3.1877 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=174 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{7 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac{173 \sqrt{1-2 x} (5 x+3)^2}{735 (3 x+2)^5}-\frac{\sqrt{1-2 x} (237807 x+146585)}{185220 (3 x+2)^4}-\frac{4369 \sqrt{1-2 x}}{1210104 (3 x+2)}-\frac{4369 \sqrt{1-2 x}}{518616 (3 x+2)^2}-\frac{4369 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{605052 \sqrt{21}} \]

[Out]

(-4369*Sqrt[1 - 2*x])/(518616*(2 + 3*x)^2) - (4369*Sqrt[1 - 2*x])/(1210104*(2 +
3*x)) - (173*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(735*(2 + 3*x)^5) - ((1 - 2*x)^(3/2)*(3
+ 5*x)^3)/(21*(2 + 3*x)^7) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(7*(2 + 3*x)^6) - (Sq
rt[1 - 2*x]*(146585 + 237807*x))/(185220*(2 + 3*x)^4) - (4369*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/(605052*Sqrt[21])

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Rubi [A]  time = 0.259958, antiderivative size = 174, 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 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{7 (3 x+2)^6}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{21 (3 x+2)^7}-\frac{173 \sqrt{1-2 x} (5 x+3)^2}{735 (3 x+2)^5}-\frac{\sqrt{1-2 x} (237807 x+146585)}{185220 (3 x+2)^4}-\frac{4369 \sqrt{1-2 x}}{1210104 (3 x+2)}-\frac{4369 \sqrt{1-2 x}}{518616 (3 x+2)^2}-\frac{4369 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{605052 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-4369*Sqrt[1 - 2*x])/(518616*(2 + 3*x)^2) - (4369*Sqrt[1 - 2*x])/(1210104*(2 +
3*x)) - (173*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(735*(2 + 3*x)^5) - ((1 - 2*x)^(3/2)*(3
+ 5*x)^3)/(21*(2 + 3*x)^7) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(7*(2 + 3*x)^6) - (Sq
rt[1 - 2*x]*(146585 + 237807*x))/(185220*(2 + 3*x)^4) - (4369*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/(605052*Sqrt[21])

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Rubi in Sympy [A]  time = 26.1178, size = 148, normalized size = 0.85 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (4981662 x + 3083724\right )}{23337720 \left (3 x + 2\right )^{5}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{49 \left (3 x + 2\right )^{6}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{21 \left (3 x + 2\right )^{7}} - \frac{4369 \sqrt{- 2 x + 1}}{1210104 \left (3 x + 2\right )} - \frac{4369 \sqrt{- 2 x + 1}}{518616 \left (3 x + 2\right )^{2}} + \frac{4369 \sqrt{- 2 x + 1}}{37044 \left (3 x + 2\right )^{3}} - \frac{4369 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{12706092} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(3/2)*(4981662*x + 3083724)/(23337720*(3*x + 2)**5) - 2*(-2*x + 1)*
*(3/2)*(5*x + 3)**2/(49*(3*x + 2)**6) - (-2*x + 1)**(3/2)*(5*x + 3)**3/(21*(3*x
+ 2)**7) - 4369*sqrt(-2*x + 1)/(1210104*(3*x + 2)) - 4369*sqrt(-2*x + 1)/(518616
*(3*x + 2)**2) + 4369*sqrt(-2*x + 1)/(37044*(3*x + 2)**3) - 4369*sqrt(21)*atanh(
sqrt(21)*sqrt(-2*x + 1)/7)/12706092

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Mathematica [A]  time = 0.165455, size = 78, normalized size = 0.45 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (15925005 x^6+76086135 x^5-42669876 x^4-182748162 x^3-98441652 x^2+606784 x+7033976\right )}{(3 x+2)^7}-43690 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{127060920} \]

Antiderivative was successfully verified.

