3.2163 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^4}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{10283 (3 x+2)^3}{6655 \sqrt{1-2 x}}-\frac{463344 \sqrt{1-2 x} (3 x+2)^2}{166375}-\frac{21 \sqrt{1-2 x} (1544625 x+4633904)}{831875}-\frac{406 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{831875 \sqrt{55}} \]

[Out]

(-463344*Sqrt[1 - 2*x]*(2 + 3*x)^2)/166375 - (10283*(2 + 3*x)^3)/(6655*Sqrt[1 -
2*x]) - (38*(2 + 3*x)^4)/(1815*Sqrt[1 - 2*x]*(3 + 5*x)) + (7*(2 + 3*x)^5)/(33*(1
- 2*x)^(3/2)*(3 + 5*x)) - (21*Sqrt[1 - 2*x]*(4633904 + 1544625*x))/831875 - (40
6*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(831875*Sqrt[55])

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Rubi [A]  time = 0.298803, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{7 (3 x+2)^5}{33 (1-2 x)^{3/2} (5 x+3)}-\frac{38 (3 x+2)^4}{1815 \sqrt{1-2 x} (5 x+3)}-\frac{10283 (3 x+2)^3}{6655 \sqrt{1-2 x}}-\frac{463344 \sqrt{1-2 x} (3 x+2)^2}{166375}-\frac{21 \sqrt{1-2 x} (1544625 x+4633904)}{831875}-\frac{406 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{831875 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-463344*Sqrt[1 - 2*x]*(2 + 3*x)^2)/166375 - (10283*(2 + 3*x)^3)/(6655*Sqrt[1 -
2*x]) - (38*(2 + 3*x)^4)/(1815*Sqrt[1 - 2*x]*(3 + 5*x)) + (7*(2 + 3*x)^5)/(33*(1
- 2*x)^(3/2)*(3 + 5*x)) - (21*Sqrt[1 - 2*x]*(4633904 + 1544625*x))/831875 - (40
6*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(831875*Sqrt[55])

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Rubi in Sympy [A]  time = 32.3527, size = 124, normalized size = 0.89 \[ - \frac{463344 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{166375} - \frac{\sqrt{- 2 x + 1} \left (1459670625 x + 4379039280\right )}{37434375} - \frac{406 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{45753125} - \frac{38 \left (3 x + 2\right )^{4}}{1815 \sqrt{- 2 x + 1} \left (5 x + 3\right )} - \frac{10283 \left (3 x + 2\right )^{3}}{6655 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{5}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-463344*sqrt(-2*x + 1)*(3*x + 2)**2/166375 - sqrt(-2*x + 1)*(1459670625*x + 4379
039280)/37434375 - 406*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/45753125 - 38*
(3*x + 2)**4/(1815*sqrt(-2*x + 1)*(5*x + 3)) - 10283*(3*x + 2)**3/(6655*sqrt(-2*
x + 1)) + 7*(3*x + 2)**5/(33*(-2*x + 1)**(3/2)*(5*x + 3))

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Mathematica [A]  time = 0.203423, size = 73, normalized size = 0.52 \[ \frac{-\frac{55 \left (72772425 x^5+480298005 x^4+2644064775 x^3-3837745731 x^2-1434109759 x+1035652776\right )}{(1-2 x)^{3/2} (5 x+3)}-1218 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{137259375} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(1035652776 - 1434109759*x - 3837745731*x^2 + 2644064775*x^3 + 480298005*x
^4 + 72772425*x^5))/((1 - 2*x)^(3/2)*(3 + 5*x)) - 1218*Sqrt[55]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/137259375

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Maple [A]  time = 0.021, size = 81, normalized size = 0.6 \[ -{\frac{729}{2000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{729}{125} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{315171}{5000}\sqrt{1-2\,x}}+{\frac{117649}{5808} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{134456}{1331}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{4159375}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{406\,\sqrt{55}}{45753125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-729/2000*(1-2*x)^(5/2)+729/125*(1-2*x)^(3/2)-315171/5000*(1-2*x)^(1/2)+117649/5
808/(1-2*x)^(3/2)-134456/1331/(1-2*x)^(1/2)+2/4159375*(1-2*x)^(1/2)/(-6/5-2*x)-4
06/45753125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49008, size = 136, normalized size = 0.97 \[ -\frac{729}{2000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{729}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{203}{45753125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{315171}{5000} \, \sqrt{-2 \, x + 1} - \frac{10084199952 \,{\left (2 \, x - 1\right )}^{2} + 48414664375 \, x - 19758729375}{19965000 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-729/2000*(-2*x + 1)^(5/2) + 729/125*(-2*x + 1)^(3/2) + 203/45753125*sqrt(55)*lo
g(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 315171/5000*sq
rt(-2*x + 1) - 1/19965000*(10084199952*(2*x - 1)^2 + 48414664375*x - 19758729375
)/(5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.221909, size = 131, normalized size = 0.94 \[ \frac{\sqrt{55}{\left (609 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{55}{\left (72772425 \, x^{5} + 480298005 \, x^{4} + 2644064775 \, x^{3} - 3837745731 \, x^{2} - 1434109759 \, x + 1035652776\right )}\right )}}{137259375 \,{\left (10 \, x^{2} + x - 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/137259375*sqrt(55)*(609*(10*x^2 + x - 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8
) + 55*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(55)*(72772425*x^5 + 480298005*x^4 + 264
4064775*x^3 - 3837745731*x^2 - 1434109759*x + 1035652776))/((10*x^2 + x - 3)*sqr
t(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.226249, size = 150, normalized size = 1.07 \[ -\frac{729}{2000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{729}{125} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{203}{45753125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{315171}{5000} \, \sqrt{-2 \, x + 1} - \frac{16807 \,{\left (768 \, x - 307\right )}}{63888 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{\sqrt{-2 \, x + 1}}{831875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-729/2000*(2*x - 1)^2*sqrt(-2*x + 1) + 729/125*(-2*x + 1)^(3/2) + 203/45753125*s
qrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1)
)) - 315171/5000*sqrt(-2*x + 1) - 16807/63888*(768*x - 307)/((2*x - 1)*sqrt(-2*x
+ 1)) - 1/831875*sqrt(-2*x + 1)/(5*x + 3)