ) }+{\frac{20149879\,\sqrt{21}}{2058}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-6050\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-486*(584179/588*(1-2*x)^(9/2)-504319/54*(1-2*x)^(7/2)[phone]/405*(1-2*x)^(5/2
)-75232787/1458*(1-2*x)^(3/2)[phone]/972*(1-2*x)^(1/2))/(-4-6*x)^5+20149879/20
58*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-6050*arctanh(1/11*55^(1/2)*(1-2*
x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.47627, size = 221, normalized size = 1.44 \[ 3025 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20149879}{4116} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{709777485 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 6672140370 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 23523155208 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 36864065630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 21667380315 \, \sqrt{-2 \, x + 1}}{1470 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3025*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
- 20149879/4116*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) + 1/1470*(709777485*(-2*x + 1)^(9/2) - 6672140370*(-2*x + 1)^(7/2) +
23523155208*(-2*x + 1)^(5/2) - 36864065630*(-2*x + 1)^(3/2) + 21667380315*sqrt(-
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 = 0.226928, size = 239, normalized size = 1.56 \[ \frac{\sqrt{21}{\left (8893500 \, \sqrt{55} \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + \sqrt{21}{\left (709777485 \, x^{4} + 1916515215 \, x^{3} + 1941349752 \, x^{2} + 874383298 \, x + 147756688\right )} \sqrt{-2 \, x + 1} + 302248185 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{61740 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/61740*sqrt(21)*(8893500*sqrt(55)*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*
x^2 + 240*x + 32)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + sqrt(21)*
(709777485*x^4 + 1916515215*x^3 + 1941349752*x^2 + 874383298*x + 147756688)*sqrt
(-2*x + 1) + 302248185*(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 + 1080*
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)**(5/2)/(2+3*x)**6/(3+5*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218011, size = 209, normalized size = 1.37 \[ 3025 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{20149879}{4116} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{709777485 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 6672140370 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 23523155208 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 36864065630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 21667380315 \, \sqrt{-2 \, x + 1}}{47040 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3025*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*
x + 1))) - 20149879/4116*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 1/47040*(709777485*(2*x - 1)^4*sqrt(-2*x + 1) + 66
72140370*(2*x - 1)^3*sqrt(-2*x + 1) + 23523155208*(2*x - 1)^2*sqrt(-2*x + 1) - 3
6864065630*(-2*x + 1)^(3/2) + 21667380315*sqrt(-2*x + 1))/(3*x + 2)^5