3.1810 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=167 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac{2 \sqrt{1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac{23717 \sqrt{1-2 x}}{9529569 (3 x+2)}+\frac{23717 \sqrt{1-2 x}}{4084101 (3 x+2)^2}+\frac{47434 \sqrt{1-2 x}}{2917215 (3 x+2)^3}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}} \]

[Out]

(47434*Sqrt[1 - 2*x])/(2917215*(2 + 3*x)^3) + (23717*Sqrt[1 - 2*x])/(4084101*(2
+ 3*x)^2) + (23717*Sqrt[1 - 2*x])/(9529569*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5
*x)^2)/(1323*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(21*(2 + 3*x)^7) - (2*Sq
rt[1 - 2*x]*(54227 + 88099*x))/(972405*(2 + 3*x)^5) + (47434*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(9529569*Sqrt[21])

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Rubi [A]  time = 0.228374, antiderivative size = 167, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{21 (3 x+2)^7}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{1323 (3 x+2)^6}-\frac{2 \sqrt{1-2 x} (88099 x+54227)}{972405 (3 x+2)^5}+\frac{23717 \sqrt{1-2 x}}{9529569 (3 x+2)}+\frac{23717 \sqrt{1-2 x}}{4084101 (3 x+2)^2}+\frac{47434 \sqrt{1-2 x}}{2917215 (3 x+2)^3}+\frac{47434 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9529569 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

(47434*Sqrt[1 - 2*x])/(2917215*(2 + 3*x)^3) + (23717*Sqrt[1 - 2*x])/(4084101*(2
+ 3*x)^2) + (23717*Sqrt[1 - 2*x])/(9529569*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5
*x)^2)/(1323*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(21*(2 + 3*x)^7) - (2*Sq
rt[1 - 2*x]*(54227 + 88099*x))/(972405*(2 + 3*x)^5) + (47434*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]])/(9529569*Sqrt[21])

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Rubi in Sympy [A]  time = 25.097, size = 148, normalized size = 0.89 \[ \frac{23717 \sqrt{- 2 x + 1}}{9529569 \left (3 x + 2\right )} + \frac{23717 \sqrt{- 2 x + 1}}{4084101 \left (3 x + 2\right )^{2}} + \frac{47434 \sqrt{- 2 x + 1}}{2917215 \left (3 x + 2\right )^{3}} - \frac{\sqrt{- 2 x + 1} \left (4228752 x + 2602896\right )}{23337720 \left (3 x + 2\right )^{5}} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{1323 \left (3 x + 2\right )^{6}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{21 \left (3 x + 2\right )^{7}} + \frac{47434 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{200120949} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**8,x)

[Out]

23717*sqrt(-2*x + 1)/(9529569*(3*x + 2)) + 23717*sqrt(-2*x + 1)/(4084101*(3*x +
2)**2) + 47434*sqrt(-2*x + 1)/(2917215*(3*x + 2)**3) - sqrt(-2*x + 1)*(4228752*x
+ 2602896)/(23337720*(3*x + 2)**5) - 53*sqrt(-2*x + 1)*(5*x + 3)**2/(1323*(3*x
+ 2)**6) - sqrt(-2*x + 1)*(5*x + 3)**3/(21*(3*x + 2)**7) + 47434*sqrt(21)*atanh(
sqrt(21)*sqrt(-2*x + 1)/7)/200120949

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Mathematica [A]  time = 0.139917, size = 78, normalized size = 0.47 \[ \frac{\frac{21 \sqrt{1-2 x} \left (86448465 x^6+413031555 x^5+863203932 x^4+473987484 x^3-306463011 x^2-361589428 x-88036937\right )}{(3 x+2)^7}+237170 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1000604745} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^8,x]

[Out]

((21*Sqrt[1 - 2*x]*(-88036937 - 361589428*x - 306463011*x^2 + 473987484*x^3 + 86
3203932*x^4 + 413031555*x^5 + 86448465*x^6))/(2 + 3*x)^7 + 237170*Sqrt[21]*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1000604745

