3.1871 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac{5}{7} (1-2 x)^{3/2} (5 x+3)^2-\frac{10}{63} (1-2 x)^{3/2} (27 x+22)+\frac{8}{9} \sqrt{1-2 x}-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(8*Sqrt[1 - 2*x])/9 + (5*(1 - 2*x)^(3/2)*(3 + 5*x)^2)/7 - ((1 - 2*x)^(3/2)*(3 +
5*x)^3)/(3*(2 + 3*x)) - (10*(1 - 2*x)^(3/2)*(22 + 27*x))/63 - (8*Sqrt[7/3]*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

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Rubi [A]  time = 0.155814, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-2 x)^{3/2} (5 x+3)^3}{3 (3 x+2)}+\frac{5}{7} (1-2 x)^{3/2} (5 x+3)^2-\frac{10}{63} (1-2 x)^{3/2} (27 x+22)+\frac{8}{9} \sqrt{1-2 x}-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(8*Sqrt[1 - 2*x])/9 + (5*(1 - 2*x)^(3/2)*(3 + 5*x)^2)/7 - ((1 - 2*x)^(3/2)*(3 +
5*x)^3)/(3*(2 + 3*x)) - (10*(1 - 2*x)^(3/2)*(22 + 27*x))/63 - (8*Sqrt[7/3]*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

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Rubi in Sympy [A]  time = 20.8721, size = 90, normalized size = 0.83 \[ \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{7} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (36450 x + 29700\right )}{8505} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{3 \left (3 x + 2\right )} + \frac{8 \sqrt{- 2 x + 1}}{9} - \frac{8 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*(-2*x + 1)**(3/2)*(5*x + 3)**2/7 - (-2*x + 1)**(3/2)*(36450*x + 29700)/8505 -
(-2*x + 1)**(3/2)*(5*x + 3)**3/(3*(3*x + 2)) + 8*sqrt(-2*x + 1)/9 - 8*sqrt(21)*a
tanh(sqrt(21)*sqrt(-2*x + 1)/7)/27

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Mathematica [A]  time = 0.118557, size = 70, normalized size = 0.65 \[ -\frac{\sqrt{1-2 x} \left (1500 x^4+780 x^3-1005 x^2-442 x+85\right )}{63 (3 x+2)}-\frac{8}{9} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(85 - 442*x - 1005*x^2 + 780*x^3 + 1500*x^4))/(63*(2 + 3*x)) - (
8*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9

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Maple [A]  time = 0.017, size = 72, normalized size = 0.7 \[{\frac{125}{126} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{145}{54} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{10}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{214}{243}\sqrt{1-2\,x}}-{\frac{14}{729}\sqrt{1-2\,x} \left ( -{\frac{4}{3}}-2\,x \right ) ^{-1}}-{\frac{8\,\sqrt{21}}{27}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/126*(1-2*x)^(7/2)-145/54*(1-2*x)^(5/2)+10/81*(1-2*x)^(3/2)+214/243*(1-2*x)^(
1/2)-14/729*(1-2*x)^(1/2)/(-4/3-2*x)-8/27*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21
^(1/2)

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Maxima [A]  time = 1.49626, size = 120, normalized size = 1.11 \[ \frac{125}{126} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{145}{54} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{10}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4}{27} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{214}{243} \, \sqrt{-2 \, x + 1} + \frac{7 \, \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/126*(-2*x + 1)^(7/2) - 145/54*(-2*x + 1)^(5/2) + 10/81*(-2*x + 1)^(3/2) + 4/
27*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) +
214/243*sqrt(-2*x + 1) + 7/243*sqrt(-2*x + 1)/(3*x + 2)

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Fricas [A]  time = 0.217633, size = 116, normalized size = 1.07 \[ \frac{\sqrt{3}{\left (28 \, \sqrt{7}{\left (3 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{3}{\left (1500 \, x^{4} + 780 \, x^{3} - 1005 \, x^{2} - 442 \, x + 85\right )} \sqrt{-2 \, x + 1}\right )}}{189 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/189*sqrt(3)*(28*sqrt(7)*(3*x + 2)*log((sqrt(3)*(3*x - 5) + 3*sqrt(7)*sqrt(-2*x
+ 1))/(3*x + 2)) - sqrt(3)*(1500*x^4 + 780*x^3 - 1005*x^2 - 442*x + 85)*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)**(3/2)*(3+5*x)**3/(2+3*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242675, size = 143, normalized size = 1.32 \[ -\frac{125}{126} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{145}{54} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{10}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4}{27} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{214}{243} \, \sqrt{-2 \, x + 1} + \frac{7 \, \sqrt{-2 \, x + 1}}{243 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/126*(2*x - 1)^3*sqrt(-2*x + 1) - 145/54*(2*x - 1)^2*sqrt(-2*x + 1) + 10/81*
(-2*x + 1)^(3/2) + 4/27*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqr
t(21) + 3*sqrt(-2*x + 1))) + 214/243*sqrt(-2*x + 1) + 7/243*sqrt(-2*x + 1)/(3*x
+ 2)