3.1933 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=114 \[ \frac{211 (1-2 x)^{7/2}}{2646 (3 x+2)^2}-\frac{(1-2 x)^{7/2}}{189 (3 x+2)^3}-\frac{887 (1-2 x)^{5/2}}{882 (3 x+2)}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{4435}{567} \sqrt{1-2 x}+\frac{4435 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

[Out]

(-4435*Sqrt[1 - 2*x])/567 - (4435*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(7/2)/(189*(
2 + 3*x)^3) + (211*(1 - 2*x)^(7/2))/(2646*(2 + 3*x)^2) - (887*(1 - 2*x)^(5/2))/(
882*(2 + 3*x)) + (4435*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Rubi [A]  time = 0.137359, antiderivative size = 114, 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 \[ \frac{211 (1-2 x)^{7/2}}{2646 (3 x+2)^2}-\frac{(1-2 x)^{7/2}}{189 (3 x+2)^3}-\frac{887 (1-2 x)^{5/2}}{882 (3 x+2)}-\frac{4435 (1-2 x)^{3/2}}{3969}-\frac{4435}{567} \sqrt{1-2 x}+\frac{4435 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(-4435*Sqrt[1 - 2*x])/567 - (4435*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(7/2)/(189*(
2 + 3*x)^3) + (211*(1 - 2*x)^(7/2))/(2646*(2 + 3*x)^2) - (887*(1 - 2*x)^(5/2))/(
882*(2 + 3*x)) + (4435*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Rubi in Sympy [A]  time = 13.3107, size = 99, normalized size = 0.87 \[ \frac{211 \left (- 2 x + 1\right )^{\frac{7}{2}}}{2646 \left (3 x + 2\right )^{2}} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{189 \left (3 x + 2\right )^{3}} - \frac{887 \left (- 2 x + 1\right )^{\frac{5}{2}}}{882 \left (3 x + 2\right )} - \frac{4435 \left (- 2 x + 1\right )^{\frac{3}{2}}}{3969} - \frac{4435 \sqrt{- 2 x + 1}}{567} + \frac{4435 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1701} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**2/(2+3*x)**4,x)

[Out]

211*(-2*x + 1)**(7/2)/(2646*(3*x + 2)**2) - (-2*x + 1)**(7/2)/(189*(3*x + 2)**3)
- 887*(-2*x + 1)**(5/2)/(882*(3*x + 2)) - 4435*(-2*x + 1)**(3/2)/3969 - 4435*sq
rt(-2*x + 1)/567 + 4435*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/1701

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Mathematica [A]  time = 0.12474, size = 68, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (3600 x^4-21240 x^3-61353 x^2-48697 x-12212\right )}{162 (3 x+2)^3}+\frac{4435 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(Sqrt[1 - 2*x]*(-12212 - 48697*x - 61353*x^2 - 21240*x^3 + 3600*x^4))/(162*(2 +
3*x)^3) + (4435*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Maple [A]  time = 0.016, size = 75, normalized size = 0.7 \[ -{\frac{100}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1480}{243}\sqrt{1-2\,x}}-{\frac{4}{9\, \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3091}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{31675}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{144305}{108}\sqrt{1-2\,x}} \right ) }+{\frac{4435\,\sqrt{21}}{1701}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^2/(2+3*x)^4,x)

[Out]

-100/243*(1-2*x)^(3/2)-1480/243*(1-2*x)^(1/2)-4/9*(-3091/12*(1-2*x)^(5/2)+31675/
27*(1-2*x)^(3/2)-144305/108*(1-2*x)^(1/2))/(-4-6*x)^3+4435/1701*arctanh(1/7*21^(
1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51505, size = 149, normalized size = 1.31 \[ -\frac{100}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{4435}{3402} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1480}{243} \, \sqrt{-2 \, x + 1} - \frac{27819 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 126700 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 144305 \, \sqrt{-2 \, x + 1}}{243 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^4,x, algorithm="maxima")

[Out]

-100/243*(-2*x + 1)^(3/2) - 4435/3402*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1)
)/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1480/243*sqrt(-2*x + 1) - 1/243*(27819*(-2*x
+ 1)^(5/2) - 126700*(-2*x + 1)^(3/2) + 144305*sqrt(-2*x + 1))/(27*(2*x - 1)^3 +
189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.212345, size = 134, normalized size = 1.18 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (3600 \, x^{4} - 21240 \, x^{3} - 61353 \, x^{2} - 48697 \, x - 12212\right )} \sqrt{-2 \, x + 1} + 4435 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{3402 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^4,x, algorithm="fricas")

[Out]

1/3402*sqrt(21)*(sqrt(21)*(3600*x^4 - 21240*x^3 - 61353*x^2 - 48697*x - 12212)*s
qrt(-2*x + 1) + 4435*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) - 21*s
qrt(-2*x + 1))/(3*x + 2)))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**(5/2)*(3+5*x)**2/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213956, size = 138, normalized size = 1.21 \[ -\frac{100}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{4435}{3402} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1480}{243} \, \sqrt{-2 \, x + 1} - \frac{27819 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 126700 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 144305 \, \sqrt{-2 \, x + 1}}{1944 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(5/2)/(3*x + 2)^4,x, algorithm="giac")

[Out]

-100/243*(-2*x + 1)^(3/2) - 4435/3402*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-
2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1480/243*sqrt(-2*x + 1) - 1/1944*(278
19*(2*x - 1)^2*sqrt(-2*x + 1) - 126700*(-2*x + 1)^(3/2) + 144305*sqrt(-2*x + 1))
/(3*x + 2)^3