3.1943 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^3}{2+3 x} \, dx\)

Optimal. Leaf size=108 \[ -\frac{125}{132} (1-2 x)^{11/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{5135}{756} (1-2 x)^{7/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{14}{729} (1-2 x)^{3/2}-\frac{98}{729} \sqrt{1-2 x}+\frac{98}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

[Out]

(-98*Sqrt[1 - 2*x])/729 - (14*(1 - 2*x)^(3/2))/729 - (2*(1 - 2*x)^(5/2))/405 - (
5135*(1 - 2*x)^(7/2))/756 + (400*(1 - 2*x)^(9/2))/81 - (125*(1 - 2*x)^(11/2))/13
2 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

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Rubi [A]  time = 0.12963, antiderivative size = 108, 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 \[ -\frac{125}{132} (1-2 x)^{11/2}+\frac{400}{81} (1-2 x)^{9/2}-\frac{5135}{756} (1-2 x)^{7/2}-\frac{2}{405} (1-2 x)^{5/2}-\frac{14}{729} (1-2 x)^{3/2}-\frac{98}{729} \sqrt{1-2 x}+\frac{98}{729} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-98*Sqrt[1 - 2*x])/729 - (14*(1 - 2*x)^(3/2))/729 - (2*(1 - 2*x)^(5/2))/405 - (
5135*(1 - 2*x)^(7/2))/756 + (400*(1 - 2*x)^(9/2))/81 - (125*(1 - 2*x)^(11/2))/13
2 + (98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/729

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Rubi in Sympy [A]  time = 12.7788, size = 95, normalized size = 0.88 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{11}{2}}}{132} + \frac{400 \left (- 2 x + 1\right )^{\frac{9}{2}}}{81} - \frac{5135 \left (- 2 x + 1\right )^{\frac{7}{2}}}{756} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{405} - \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{729} - \frac{98 \sqrt{- 2 x + 1}}{729} + \frac{98 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{2187} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(-2*x + 1)**(11/2)/132 + 400*(-2*x + 1)**(9/2)/81 - 5135*(-2*x + 1)**(7/2)/
756 - 2*(-2*x + 1)**(5/2)/405 - 14*(-2*x + 1)**(3/2)/729 - 98*sqrt(-2*x + 1)/729
+ 98*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/2187

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Mathematica [A]  time = 0.101016, size = 66, normalized size = 0.61 \[ \frac{3 \sqrt{1-2 x} \left (8505000 x^5+913500 x^4-7838550 x^3-249219 x^2+3024349 x-830656\right )+37730 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{841995} \]

Antiderivative was successfully verified.

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

[Out]

(3*Sqrt[1 - 2*x]*(-830656 + 3024349*x - 249219*x^2 - 7838550*x^3 + 913500*x^4 +
8505000*x^5) + 37730*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/841995

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Maple [A]  time = 0.01, size = 74, normalized size = 0.7 \[ -{\frac{14}{729} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2}{405} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{5135}{756} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{400}{81} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{125}{132} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}+{\frac{98\,\sqrt{21}}{2187}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{98}{729}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14/729*(1-2*x)^(3/2)-2/405*(1-2*x)^(5/2)-5135/756*(1-2*x)^(7/2)+400/81*(1-2*x)^
(9/2)-125/132*(1-2*x)^(11/2)+98/2187*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2
)-98/729*(1-2*x)^(1/2)

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Maxima [A]  time = 1.50502, size = 123, normalized size = 1.14 \[ -\frac{125}{132} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{400}{81} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{5135}{756} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{2}{405} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{14}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{2187} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{98}{729} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125/132*(-2*x + 1)^(11/2) + 400/81*(-2*x + 1)^(9/2) - 5135/756*(-2*x + 1)^(7/2)
- 2/405*(-2*x + 1)^(5/2) - 14/729*(-2*x + 1)^(3/2) - 49/2187*sqrt(21)*log(-(sqr
t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 98/729*sqrt(-2*x + 1)

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Fricas [A]  time = 0.21647, size = 105, normalized size = 0.97 \[ \frac{1}{841995} \, \sqrt{3}{\left (\sqrt{3}{\left (8505000 \, x^{5} + 913500 \, x^{4} - 7838550 \, x^{3} - 249219 \, x^{2} + 3024349 \, x - 830656\right )} \sqrt{-2 \, x + 1} + 18865 \, \sqrt{7} \log \left (\frac{\sqrt{3}{\left (3 \, x - 5\right )} - 3 \, \sqrt{7} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/841995*sqrt(3)*(sqrt(3)*(8505000*x^5 + 913500*x^4 - 7838550*x^3 - 249219*x^2 +
3024349*x - 830656)*sqrt(-2*x + 1) + 18865*sqrt(7)*log((sqrt(3)*(3*x - 5) - 3*s
qrt(7)*sqrt(-2*x + 1))/(3*x + 2)))

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Sympy [A]  time = 18.1488, size = 134, normalized size = 1.24 \[ - \frac{125 \left (- 2 x + 1\right )^{\frac{11}{2}}}{132} + \frac{400 \left (- 2 x + 1\right )^{\frac{9}{2}}}{81} - \frac{5135 \left (- 2 x + 1\right )^{\frac{7}{2}}}{756} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}}}{405} - \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}}}{729} - \frac{98 \sqrt{- 2 x + 1}}{729} - \frac{686 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 > \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{21} & \text{for}\: - 2 x + 1 < \frac{7}{3} \end{cases}\right )}{729} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(-2*x + 1)**(11/2)/132 + 400*(-2*x + 1)**(9/2)/81 - 5135*(-2*x + 1)**(7/2)/
756 - 2*(-2*x + 1)**(5/2)/405 - 14*(-2*x + 1)**(3/2)/729 - 98*sqrt(-2*x + 1)/729
- 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))/729

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GIAC/XCAS [A]  time = 0.212836, size = 165, normalized size = 1.53 \[ \frac{125}{132} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{400}{81} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{5135}{756} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{2}{405} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{14}{729} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{49}{2187} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{98}{729} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/132*(2*x - 1)^5*sqrt(-2*x + 1) + 400/81*(2*x - 1)^4*sqrt(-2*x + 1) + 5135/75
6*(2*x - 1)^3*sqrt(-2*x + 1) - 2/405*(2*x - 1)^2*sqrt(-2*x + 1) - 14/729*(-2*x +
1)^(3/2) - 49/2187*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) - 98/729*sqrt(-2*x + 1)