3.1809 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=147 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{18 (3 x+2)^6}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{945 (3 x+2)^5}-\frac{\sqrt{1-2 x} (160029 x+98995)}{476280 (3 x+2)^4}+\frac{43957 \sqrt{1-2 x}}{3111696 (3 x+2)}+\frac{43957 \sqrt{1-2 x}}{1333584 (3 x+2)^2}+\frac{43957 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}} \]

[Out]

(43957*Sqrt[1 - 2*x])/(1333584*(2 + 3*x)^2) + (43957*Sqrt[1 - 2*x])/(3111696*(2
+ 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(945*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 +
5*x)^3)/(18*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(98995 + 160029*x))/(476280*(2 + 3*x)
^4) + (43957*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1555848*Sqrt[21])

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Rubi [A]  time = 0.20289, antiderivative size = 147, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{18 (3 x+2)^6}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{945 (3 x+2)^5}-\frac{\sqrt{1-2 x} (160029 x+98995)}{476280 (3 x+2)^4}+\frac{43957 \sqrt{1-2 x}}{3111696 (3 x+2)}+\frac{43957 \sqrt{1-2 x}}{1333584 (3 x+2)^2}+\frac{43957 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1555848 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(43957*Sqrt[1 - 2*x])/(1333584*(2 + 3*x)^2) + (43957*Sqrt[1 - 2*x])/(3111696*(2
+ 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(945*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(3 +
5*x)^3)/(18*(2 + 3*x)^6) - (Sqrt[1 - 2*x]*(98995 + 160029*x))/(476280*(2 + 3*x)
^4) + (43957*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1555848*Sqrt[21])

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Rubi in Sympy [A]  time = 23.1395, size = 129, normalized size = 0.88 \[ \frac{43957 \sqrt{- 2 x + 1}}{3111696 \left (3 x + 2\right )} + \frac{43957 \sqrt{- 2 x + 1}}{1333584 \left (3 x + 2\right )^{2}} - \frac{\sqrt{- 2 x + 1} \left (3360609 x + 2078895\right )}{10001880 \left (3 x + 2\right )^{4}} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{945 \left (3 x + 2\right )^{5}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{18 \left (3 x + 2\right )^{6}} + \frac{43957 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{32672808} \]

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)**7,x)

[Out]

43957*sqrt(-2*x + 1)/(3111696*(3*x + 2)) + 43957*sqrt(-2*x + 1)/(1333584*(3*x +
2)**2) - sqrt(-2*x + 1)*(3360609*x + 2078895)/(10001880*(3*x + 2)**4) - 53*sqrt(
-2*x + 1)*(5*x + 3)**2/(945*(3*x + 2)**5) - sqrt(-2*x + 1)*(5*x + 3)**3/(18*(3*x
+ 2)**6) + 43957*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/32672808

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Mathematica [A]  time = 0.123963, size = 73, normalized size = 0.5 \[ \frac{\frac{21 \sqrt{1-2 x} \left (53407755 x^5+219565215 x^4+127601514 x^3-139462938 x^2-150340360 x-36741296\right )}{(3 x+2)^6}+439570 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{326728080} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(-36741296 - 150340360*x - 139462938*x^2 + 127601514*x^3 + 21
9565215*x^4 + 53407755*x^5))/(2 + 3*x)^6 + 439570*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqr
t[1 - 2*x]])/326728080

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Maple [A]  time = 0.018, size = 84, normalized size = 0.6 \[ -11664\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ({\frac{43957\, \left ( 1-2\,x \right ) ^{11/2}}{74680704}}-{\frac{747269\, \left ( 1-2\,x \right ) ^{9/2}}{96018048}}+{\frac{1058581\, \left ( 1-2\,x \right ) ^{7/2}}{34292160}}-{\frac{1354639\, \left ( 1-2\,x \right ) ^{5/2}}{34292160}}-{\frac{630947\, \left ( 1-2\,x \right ) ^{3/2}}{52907904}}+{\frac{307699\,\sqrt{1-2\,x}}{7558272}} \right ) }+{\frac{43957\,\sqrt{21}}{32672808}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11664*(43957/74680704*(1-2*x)^(11/2)-747269/96018048*(1-2*x)^(9/2)+1058581/3429
2160*(1-2*x)^(7/2)-1354639/34292160*(1-2*x)^(5/2)-630947/52907904*(1-2*x)^(3/2)+
307699/7558272*(1-2*x)^(1/2))/(-4-6*x)^6+43957/32672808*arctanh(1/7*21^(1/2)*(1-
2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.53615, size = 197, normalized size = 1.34 \[ -\frac{43957}{65345616} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{53407755 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 706169205 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2801005326 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 3584374794 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 1082074105 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 3693926495 \, \sqrt{-2 \, x + 1}}{7779240 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-43957/65345616*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 1/7779240*(53407755*(-2*x + 1)^(11/2) - 706169205*(-2*x + 1)^(9/2)
+ 2801005326*(-2*x + 1)^(7/2) - 3584374794*(-2*x + 1)^(5/2) - 1082074105*(-2*x +
1)^(3/2) + 3693926495*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59
535*(2*x - 1)^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]  time = 0.21231, size = 181, normalized size = 1.23 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (53407755 \, x^{5} + 219565215 \, x^{4} + 127601514 \, x^{3} - 139462938 \, x^{2} - 150340360 \, x - 36741296\right )} \sqrt{-2 \, x + 1} + 219785 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{326728080 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/326728080*sqrt(21)*(sqrt(21)*(53407755*x^5 + 219565215*x^4 + 127601514*x^3 - 1
39462938*x^2 - 150340360*x - 36741296)*sqrt(-2*x + 1) + 219785*(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*s
qrt(-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((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222565, size = 178, normalized size = 1.21 \[ -\frac{43957}{65345616} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{53407755 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 706169205 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 2801005326 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 3584374794 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 1082074105 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3693926495 \, \sqrt{-2 \, x + 1}}{497871360 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-43957/65345616*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) +
3*sqrt(-2*x + 1))) + 1/497871360*(53407755*(2*x - 1)^5*sqrt(-2*x + 1) + 70616920
5*(2*x - 1)^4*sqrt(-2*x + 1) + 2801005326*(2*x - 1)^3*sqrt(-2*x + 1) + 358437479
4*(2*x - 1)^2*sqrt(-2*x + 1) + 1082074105*(-2*x + 1)^(3/2) - 3693926495*sqrt(-2*
x + 1))/(3*x + 2)^6