3.1918 \(\int \frac{(1-2 x)^{5/2} (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=82 \[ -\frac{5}{21} (1-2 x)^{7/2}-\frac{2}{45} (1-2 x)^{5/2}-\frac{14}{81} (1-2 x)^{3/2}-\frac{98}{81} \sqrt{1-2 x}+\frac{98}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-98*Sqrt[1 - 2*x])/81 - (14*(1 - 2*x)^(3/2))/81 - (2*(1 - 2*x)^(5/2))/45 - (5*(
1 - 2*x)^(7/2))/21 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi [A]  time = 0.0954778, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5}{21} (1-2 x)^{7/2}-\frac{2}{45} (1-2 x)^{5/2}-\frac{14}{81} (1-2 x)^{3/2}-\frac{98}{81} \sqrt{1-2 x}+\frac{98}{81} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-98*Sqrt[1 - 2*x])/81 - (14*(1 - 2*x)^(3/2))/81 - (2*(1 - 2*x)^(5/2))/45 - (5*(
1 - 2*x)^(7/2))/21 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/81

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Rubi in Sympy [A]  time = 9.16496, size = 71, normalized size = 0.87 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{7}{2}}}{21} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{45} - \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{98 \sqrt{- 2 x + 1}}{81} + \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(7/2)/21 - 2*(-2*x + 1)**(5/2)/45 - 14*(-2*x + 1)**(3/2)/81 - 98*
sqrt(-2*x + 1)/81 + 98*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/243

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Mathematica [A]  time = 0.0685167, size = 56, normalized size = 0.68 \[ \frac{3 \sqrt{1-2 x} \left (5400 x^3-8604 x^2+5534 x-4721\right )+3430 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{8505} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-4721 + 5534*x - 8604*x^2 + 5400*x^3) + 3430*Sqrt[21]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/8505

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Maple [A]  time = 0.008, size = 56, normalized size = 0.7 \[ -{\frac{14}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2}{45} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{21} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{98\,\sqrt{21}}{243}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{98}{81}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14/81*(1-2*x)^(3/2)-2/45*(1-2*x)^(5/2)-5/21*(1-2*x)^(7/2)+98/243*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)-98/81*(1-2*x)^(1/2)

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Maxima [A]  time = 1.50144, size = 99, normalized size = 1.21 \[ -\frac{5}{21} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2}{45} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{14}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{243} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{98}{81} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/21*(-2*x + 1)^(7/2) - 2/45*(-2*x + 1)^(5/2) - 14/81*(-2*x + 1)^(3/2) - 49/243
*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98
/81*sqrt(-2*x + 1)

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Fricas [A]  time = 0.217915, size = 92, normalized size = 1.12 \[ \frac{1}{8505} \, \sqrt{3}{\left (\sqrt{3}{\left (5400 \, x^{3} - 8604 \, x^{2} + 5534 \, x - 4721\right )} \sqrt{-2 \, x + 1} + 1715 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8505*sqrt(3)*(sqrt(3)*(5400*x^3 - 8604*x^2 + 5534*x - 4721)*sqrt(-2*x + 1) + 1
715*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 = 10.0818, size = 112, normalized size = 1.37 \[ - \frac{5 \left (- 2 x + 1\right )^{\frac{7}{2}}}{21} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{45} - \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{81} - \frac{98 \sqrt{- 2 x + 1}}{81} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-2*x + 1)**(7/2)/21 - 2*(-2*x + 1)**(5/2)/45 - 14*(-2*x + 1)**(3/2)/81 - 98*
sqrt(-2*x + 1)/81 - 686*Piecewise((-sqrt(21)*acoth(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))/81

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GIAC/XCAS [A]  time = 0.210562, size = 122, normalized size = 1.49 \[ \frac{5}{21} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2}{45} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{14}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{243} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{98}{81} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/21*(2*x - 1)^3*sqrt(-2*x + 1) - 2/45*(2*x - 1)^2*sqrt(-2*x + 1) - 14/81*(-2*x
+ 1)^(3/2) - 49/243*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) - 98/81*sqrt(-2*x + 1)