3.1901 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx\)

Optimal. Leaf size=181 \[ -\frac{8110915 \sqrt{1-2 x}}{1176 (5 x+3)}+\frac{302668 \sqrt{1-2 x}}{441 (3 x+2) (5 x+3)}+\frac{23173 \sqrt{1-2 x}}{504 (3 x+2)^2 (5 x+3)}+\frac{83 \sqrt{1-2 x}}{18 (3 x+2)^3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-8110915*Sqrt[1 - 2*x])/(1176*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*(3
+ 5*x)) + (83*Sqrt[1 - 2*x])/(18*(2 + 3*x)^3*(3 + 5*x)) + (23173*Sqrt[1 - 2*x])
/(504*(2 + 3*x)^2*(3 + 5*x)) + (302668*Sqrt[1 - 2*x])/(441*(2 + 3*x)*(3 + 5*x))
- (55953383*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(196*Sqrt[21]) + 8400*Sqrt[55]*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.37468, antiderivative size = 181, 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 \[ -\frac{8110915 \sqrt{1-2 x}}{1176 (5 x+3)}+\frac{302668 \sqrt{1-2 x}}{441 (3 x+2) (5 x+3)}+\frac{23173 \sqrt{1-2 x}}{504 (3 x+2)^2 (5 x+3)}+\frac{83 \sqrt{1-2 x}}{18 (3 x+2)^3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-8110915*Sqrt[1 - 2*x])/(1176*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*(3
+ 5*x)) + (83*Sqrt[1 - 2*x])/(18*(2 + 3*x)^3*(3 + 5*x)) + (23173*Sqrt[1 - 2*x])
/(504*(2 + 3*x)^2*(3 + 5*x)) + (302668*Sqrt[1 - 2*x])/(441*(2 + 3*x)*(3 + 5*x))
- (55953383*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(196*Sqrt[21]) + 8400*Sqrt[55]*Arc
Tanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 42.4618, size = 153, normalized size = 0.85 \[ - \frac{1622183 \sqrt{- 2 x + 1}}{392 \left (3 x + 2\right )} - \frac{69863 \sqrt{- 2 x + 1}}{168 \left (3 x + 2\right )^{2}} - \frac{3335 \sqrt{- 2 x + 1}}{36 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} + \frac{83 \sqrt{- 2 x + 1}}{18 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} + \frac{7 \sqrt{- 2 x + 1}}{12 \left (3 x + 2\right )^{4} \left (5 x + 3\right )} - \frac{55953383 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{4116} + 8400 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1622183*sqrt(-2*x + 1)/(392*(3*x + 2)) - 69863*sqrt(-2*x + 1)/(168*(3*x + 2)**2
) - 3335*sqrt(-2*x + 1)/(36*(3*x + 2)**2*(5*x + 3)) + 83*sqrt(-2*x + 1)/(18*(3*x
+ 2)**3*(5*x + 3)) + 7*sqrt(-2*x + 1)/(12*(3*x + 2)**4*(5*x + 3)) - 55953383*sq
rt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/4116 + 8400*sqrt(55)*atanh(sqrt(55)*sqrt
(-2*x + 1)/11)

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Mathematica [A]  time = 0.176049, size = 100, normalized size = 0.55 \[ -\frac{\sqrt{1-2 x} \left (218994705 x^4+576721848 x^3+569295605 x^2+249642200 x+41029970\right )}{392 (3 x+2)^4 (5 x+3)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(41029970 + 249642200*x + 569295605*x^2 + 576721848*x^3 + 218994
705*x^4))/(392*(2 + 3*x)^4*(3 + 5*x)) - (55953383*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x
]])/(196*Sqrt[21]) + 8400*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.021, size = 100, normalized size = 0.6 \[ 162\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{1298783\, \left ( 1-2\,x \right ) ^{7/2}}{1176}}-{\frac{11773333\, \left ( 1-2\,x \right ) ^{5/2}}{1512}}+{\frac{11859787\, \left ( 1-2\,x \right ) ^{3/2}}{648}}-{\frac{344197\,\sqrt{1-2\,x}}{24}} \right ) }-{\frac{55953383\,\sqrt{21}}{4116}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+550\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+8400\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

162*(1298783/1176*(1-2*x)^(7/2)-11773333/1512*(1-2*x)^(5/2)[phone]/648*(1-2*x)
^(3/2)-344197/24*(1-2*x)^(1/2))/(-4-6*x)^4-55953383/4116*arctanh(1/7*21^(1/2)*(1
-2*x)^(1/2))*21^(1/2)+550*(1-2*x)^(1/2)/(-6/5-2*x)+8400*arctanh(1/11*55^(1/2)*(1
-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51201, size = 221, normalized size = 1.22 \[ -4200 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{55953383}{8232} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{218994705 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2029422516 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 7051481738 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10887812348 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6303237941 \, \sqrt{-2 \, x + 1}}{196 \,{\left (405 \,{\left (2 \, x - 1\right )}^{5} + 4671 \,{\left (2 \, x - 1\right )}^{4} + 21546 \,{\left (2 \, x - 1\right )}^{3} + 49686 \,{\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4200*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)))
+ 55953383/8232*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 1/196*(218994705*(-2*x + 1)^(9/2) - 2029422516*(-2*x + 1)^(7/2) +
7051481738*(-2*x + 1)^(5/2) - 10887812348*(-2*x + 1)^(3/2) + 6303237941*sqrt(-2*
x + 1))/(405*(2*x - 1)^5 + 4671*(2*x - 1)^4 + 21546*(2*x - 1)^3 + 49686*(2*x - 1
)^2 + 114562*x - 30870)

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Fricas [A]  time = 0.221664, size = 242, normalized size = 1.34 \[ \frac{\sqrt{21}{\left (1646400 \, \sqrt{55} \sqrt{21}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - \sqrt{21}{\left (218994705 \, x^{4} + 576721848 \, x^{3} + 569295605 \, x^{2} + 249642200 \, x + 41029970\right )} \sqrt{-2 \, x + 1} + 55953383 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{8232 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8232*sqrt(21)*(1646400*sqrt(55)*sqrt(21)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128
*x^2 + 368*x + 48)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - sqrt(21)
*(218994705*x^4 + 576721848*x^3 + 569295605*x^2 + 249642200*x + 41029970)*sqrt(-
2*x + 1) + 55953383*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(
(sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(405*x^5 + 1323*x^4 + 1728*
x^3 + 1128*x^2 + 368*x + 48)

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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)**5/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216211, size = 209, normalized size = 1.15 \[ -4200 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{55953383}{8232} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1375 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{35067141 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 247239993 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 581129563 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 455372631 \, \sqrt{-2 \, x + 1}}{3136 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4200*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2
*x + 1))) + 55953383/8232*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) - 1375*sqrt(-2*x + 1)/(5*x + 3) - 1/3136*(35067141*
(2*x - 1)^3*sqrt(-2*x + 1) + 247239993*(2*x - 1)^2*sqrt(-2*x + 1) - 581129563*(-
2*x + 1)^(3/2) + 455372631*sqrt(-2*x + 1))/(3*x + 2)^4