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

[Out]

((-21*Sqrt[1 - 2*x]*(7033976 + 606784*x - 98441652*x^2 - 182748162*x^3 - 4266987
6*x^4 + 76086135*x^5 + 15925005*x^6))/(2 + 3*x)^7 - 43690*Sqrt[21]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/127060920

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Maple [A]  time = 0.019, size = 93, normalized size = 0.5 \[ 69984\,{\frac{1}{ \left ( -4-6\,x \right ) ^{7}} \left ({\frac{4369\, \left ( 1-2\,x \right ) ^{13/2}}{58084992}}-{\frac{21845\, \left ( 1-2\,x \right ) ^{11/2}}{18670176}}+{\frac{5639843\, \left ( 1-2\,x \right ) ^{9/2}}{1440270720}}+{\frac{1798\, \left ( 1-2\,x \right ) ^{7/2}}{1250235}}-{\frac{725323\, \left ( 1-2\,x \right ) ^{5/2}}{29393280}}+{\frac{21845\, \left ( 1-2\,x \right ) ^{3/2}}{629856}}-{\frac{30583\,\sqrt{1-2\,x}}{2519424}} \right ) }-{\frac{4369\,\sqrt{21}}{12706092}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

69984*(4369/58084992*(1-2*x)^(13/2)-21845/18670176*(1-2*x)^(11/2)+5639843/144027
0720*(1-2*x)^(9/2)+1798/1250235*(1-2*x)^(7/2)-725323/29393280*(1-2*x)^(5/2)+2184
5/629856*(1-2*x)^(3/2)-30583/2519424*(1-2*x)^(1/2))/(-4-6*x)^7-4369/12706092*arc
tanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49496, size = 221, normalized size = 1.27 \[ \frac{4369}{25412184} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{15925005 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 247722300 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 829056921 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 304480512 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 5224501569 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 7342978300 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2570042405 \, \sqrt{-2 \, x + 1}}{3025260 \,{\left (2187 \,{\left (2 \, x - 1\right )}^{7} + 35721 \,{\left (2 \, x - 1\right )}^{6} + 250047 \,{\left (2 \, x - 1\right )}^{5} + 972405 \,{\left (2 \, x - 1\right )}^{4} + 2268945 \,{\left (2 \, x - 1\right )}^{3} + 3176523 \,{\left (2 \, x - 1\right )}^{2} + 4941258 \, x - 1647086\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4369/25412184*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) - 1/3025260*(15925005*(-2*x + 1)^(13/2) - 247722300*(-2*x + 1)^(11/2) +
829056921*(-2*x + 1)^(9/2) + 304480512*(-2*x + 1)^(7/2) - 5224501569*(-2*x + 1)
^(5/2) + 7342978300*(-2*x + 1)^(3/2) - 2570042405*sqrt(-2*x + 1))/(2187*(2*x - 1
)^7 + 35721*(2*x - 1)^6 + 250047*(2*x - 1)^5 + 972405*(2*x - 1)^4 + 2268945*(2*x
- 1)^3 + 3176523*(2*x - 1)^2 + 4941258*x - 1647086)

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Fricas [A]  time = 0.208838, size = 201, normalized size = 1.16 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (15925005 \, x^{6} + 76086135 \, x^{5} - 42669876 \, x^{4} - 182748162 \, x^{3} - 98441652 \, x^{2} + 606784 \, x + 7033976\right )} \sqrt{-2 \, x + 1} - 21845 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{127060920 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/127060920*sqrt(21)*(sqrt(21)*(15925005*x^6 + 76086135*x^5 - 42669876*x^4 - 18
2748162*x^3 - 98441652*x^2 + 606784*x + 7033976)*sqrt(-2*x + 1) - 21845*(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 + 20
412*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((1-2*x)**(3/2)*(3+5*x)**3/(2+3*x)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215005, size = 200, normalized size = 1.15 \[ \frac{4369}{25412184} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{15925005 \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + 247722300 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 829056921 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 304480512 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 5224501569 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 7342978300 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2570042405 \, \sqrt{-2 \, x + 1}}{387233280 \,{\left (3 \, x + 2\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4369/25412184*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) - 1/387233280*(15925005*(2*x - 1)^6*sqrt(-2*x + 1) + 247722300*
(2*x - 1)^5*sqrt(-2*x + 1) + 829056921*(2*x - 1)^4*sqrt(-2*x + 1) - 304480512*(2
*x - 1)^3*sqrt(-2*x + 1) - 5224501569*(2*x - 1)^2*sqrt(-2*x + 1) + 7342978300*(-
2*x + 1)^(3/2) - 2570042405*sqrt(-2*x + 1))/(3*x + 2)^7