3.2098 \(\int \frac{(2+3 x)^6}{(1-2 x)^{3/2} (3+5 x)^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^4}{605 (5 x+3)}+\frac{14517 \sqrt{1-2 x} (3 x+2)^3}{21175}+\frac{217152 \sqrt{1-2 x} (3 x+2)^2}{75625}+\frac{9 \sqrt{1-2 x} (1688625 x+5065808)}{378125}-\frac{402 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

[Out]

(217152*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 + (14517*Sqrt[1 - 2*x]*(2 + 3*x)^3)/211
75 - (36*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(605*(3 + 5*x)) + (7*(2 + 3*x)^5)/(11*Sqrt[1
- 2*x]*(3 + 5*x)) + (9*Sqrt[1 - 2*x]*(5065808 + 1688625*x))/378125 - (402*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(378125*Sqrt[55])

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Rubi [A]  time = 0.282254, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (3 x+2)^5}{11 \sqrt{1-2 x} (5 x+3)}-\frac{36 \sqrt{1-2 x} (3 x+2)^4}{605 (5 x+3)}+\frac{14517 \sqrt{1-2 x} (3 x+2)^3}{21175}+\frac{217152 \sqrt{1-2 x} (3 x+2)^2}{75625}+\frac{9 \sqrt{1-2 x} (1688625 x+5065808)}{378125}-\frac{402 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{378125 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(217152*Sqrt[1 - 2*x]*(2 + 3*x)^2)/75625 + (14517*Sqrt[1 - 2*x]*(2 + 3*x)^3)/211
75 - (36*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(605*(3 + 5*x)) + (7*(2 + 3*x)^5)/(11*Sqrt[1
- 2*x]*(3 + 5*x)) + (9*Sqrt[1 - 2*x]*(5065808 + 1688625*x))/378125 - (402*ArcTa
nh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(378125*Sqrt[55])

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Rubi in Sympy [A]  time = 32.8866, size = 122, normalized size = 0.87 \[ - \frac{36 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{605 \left (5 x + 3\right )} + \frac{14517 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{21175} + \frac{217152 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{75625} + \frac{\sqrt{- 2 x + 1} \left (1595750625 x + 4787188560\right )}{39703125} - \frac{402 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{20796875} + \frac{7 \left (3 x + 2\right )^{5}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36*sqrt(-2*x + 1)*(3*x + 2)**4/(605*(5*x + 3)) + 14517*sqrt(-2*x + 1)*(3*x + 2)
**3/21175 + 217152*sqrt(-2*x + 1)*(3*x + 2)**2/75625 + sqrt(-2*x + 1)*(159575062
5*x + 4787188560)/39703125 - 402*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/2079
6875 + 7*(3*x + 2)**5/(11*sqrt(-2*x + 1)*(5*x + 3))

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Mathematica [A]  time = 0.156136, size = 76, normalized size = 0.54 \[ \frac{\frac{55 \sqrt{1-2 x} \left (55130625 x^5+293294925 x^4+795400155 x^3+2195407665 x^2-818846961 x-1143572552\right )}{10 x^2+x-3}-2814 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{145578125} \]

Antiderivative was successfully verified.

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

[Out]

((55*Sqrt[1 - 2*x]*(-1143572552 - 818846961*x + 2195407665*x^2 + 795400155*x^3 +
293294925*x^4 + 55130625*x^5))/(-3 + x + 10*x^2) - 2814*Sqrt[55]*ArcTanh[Sqrt[5
/11]*Sqrt[1 - 2*x]])/145578125

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Maple [A]  time = 0.021, size = 81, normalized size = 0.6 \[ -{\frac{729}{2800} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{2187}{625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{105057}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{315684}{3125}\sqrt{1-2\,x}}+{\frac{117649}{1936}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{2}{1890625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{402\,\sqrt{55}}{20796875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-729/2800*(1-2*x)^(7/2)+2187/625*(1-2*x)^(5/2)-105057/5000*(1-2*x)^(3/2)+315684/
3125*(1-2*x)^(1/2)+117649/1936/(1-2*x)^(1/2)+2/1890625*(1-2*x)^(1/2)/(-6/5-2*x)-
402/20796875*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49929, size = 136, normalized size = 0.97 \[ -\frac{729}{2800} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2187}{625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{105057}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{201}{20796875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{315684}{3125} \, \sqrt{-2 \, x + 1} - \frac{1838265657 \, x + 1102959359}{3025000 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-729/2800*(-2*x + 1)^(7/2) + 2187/625*(-2*x + 1)^(5/2) - 105057/5000*(-2*x + 1)^
(3/2) + 201/20796875*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) + 315684/3125*sqrt(-2*x + 1) - 1/3025000*(1838265657*x + 1102959
359)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))

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Fricas [A]  time = 0.27582, size = 124, normalized size = 0.89 \[ \frac{\sqrt{55}{\left (1407 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (55130625 \, x^{5} + 293294925 \, x^{4} + 795400155 \, x^{3} + 2195407665 \, x^{2} - 818846961 \, x - 1143572552\right )}\right )}}{145578125 \,{\left (5 \, x + 3\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/145578125*sqrt(55)*(1407*(5*x + 3)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x - 8) + 55
*sqrt(-2*x + 1))/(5*x + 3)) - sqrt(55)*(55130625*x^5 + 293294925*x^4 + 795400155
*x^3 + 2195407665*x^2 - 818846961*x - 1143572552))/((5*x + 3)*sqrt(-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)**(3/2)/(3+5*x)**2,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.222525, size = 159, normalized size = 1.14 \[ \frac{729}{2800} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2187}{625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{105057}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{201}{20796875} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{315684}{3125} \, \sqrt{-2 \, x + 1} - \frac{1838265657 \, x + 1102959359}{3025000 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

729/2800*(2*x - 1)^3*sqrt(-2*x + 1) + 2187/625*(2*x - 1)^2*sqrt(-2*x + 1) - 1050
57/5000*(-2*x + 1)^(3/2) + 201/20796875*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqr
t(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 315684/3125*sqrt(-2*x + 1) - 1/302
5000*(1838265657*x + 1102959359)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1))