3.2087 \(\int \frac{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=106 \[ \frac{81}{160} (1-2 x)^{9/2}-\frac{43011 (1-2 x)^{7/2}}{5600}+\frac{507627 (1-2 x)^{5/2}}{10000}-\frac{1997451 (1-2 x)^{3/2}}{10000}+\frac{70752609 \sqrt{1-2 x}}{100000}+\frac{117649}{352 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

[Out]

117649/(352*Sqrt[1 - 2*x]) + (70752609*Sqrt[1 - 2*x])/100000 - (1997451*(1 - 2*x
)^(3/2))/10000 + (507627*(1 - 2*x)^(5/2))/10000 - (43011*(1 - 2*x)^(7/2))/5600 +
(81*(1 - 2*x)^(9/2))/160 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(34375*Sqrt[55
])

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Rubi [A]  time = 0.218816, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{81}{160} (1-2 x)^{9/2}-\frac{43011 (1-2 x)^{7/2}}{5600}+\frac{507627 (1-2 x)^{5/2}}{10000}-\frac{1997451 (1-2 x)^{3/2}}{10000}+\frac{70752609 \sqrt{1-2 x}}{100000}+\frac{117649}{352 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

117649/(352*Sqrt[1 - 2*x]) + (70752609*Sqrt[1 - 2*x])/100000 - (1997451*(1 - 2*x
)^(3/2))/10000 + (507627*(1 - 2*x)^(5/2))/10000 - (43011*(1 - 2*x)^(7/2))/5600 +
(81*(1 - 2*x)^(9/2))/160 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(34375*Sqrt[55
])

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Rubi in Sympy [A]  time = 18.2067, size = 95, normalized size = 0.9 \[ \frac{81 \left (- 2 x + 1\right )^{\frac{9}{2}}}{160} - \frac{43011 \left (- 2 x + 1\right )^{\frac{7}{2}}}{5600} + \frac{507627 \left (- 2 x + 1\right )^{\frac{5}{2}}}{10000} - \frac{1997451 \left (- 2 x + 1\right )^{\frac{3}{2}}}{10000} + \frac{70752609 \sqrt{- 2 x + 1}}{100000} - \frac{2 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1890625} + \frac{117649}{352 \sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81*(-2*x + 1)**(9/2)/160 - 43011*(-2*x + 1)**(7/2)/5600 + 507627*(-2*x + 1)**(5/
2)/10000 - 1997451*(-2*x + 1)**(3/2)/10000 + 70752609*sqrt(-2*x + 1)/100000 - 2*
sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1890625 + 117649/(352*sqrt(-2*x + 1))

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Mathematica [A]  time = 0.179148, size = 66, normalized size = 0.62 \[ \frac{-\frac{55 \left (3898125 x^5+19824750 x^4+48323385 x^3+85159800 x^2+207964053 x-213097384\right )}{\sqrt{1-2 x}}-14 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{13234375} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(-213097384 + 207964053*x + 85159800*x^2 + 48323385*x^3 + 19824750*x^4 + 3
898125*x^5))/Sqrt[1 - 2*x] - 14*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1323
4375

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Maple [A]  time = 0.013, size = 74, normalized size = 0.7 \[ -{\frac{1997451}{10000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{507627}{10000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{43011}{5600} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{81}{160} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2\,\sqrt{55}}{1890625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{117649}{352}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{70752609}{100000}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1997451/10000*(1-2*x)^(3/2)+507627/10000*(1-2*x)^(5/2)-43011/5600*(1-2*x)^(7/2)
+81/160*(1-2*x)^(9/2)-2/1890625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+11
7649/352/(1-2*x)^(1/2)[phone]/100000*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49526, size = 123, normalized size = 1.16 \[ \frac{81}{160} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{43011}{5600} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{507627}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{1997451}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{1890625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{70752609}{100000} \, \sqrt{-2 \, x + 1} + \frac{117649}{352 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/160*(-2*x + 1)^(9/2) - 43011/5600*(-2*x + 1)^(7/2) + 507627/10000*(-2*x + 1)^
(5/2) - 1997451/10000*(-2*x + 1)^(3/2) + 1/1890625*sqrt(55)*log(-(sqrt(55) - 5*s
qrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70752609/100000*sqrt(-2*x + 1) +
117649/352/sqrt(-2*x + 1)

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Fricas [A]  time = 0.237373, size = 107, normalized size = 1.01 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (3898125 \, x^{5} + 19824750 \, x^{4} + 48323385 \, x^{3} + 85159800 \, x^{2} + 207964053 \, x - 213097384\right )} - 7 \, \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{13234375 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13234375*sqrt(55)*(sqrt(55)*(3898125*x^5 + 19824750*x^4 + 48323385*x^3 + 8515
9800*x^2 + 207964053*x - 213097384) - 7*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) +
55*sqrt(-2*x + 1))/(5*x + 3)))/sqrt(-2*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{6}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((3*x + 2)**6/((-2*x + 1)**(3/2)*(5*x + 3)), x)

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GIAC/XCAS [A]  time = 0.218476, size = 155, normalized size = 1.46 \[ \frac{81}{160} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{43011}{5600} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{507627}{10000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{1997451}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{1890625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{70752609}{100000} \, \sqrt{-2 \, x + 1} + \frac{117649}{352 \, \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/160*(2*x - 1)^4*sqrt(-2*x + 1) + 43011/5600*(2*x - 1)^3*sqrt(-2*x + 1) + 5076
27/10000*(2*x - 1)^2*sqrt(-2*x + 1) - 1997451/10000*(-2*x + 1)^(3/2) + 1/1890625
*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) + 70752609/100000*sqrt(-2*x + 1) + 117649/352/sqrt(-2*x + 1)