3.1870 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{2+3 x} \, dx\)

Optimal. Leaf size=95 \[ -\frac{125}{108} (1-2 x)^{9/2}+\frac{400}{63} (1-2 x)^{7/2}-\frac{1027}{108} (1-2 x)^{5/2}-\frac{2}{243} (1-2 x)^{3/2}-\frac{14}{243} \sqrt{1-2 x}+\frac{14}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-14*Sqrt[1 - 2*x])/243 - (2*(1 - 2*x)^(3/2))/243 - (1027*(1 - 2*x)^(5/2))/108 +
(400*(1 - 2*x)^(7/2))/63 - (125*(1 - 2*x)^(9/2))/108 + (14*Sqrt[7/3]*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/243

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Rubi [A]  time = 0.103802, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{125}{108} (1-2 x)^{9/2}+\frac{400}{63} (1-2 x)^{7/2}-\frac{1027}{108} (1-2 x)^{5/2}-\frac{2}{243} (1-2 x)^{3/2}-\frac{14}{243} \sqrt{1-2 x}+\frac{14}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-14*Sqrt[1 - 2*x])/243 - (2*(1 - 2*x)^(3/2))/243 - (1027*(1 - 2*x)^(5/2))/108 +
(400*(1 - 2*x)^(7/2))/63 - (125*(1 - 2*x)^(9/2))/108 + (14*Sqrt[7/3]*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/243

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Rubi in Sympy [A]  time = 11.7621, size = 83, normalized size = 0.87 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{9}{2}}}{108} + \frac{400 \left (- 2 x + 1\right )^{\frac{7}{2}}}{63} - \frac{1027 \left (- 2 x + 1\right )^{\frac{5}{2}}}{108} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{243} - \frac{14 \sqrt{- 2 x + 1}}{243} + \frac{14 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{729} \]

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),x)

[Out]

-125*(-2*x + 1)**(9/2)/108 + 400*(-2*x + 1)**(7/2)/63 - 1027*(-2*x + 1)**(5/2)/1
08 - 2*(-2*x + 1)**(3/2)/243 - 14*sqrt(-2*x + 1)/243 + 14*sqrt(21)*atanh(sqrt(21
)*sqrt(-2*x + 1)/7)/729

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Mathematica [A]  time = 0.0881326, size = 61, normalized size = 0.64 \[ \frac{98 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-3 \sqrt{1-2 x} \left (31500 x^4+23400 x^3-17649 x^2-15679 x+7456\right )}{5103} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[1 - 2*x]*(7456 - 15679*x - 17649*x^2 + 23400*x^3 + 31500*x^4) + 98*Sqrt
[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/5103

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Maple [A]  time = 0.01, size = 65, normalized size = 0.7 \[ -{\frac{2}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1027}{108} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{400}{63} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{125}{108} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{14\,\sqrt{21}}{729}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{14}{243}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/243*(1-2*x)^(3/2)-1027/108*(1-2*x)^(5/2)+400/63*(1-2*x)^(7/2)-125/108*(1-2*x)
^(9/2)+14/729*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-14/243*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49875, size = 111, normalized size = 1.17 \[ -\frac{125}{108} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{400}{63} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1027}{108} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{2}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7}{729} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{14}{243} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/108*(-2*x + 1)^(9/2) + 400/63*(-2*x + 1)^(7/2) - 1027/108*(-2*x + 1)^(5/2)
- 2/243*(-2*x + 1)^(3/2) - 7/729*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) - 14/243*sqrt(-2*x + 1)

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Fricas [A]  time = 0.216106, size = 99, normalized size = 1.04 \[ -\frac{1}{5103} \, \sqrt{3}{\left (\sqrt{3}{\left (31500 \, x^{4} + 23400 \, x^{3} - 17649 \, x^{2} - 15679 \, x + 7456\right )} \sqrt{-2 \, x + 1} - 49 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5103*sqrt(3)*(sqrt(3)*(31500*x^4 + 23400*x^3 - 17649*x^2 - 15679*x + 7456)*sq
rt(-2*x + 1) - 49*sqrt(7)*log((sqrt(3)*(3*x - 5) - 3*sqrt(7)*sqrt(-2*x + 1))/(3*
x + 2)))

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Sympy [A]  time = 13.3597, size = 122, normalized size = 1.28 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{9}{2}}}{108} + \frac{400 \left (- 2 x + 1\right )^{\frac{7}{2}}}{63} - \frac{1027 \left (- 2 x + 1\right )^{\frac{5}{2}}}{108} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{243} - \frac{14 \sqrt{- 2 x + 1}}{243} - \frac{98 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(-2*x + 1)**(9/2)/108 + 400*(-2*x + 1)**(7/2)/63 - 1027*(-2*x + 1)**(5/2)/1
08 - 2*(-2*x + 1)**(3/2)/243 - 14*sqrt(-2*x + 1)/243 - 98*Piecewise((-sqrt(21)*a
coth(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*s
qrt(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/243

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GIAC/XCAS [A]  time = 0.233508, size = 143, normalized size = 1.51 \[ -\frac{125}{108} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{400}{63} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1027}{108} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{2}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7}{729} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{14}{243} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/108*(2*x - 1)^4*sqrt(-2*x + 1) - 400/63*(2*x - 1)^3*sqrt(-2*x + 1) - 1027/1
08*(2*x - 1)^2*sqrt(-2*x + 1) - 2/243*(-2*x + 1)^(3/2) - 7/729*sqrt(21)*ln(1/2*a
bs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 14/243*sqrt(
-2*x + 1)