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Maple [A]  time = 0.018, size = 93, normalized size = 0.6 \[ 69984\,{\frac{1}{ \left ( -4-6\,x \right ) ^{7}} \left ( -{\frac{23717\, \left ( 1-2\,x \right ) ^{13/2}}{457419312}}+{\frac{118585\, \left ( 1-2\,x \right ) ^{11/2}}{147027636}}-{\frac{6711911\, \left ( 1-2\,x \right ) ^{9/2}}{1260236880}}+{\frac{1303513\, \left ( 1-2\,x \right ) ^{7/2}}{78764805}}-{\frac{5101561\, \left ( 1-2\,x \right ) ^{5/2}}{231472080}}+{\frac{25163\, \left ( 1-2\,x \right ) ^{3/2}}{4960116}}+{\frac{23717\,\sqrt{1-2\,x}}{2834352}} \right ) }+{\frac{47434\,\sqrt{21}}{200120949}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^8,x)

[Out]

69984*(-23717/457419312*(1-2*x)^(13/2)+118585/147027636*(1-2*x)^(11/2)-6711911/1
260236880*(1-2*x)^(9/2)+1303513/78764805*(1-2*x)^(7/2)-5101561/231472080*(1-2*x)
^(5/2)+25163/4960116*(1-2*x)^(3/2)+23717/2834352*(1-2*x)^(1/2))/(-4-6*x)^7+47434
/200120949*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.51992, size = 221, normalized size = 1.32 \[ -\frac{23717}{200120949} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (86448465 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 1344753900 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 8879858253 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 27592763184 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 36746543883 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 8458290820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13951406665 \, \sqrt{-2 \, x + 1}\right )}}{47647845 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^8,x, algorithm="maxima")

[Out]

-23717/200120949*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 2/47647845*(86448465*(-2*x + 1)^(13/2) - 1344753900*(-2*x + 1)^(11
/2) + 8879858253*(-2*x + 1)^(9/2) - 27592763184*(-2*x + 1)^(7/2) + 36746543883*(
-2*x + 1)^(5/2) - 8458290820*(-2*x + 1)^(3/2) - 13951406665*sqrt(-2*x + 1))/(218
7*(2*x - 1)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 22
68945*(2*x - 1)^3 + 3176523*(2*x - 1)^2 + 4941258*x - 1647086)

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Fricas [A]  time = 0.21302, size = 201, normalized size = 1.2 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (86448465 \, x^{6} + 413031555 \, x^{5} + 863203932 \, x^{4} + 473987484 \, x^{3} - 306463011 \, x^{2} - 361589428 \, x - 88036937\right )} \sqrt{-2 \, x + 1} + 118585 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{1000604745 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^8,x, algorithm="fricas")

[Out]

1/1000604745*sqrt(21)*(sqrt(21)*(86448465*x^6 + 413031555*x^5 + 863203932*x^4 +
473987484*x^3 - 306463011*x^2 - 361589428*x - 88036937)*sqrt(-2*x + 1) + 118585*
(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x +
128)*log((sqrt(21)*(3*x - 5) - 21*sqrt(-2*x + 1))/(3*x + 2)))/(2187*x^7 + 10206*
x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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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)**3*(1-2*x)**(1/2)/(2+3*x)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.227349, size = 200, normalized size = 1.2 \[ -\frac{23717}{200120949} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{86448465 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + 1344753900 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 8879858253 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 27592763184 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 36746543883 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 8458290820 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 13951406665 \, \sqrt{-2 \, x + 1}}{3049462080 \,{\left (3 \, x + 2\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^8,x, algorithm="giac")

[Out]

-23717/200120949*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 1/3049462080*(86448465*(2*x - 1)^6*sqrt(-2*x + 1) + 134475
3900*(2*x - 1)^5*sqrt(-2*x + 1) + 8879858253*(2*x - 1)^4*sqrt(-2*x + 1) + 275927
63184*(2*x - 1)^3*sqrt(-2*x + 1) + 36746543883*(2*x - 1)^2*sqrt(-2*x + 1) - 8458
290820*(-2*x + 1)^(3/2) - 13951406665*sqrt(-2*x + 1))/(3*x + 2)^7