3.1931 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac{(1-2 x)^{7/2}}{63 (3 x+2)}-\frac{25}{63} (1-2 x)^{7/2}-\frac{10}{63} (1-2 x)^{5/2}-\frac{50}{81} (1-2 x)^{3/2}-\frac{350}{81} \sqrt{1-2 x}+\frac{350}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-350*Sqrt[1 - 2*x])/81 - (50*(1 - 2*x)^(3/2))/81 - (10*(1 - 2*x)^(5/2))/63 - (2
5*(1 - 2*x)^(7/2))/63 - (1 - 2*x)^(7/2)/(63*(2 + 3*x)) + (350*Sqrt[7/3]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi [A]  time = 0.129634, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(1-2 x)^{7/2}}{63 (3 x+2)}-\frac{25}{63} (1-2 x)^{7/2}-\frac{10}{63} (1-2 x)^{5/2}-\frac{50}{81} (1-2 x)^{3/2}-\frac{350}{81} \sqrt{1-2 x}+\frac{350}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-350*Sqrt[1 - 2*x])/81 - (50*(1 - 2*x)^(3/2))/81 - (10*(1 - 2*x)^(5/2))/63 - (2
5*(1 - 2*x)^(7/2))/63 - (1 - 2*x)^(7/2)/(63*(2 + 3*x)) + (350*Sqrt[7/3]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi in Sympy [A]  time = 12.3821, size = 85, normalized size = 0.83 \[ - \frac{25 \left (- 2 x + 1\right )^{\frac{7}{2}}}{63} - \frac{\left (- 2 x + 1\right )^{\frac{7}{2}}}{63 \left (3 x + 2\right )} - \frac{10 \left (- 2 x + 1\right )^{\frac{5}{2}}}{63} - \frac{50 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{350 \sqrt{- 2 x + 1}}{81} + \frac{350 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

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)**2,x)

[Out]

-25*(-2*x + 1)**(7/2)/63 - (-2*x + 1)**(7/2)/(63*(3*x + 2)) - 10*(-2*x + 1)**(5/
2)/63 - 50*(-2*x + 1)**(3/2)/81 - 350*sqrt(-2*x + 1)/81 + 350*sqrt(21)*atanh(sqr
t(21)*sqrt(-2*x + 1)/7)/243

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Mathematica [A]  time = 0.113314, size = 70, normalized size = 0.69 \[ \frac{\sqrt{1-2 x} \left (5400 x^4-5508 x^3+1002 x^2-4471 x-6239\right )}{567 (3 x+2)}+\frac{350}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(-6239 - 4471*x + 1002*x^2 - 5508*x^3 + 5400*x^4))/(567*(2 + 3*x)
) + (350*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Maple [A]  time = 0.016, size = 72, normalized size = 0.7 \[ -{\frac{25}{63} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{4}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{16}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1036}{243}\sqrt{1-2\,x}}+{\frac{98}{729}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}+{\frac{350\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/63*(1-2*x)^(7/2)-4/27*(1-2*x)^(5/2)-16/27*(1-2*x)^(3/2)-1036/243*(1-2*x)^(1/
2)+98/729*(1-2*x)^(1/2)/(-4/3-2*x)+350/243*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)

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Maxima [A]  time = 1.58291, size = 120, normalized size = 1.18 \[ -\frac{25}{63} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{4}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{16}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{175}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1036}{243} \, \sqrt{-2 \, x + 1} - \frac{49 \, \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/63*(-2*x + 1)^(7/2) - 4/27*(-2*x + 1)^(5/2) - 16/27*(-2*x + 1)^(3/2) - 175/2
43*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) -
1036/243*sqrt(-2*x + 1) - 49/243*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.22135, size = 115, normalized size = 1.13 \[ \frac{\sqrt{3}{\left (1225 \, \sqrt{7}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{3}{\left (5400 \, x^{4} - 5508 \, x^{3} + 1002 \, x^{2} - 4471 \, x - 6239\right )} \sqrt{-2 \, x + 1}\right )}}{1701 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1701*sqrt(3)*(1225*sqrt(7)*(3*x + 2)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-
2*x + 1))/(3*x + 2)) + sqrt(3)*(5400*x^4 - 5508*x^3 + 1002*x^2 - 4471*x - 6239)*
sqrt(-2*x + 1))/(3*x + 2)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212413, size = 143, normalized size = 1.4 \[ \frac{25}{63} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{4}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{16}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{175}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1036}{243} \, \sqrt{-2 \, x + 1} - \frac{49 \, \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/63*(2*x - 1)^3*sqrt(-2*x + 1) - 4/27*(2*x - 1)^2*sqrt(-2*x + 1) - 16/27*(-2*x
+ 1)^(3/2) - 175/243*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) - 1036/243*sqrt(-2*x + 1) - 49/243*sqrt(-2*x + 1)/(3*x
+ 2)