3.1930 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^2}{2+3 x} \, dx\)

Optimal. Leaf size=95 \[ \frac{25}{54} (1-2 x)^{9/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{2}{135} (1-2 x)^{5/2}+\frac{14}{243} (1-2 x)^{3/2}+\frac{98}{243} \sqrt{1-2 x}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(98*Sqrt[1 - 2*x])/243 + (14*(1 - 2*x)^(3/2))/243 + (2*(1 - 2*x)^(5/2))/135 - (1
55*(1 - 2*x)^(7/2))/126 + (25*(1 - 2*x)^(9/2))/54 - (98*Sqrt[7/3]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/243

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Rubi [A]  time = 0.123307, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{25}{54} (1-2 x)^{9/2}-\frac{155}{126} (1-2 x)^{7/2}+\frac{2}{135} (1-2 x)^{5/2}+\frac{14}{243} (1-2 x)^{3/2}+\frac{98}{243} \sqrt{1-2 x}-\frac{98}{243} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(98*Sqrt[1 - 2*x])/243 + (14*(1 - 2*x)^(3/2))/243 + (2*(1 - 2*x)^(5/2))/135 - (1
55*(1 - 2*x)^(7/2))/126 + (25*(1 - 2*x)^(9/2))/54 - (98*Sqrt[7/3]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/243

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Rubi in Sympy [A]  time = 11.586, size = 83, normalized size = 0.87 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{9}{2}}}{54} - \frac{155 \left (- 2 x + 1\right )^{\frac{7}{2}}}{126} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{135} + \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{243} + \frac{98 \sqrt{- 2 x + 1}}{243} - \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{729} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(9/2)/54 - 155*(-2*x + 1)**(7/2)/126 + 2*(-2*x + 1)**(5/2)/135 +
14*(-2*x + 1)**(3/2)/243 + 98*sqrt(-2*x + 1)/243 - 98*sqrt(21)*atanh(sqrt(21)*sq
rt(-2*x + 1)/7)/729

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Mathematica [A]  time = 0.0850716, size = 61, normalized size = 0.64 \[ \frac{3 \sqrt{1-2 x} \left (63000 x^4-42300 x^3-30546 x^2+29791 x-2479\right )-3430 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{25515} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-2479 + 29791*x - 30546*x^2 - 42300*x^3 + 63000*x^4) - 3430*Sq
rt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/25515

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Maple [A]  time = 0.01, size = 65, normalized size = 0.7 \[{\frac{14}{243} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{135} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{155}{126} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{25}{54} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{98\,\sqrt{21}}{729}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{98}{243}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

14/243*(1-2*x)^(3/2)+2/135*(1-2*x)^(5/2)-155/126*(1-2*x)^(7/2)+25/54*(1-2*x)^(9/
2)-98/729*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+98/243*(1-2*x)^(1/2)

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Maxima [A]  time = 1.50745, size = 111, normalized size = 1.17 \[ \frac{25}{54} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{155}{126} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{2}{135} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{14}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{49}{729} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{98}{243} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/54*(-2*x + 1)^(9/2) - 155/126*(-2*x + 1)^(7/2) + 2/135*(-2*x + 1)^(5/2) + 14/
243*(-2*x + 1)^(3/2) + 49/729*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) + 98/243*sqrt(-2*x + 1)

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Fricas [A]  time = 0.22473, size = 99, normalized size = 1.04 \[ \frac{1}{25515} \, \sqrt{3}{\left (\sqrt{3}{\left (63000 \, x^{4} - 42300 \, x^{3} - 30546 \, x^{2} + 29791 \, x - 2479\right )} \sqrt{-2 \, x + 1} + 1715 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} + 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25515*sqrt(3)*(sqrt(3)*(63000*x^4 - 42300*x^3 - 30546*x^2 + 29791*x - 2479)*sq
rt(-2*x + 1) + 1715*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 = 13.7211, size = 122, normalized size = 1.28 \[ \frac{25 \left (- 2 x + 1\right )^{\frac{9}{2}}}{54} - \frac{155 \left (- 2 x + 1\right )^{\frac{7}{2}}}{126} + \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{135} + \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{243} + \frac{98 \sqrt{- 2 x + 1}}{243} + \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*(-2*x + 1)**(9/2)/54 - 155*(-2*x + 1)**(7/2)/126 + 2*(-2*x + 1)**(5/2)/135 +
14*(-2*x + 1)**(3/2)/243 + 98*sqrt(-2*x + 1)/243 + 686*Piecewise((-sqrt(21)*acot
h(sqrt(21)*sqrt(-2*x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt
(-2*x + 1)/7)/21, -2*x + 1 < 7/3))/243

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GIAC/XCAS [A]  time = 0.212845, size = 143, normalized size = 1.51 \[ \frac{25}{54} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{155}{126} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{2}{135} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{14}{243} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{49}{729} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{98}{243} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/54*(2*x - 1)^4*sqrt(-2*x + 1) + 155/126*(2*x - 1)^3*sqrt(-2*x + 1) + 2/135*(2
*x - 1)^2*sqrt(-2*x + 1) + 14/243*(-2*x + 1)^(3/2) + 49/729*sqrt(21)*ln(1/2*abs(
-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 98/243*sqrt(-2*
x + 1)