3.1876 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=154 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}-\frac{653 \sqrt{1-2 x} (5 x+3)^2}{2520 (3 x+2)^4}-\frac{\sqrt{1-2 x} (664915 x+413424)}{317520 (3 x+2)^3}-\frac{15313 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{15313 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

[Out]

(-15313*Sqrt[1 - 2*x])/(444528*(2 + 3*x)) - (653*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(252
0*(2 + 3*x)^4) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^6) + (2*Sqrt[1 - 2*
x]*(3 + 5*x)^3)/(5*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(413424 + 664915*x))/(317520*(2
+ 3*x)^3) - (15313*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

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Rubi [A]  time = 0.233413, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^3}{5 (3 x+2)^5}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{18 (3 x+2)^6}-\frac{653 \sqrt{1-2 x} (5 x+3)^2}{2520 (3 x+2)^4}-\frac{\sqrt{1-2 x} (664915 x+413424)}{317520 (3 x+2)^3}-\frac{15313 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{15313 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-15313*Sqrt[1 - 2*x])/(444528*(2 + 3*x)) - (653*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(252
0*(2 + 3*x)^4) - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(18*(2 + 3*x)^6) + (2*Sqrt[1 - 2*
x]*(3 + 5*x)^3)/(5*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(413424 + 664915*x))/(317520*(2
+ 3*x)^3) - (15313*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

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Rubi in Sympy [A]  time = 23.5592, size = 129, normalized size = 0.84 \[ - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3918915 x + 2436651\right )}{10001880 \left (3 x + 2\right )^{4}} - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{2}}{35 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{3}}{18 \left (3 x + 2\right )^{6}} - \frac{15313 \sqrt{- 2 x + 1}}{444528 \left (3 x + 2\right )} + \frac{15313 \sqrt{- 2 x + 1}}{63504 \left (3 x + 2\right )^{2}} - \frac{15313 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{4667544} \]

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)**7,x)

[Out]

-(-2*x + 1)**(3/2)*(3918915*x + 2436651)/(10001880*(3*x + 2)**4) - 2*(-2*x + 1)*
*(3/2)*(5*x + 3)**2/(35*(3*x + 2)**5) - (-2*x + 1)**(3/2)*(5*x + 3)**3/(18*(3*x
+ 2)**6) - 15313*sqrt(-2*x + 1)/(444528*(3*x + 2)) + 15313*sqrt(-2*x + 1)/(63504
*(3*x + 2)**2) - 15313*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/4667544

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Mathematica [A]  time = 0.137855, size = 73, normalized size = 0.47 \[ \frac{-\frac{63 \sqrt{1-2 x} \left (18605295 x^5-46991565 x^4-122053374 x^3-75153042 x^2-10947400 x+1660816\right )}{(3 x+2)^6}-459390 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{140026320} \]

Antiderivative was successfully verified.

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

[Out]

((-63*Sqrt[1 - 2*x]*(1660816 - 10947400*x - 75153042*x^2 - 122053374*x^3 - 46991
565*x^4 + 18605295*x^5))/(2 + 3*x)^6 - 459390*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/140026320

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Maple [A]  time = 0.018, size = 84, normalized size = 0.6 \[ -11664\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ( -{\frac{15313\, \left ( 1-2\,x \right ) ^{11/2}}{10668672}}-{\frac{3037\, \left ( 1-2\,x \right ) ^{9/2}}{41150592}}+{\frac{256271\, \left ( 1-2\,x \right ) ^{7/2}}{4898880}}-{\frac{923549\, \left ( 1-2\,x \right ) ^{5/2}}{4898880}}+{\frac{1822247\, \left ( 1-2\,x \right ) ^{3/2}}{7558272}}-{\frac{750337\,\sqrt{1-2\,x}}{7558272}} \right ) }-{\frac{15313\,\sqrt{21}}{4667544}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11664*(-15313/10668672*(1-2*x)^(11/2)-3037/41150592*(1-2*x)^(9/2)+256271/489888
0*(1-2*x)^(7/2)-923549/4898880*(1-2*x)^(5/2)+1822247/7558272*(1-2*x)^(3/2)-75033
7/7558272*(1-2*x)^(1/2))/(-4-6*x)^6-15313/4667544*arctanh(1/7*21^(1/2)*(1-2*x)^(
1/2))*21^(1/2)

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Maxima [A]  time = 1.51746, size = 197, normalized size = 1.28 \[ \frac{15313}{9335088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18605295 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 956655 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 678093066 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2443710654 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 3125153605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1286827955 \, \sqrt{-2 \, x + 1}}{1111320 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15313/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 1/1111320*(18605295*(-2*x + 1)^(11/2) + 956655*(-2*x + 1)^(9/2) - 678
093066*(-2*x + 1)^(7/2) + 2443710654*(-2*x + 1)^(5/2) - 3125153605*(-2*x + 1)^(3
/2) + 1286827955*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 = 0.211174, size = 181, normalized size = 1.18 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (18605295 \, x^{5} - 46991565 \, x^{4} - 122053374 \, x^{3} - 75153042 \, x^{2} - 10947400 \, x + 1660816\right )} \sqrt{-2 \, x + 1} - 76565 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{46675440 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/46675440*sqrt(21)*(sqrt(21)*(18605295*x^5 - 46991565*x^4 - 122053374*x^3 - 75
153042*x^2 - 10947400*x + 1660816)*sqrt(-2*x + 1) - 76565*(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 + 64)

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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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225271, size = 178, normalized size = 1.16 \[ \frac{15313}{9335088} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{18605295 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - 956655 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - 678093066 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 2443710654 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 3125153605 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1286827955 \, \sqrt{-2 \, x + 1}}{71124480 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15313/9335088*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) - 1/71124480*(18605295*(2*x - 1)^5*sqrt(-2*x + 1) - 956655*(2*x
- 1)^4*sqrt(-2*x + 1) - 678093066*(2*x - 1)^3*sqrt(-2*x + 1) - 2443710654*(2*x
- 1)^2*sqrt(-2*x + 1) + 3125153605*(-2*x + 1)^(3/2) - 1286827955*sqrt(-2*x + 1))
/(3*x + 2)^6