3.1797 \(\int \frac{\sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac{137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac{(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac{1613 \sqrt{1-2 x}}{1037232 (3 x+2)}+\frac{1613 \sqrt{1-2 x}}{444528 (3 x+2)^2}+\frac{1613 \sqrt{1-2 x}}{158760 (3 x+2)^3}-\frac{1613 \sqrt{1-2 x}}{7560 (3 x+2)^4}+\frac{1613 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{518616 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(3/2)/(378*(2 + 3*x)^6) + (137*(1 - 2*x)^(3/2))/(4410*(2 + 3*x)^5) -
(1613*Sqrt[1 - 2*x])/(7560*(2 + 3*x)^4) + (1613*Sqrt[1 - 2*x])/(158760*(2 + 3*x)
^3) + (1613*Sqrt[1 - 2*x])/(444528*(2 + 3*x)^2) + (1613*Sqrt[1 - 2*x])/(1037232*
(2 + 3*x)) + (1613*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(518616*Sqrt[21])

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Rubi [A]  time = 0.16292, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac{(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac{1613 \sqrt{1-2 x}}{1037232 (3 x+2)}+\frac{1613 \sqrt{1-2 x}}{444528 (3 x+2)^2}+\frac{1613 \sqrt{1-2 x}}{158760 (3 x+2)^3}-\frac{1613 \sqrt{1-2 x}}{7560 (3 x+2)^4}+\frac{1613 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{518616 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-(1 - 2*x)^(3/2)/(378*(2 + 3*x)^6) + (137*(1 - 2*x)^(3/2))/(4410*(2 + 3*x)^5) -
(1613*Sqrt[1 - 2*x])/(7560*(2 + 3*x)^4) + (1613*Sqrt[1 - 2*x])/(158760*(2 + 3*x)
^3) + (1613*Sqrt[1 - 2*x])/(444528*(2 + 3*x)^2) + (1613*Sqrt[1 - 2*x])/(1037232*
(2 + 3*x)) + (1613*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(518616*Sqrt[21])

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Rubi in Sympy [A]  time = 16.1669, size = 131, normalized size = 0.89 \[ \frac{137 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4410 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{378 \left (3 x + 2\right )^{6}} + \frac{1613 \sqrt{- 2 x + 1}}{1037232 \left (3 x + 2\right )} + \frac{1613 \sqrt{- 2 x + 1}}{444528 \left (3 x + 2\right )^{2}} + \frac{1613 \sqrt{- 2 x + 1}}{158760 \left (3 x + 2\right )^{3}} - \frac{1613 \sqrt{- 2 x + 1}}{7560 \left (3 x + 2\right )^{4}} + \frac{1613 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{10890936} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

137*(-2*x + 1)**(3/2)/(4410*(3*x + 2)**5) - (-2*x + 1)**(3/2)/(378*(3*x + 2)**6)
+ 1613*sqrt(-2*x + 1)/(1037232*(3*x + 2)) + 1613*sqrt(-2*x + 1)/(444528*(3*x +
2)**2) + 1613*sqrt(-2*x + 1)/(158760*(3*x + 2)**3) - 1613*sqrt(-2*x + 1)/(7560*(
3*x + 2)**4) + 1613*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/10890936

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Mathematica [A]  time = 0.124675, size = 73, normalized size = 0.49 \[ \frac{\frac{21 \sqrt{1-2 x} \left (1959795 x^5+8056935 x^4+14197626 x^3+1791558 x^2-7772840 x-3136864\right )}{(3 x+2)^6}+16130 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{108909360} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

((21*Sqrt[1 - 2*x]*(-3136864 - 7772840*x + 1791558*x^2 + 14197626*x^3 + 8056935*
x^4 + 1959795*x^5))/(2 + 3*x)^6 + 16130*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/108909360

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Maple [A]  time = 0.017, size = 84, normalized size = 0.6 \[ 23328\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ( -{\frac{1613\, \left ( 1-2\,x \right ) ^{11/2}}{49787136}}+{\frac{27421\, \left ( 1-2\,x \right ) ^{9/2}}{64012032}}-{\frac{17743\, \left ( 1-2\,x \right ) ^{7/2}}{7620480}}+{\frac{4213\, \left ( 1-2\,x \right ) ^{5/2}}{846720}}-{\frac{86837\, \left ( 1-2\,x \right ) ^{3/2}}{35271936}}-{\frac{11291\,\sqrt{1-2\,x}}{5038848}} \right ) }+{\frac{1613\,\sqrt{21}}{10890936}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x)

[Out]

23328*(-1613/49787136*(1-2*x)^(11/2)+27421/64012032*(1-2*x)^(9/2)-17743/7620480*
(1-2*x)^(7/2)+4213/846720*(1-2*x)^(5/2)-86837/35271936*(1-2*x)^(3/2)-11291/50388
48*(1-2*x)^(1/2))/(-4-6*x)^6+1613/10890936*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)

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Maxima [A]  time = 1.52931, size = 197, normalized size = 1.33 \[ -\frac{1613}{21781872} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1959795 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 25912845 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 140843934 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 300985146 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 148925455 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 135548455 \, \sqrt{-2 \, x + 1}}{2593080 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*sqrt(-2*x + 1)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

-1613/21781872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) - 1/2593080*(1959795*(-2*x + 1)^(11/2) - 25912845*(-2*x + 1)^(9/2) + 1
40843934*(-2*x + 1)^(7/2) - 300985146*(-2*x + 1)^(5/2) + 148925455*(-2*x + 1)^(3
/2) + 135548455*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*
x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]  time = 0.211783, size = 181, normalized size = 1.22 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (1959795 \, x^{5} + 8056935 \, x^{4} + 14197626 \, x^{3} + 1791558 \, x^{2} - 7772840 \, x - 3136864\right )} \sqrt{-2 \, x + 1} + 8065 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{108909360 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*sqrt(-2*x + 1)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

1/108909360*sqrt(21)*(sqrt(21)*(1959795*x^5 + 8056935*x^4 + 14197626*x^3 + 17915
58*x^2 - 7772840*x - 3136864)*sqrt(-2*x + 1) + 8065*(729*x^6 + 2916*x^5 + 4860*x
^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x +
1))/(3*x + 2)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 6
4)

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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((3+5*x)**2*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222038, size = 178, normalized size = 1.2 \[ -\frac{1613}{21781872} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1959795 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 25912845 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 140843934 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 300985146 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 148925455 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 135548455 \, \sqrt{-2 \, x + 1}}{165957120 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*sqrt(-2*x + 1)/(3*x + 2)^7,x, algorithm="giac")

[Out]

-1613/21781872*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3
*sqrt(-2*x + 1))) + 1/165957120*(1959795*(2*x - 1)^5*sqrt(-2*x + 1) + 25912845*(
2*x - 1)^4*sqrt(-2*x + 1) + 140843934*(2*x - 1)^3*sqrt(-2*x + 1) + 300985146*(2*
x - 1)^2*sqrt(-2*x + 1) - 148925455*(-2*x + 1)^(3/2) - 135548455*sqrt(-2*x + 1))
/(3*x + 2)^6