3.1909 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=147 \[ \frac{1020 \sqrt{1-2 x}}{5 x+3}-\frac{1015 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{45 \sqrt{1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-13665 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-1015*Sqrt[1 - 2*x])/(6*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*
x)^2) + (45*Sqrt[1 - 2*x])/(2*(2 + 3*x)*(3 + 5*x)^2) + (1020*Sqrt[1 - 2*x])/(3 +
5*x) + 14073*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13665*Sqrt[5/11]*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.320903, antiderivative size = 147, 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 \[ \frac{1020 \sqrt{1-2 x}}{5 x+3}-\frac{1015 \sqrt{1-2 x}}{6 (5 x+3)^2}+\frac{45 \sqrt{1-2 x}}{2 (3 x+2) (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-13665 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1015*Sqrt[1 - 2*x])/(6*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*
x)^2) + (45*Sqrt[1 - 2*x])/(2*(2 + 3*x)*(3 + 5*x)^2) + (1020*Sqrt[1 - 2*x])/(3 +
5*x) + 14073*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13665*Sqrt[5/11]*ArcT
anh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 34.5174, size = 133, normalized size = 0.9 \[ \frac{1020 \sqrt{- 2 x + 1}}{5 x + 3} - \frac{1015 \sqrt{- 2 x + 1}}{6 \left (5 x + 3\right )^{2}} + \frac{45 \sqrt{- 2 x + 1}}{2 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{7 \sqrt{- 2 x + 1}}{6 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{14073 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7} - \frac{13665 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1020*sqrt(-2*x + 1)/(5*x + 3) - 1015*sqrt(-2*x + 1)/(6*(5*x + 3)**2) + 45*sqrt(-
2*x + 1)/(2*(3*x + 2)*(5*x + 3)**2) + 7*sqrt(-2*x + 1)/(6*(3*x + 2)**2*(5*x + 3)
**2) + 14073*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/7 - 13665*sqrt(55)*atanh(
sqrt(55)*sqrt(-2*x + 1)/11)/11

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Mathematica [A]  time = 0.149535, size = 97, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (91800 x^3+174435 x^2+110315 x+23219\right )}{2 (3 x+2)^2 (5 x+3)^2}+14073 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-13665 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(23219 + 110315*x + 174435*x^2 + 91800*x^3))/(2*(2 + 3*x)^2*(3 +
5*x)^2) + 14073*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - 13665*Sqrt[5/11]*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.02, size = 94, normalized size = 0.6 \[ -324\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{205\, \left ( 1-2\,x \right ) ^{3/2}}{36}}-{\frac{161\,\sqrt{1-2\,x}}{12}} \right ) }+{\frac{14073\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{203\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{2211\,\sqrt{1-2\,x}}{100}} \right ) }-{\frac{13665\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-324*(205/36*(1-2*x)^(3/2)-161/12*(1-2*x)^(1/2))/(-4-6*x)^2+14073/7*arctanh(1/7*
21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+500*(-203/20*(1-2*x)^(3/2)+2211/100*(1-2*x)^(1/
2))/(-6-10*x)^2-13665/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50601, size = 197, normalized size = 1.34 \[ \frac{13665}{22} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{14073}{14} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (45900 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 312135 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 707200 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 533841 \, \sqrt{-2 \, x + 1}\right )}}{225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13665/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1
))) - 14073/14*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 2*(45900*(-2*x + 1)^(7/2) - 312135*(-2*x + 1)^(5/2) + 707200*(-2*x +
1)^(3/2) - 533841*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*(2
*x - 1)^2 + 20944*x - 4543)

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Fricas [A]  time = 0.226305, size = 240, normalized size = 1.63 \[ \frac{\sqrt{11} \sqrt{7}{\left (13665 \, \sqrt{7} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 14073 \, \sqrt{11} \sqrt{3}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (91800 \, x^{3} + 174435 \, x^{2} + 110315 \, x + 23219\right )} \sqrt{-2 \, x + 1}\right )}}{154 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/154*sqrt(11)*sqrt(7)*(13665*sqrt(7)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228
*x + 36)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 14073
*sqrt(11)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(7)*(3*x -
5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(91800*x^3 + 17443
5*x^2 + 110315*x + 23219)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x +
36)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239226, size = 200, normalized size = 1.36 \[ \frac{13665}{22} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{14073}{14} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (45900 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 312135 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 707200 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 533841 \, \sqrt{-2 \, x + 1}\right )}}{{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13665/22*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt
(-2*x + 1))) - 14073/14*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 2*(45900*(2*x - 1)^3*sqrt(-2*x + 1) + 312135*(2*x -
1)^2*sqrt(-2*x + 1) - 707200*(-2*x + 1)^(3/2) + 533841*sqrt(-2*x + 1))/(15*(2*x
- 1)^2 + 136*x + 9)^2