3.2063 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=108 \[ -\frac{5 \sqrt{1-2 x}}{49 (3 x+2)}-\frac{5 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{4}{9 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{63 \sqrt{1-2 x} (3 x+2)^3}-\frac{10 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

[Out]

1/(63*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 4/(9*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[1 -
2*x])/(21*(2 + 3*x)^2) - (5*Sqrt[1 - 2*x])/(49*(2 + 3*x)) - (10*ArcTanh[Sqrt[3/
7]*Sqrt[1 - 2*x]])/(49*Sqrt[21])

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Rubi [A]  time = 0.114875, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 \sqrt{1-2 x}}{49 (3 x+2)}-\frac{5 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{4}{9 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{63 \sqrt{1-2 x} (3 x+2)^3}-\frac{10 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

1/(63*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 4/(9*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[1 -
2*x])/(21*(2 + 3*x)^2) - (5*Sqrt[1 - 2*x])/(49*(2 + 3*x)) - (10*ArcTanh[Sqrt[3/
7]*Sqrt[1 - 2*x]])/(49*Sqrt[21])

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Rubi in Sympy [A]  time = 11.6641, size = 94, normalized size = 0.87 \[ - \frac{5 \sqrt{- 2 x + 1}}{49 \left (3 x + 2\right )} - \frac{10 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1029} + \frac{10}{63 \sqrt{- 2 x + 1} \left (3 x + 2\right )} - \frac{1}{9 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{1}{63 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)/(1-2*x)**(3/2)/(2+3*x)**4,x)

[Out]

-5*sqrt(-2*x + 1)/(49*(3*x + 2)) - 10*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/
1029 + 10/(63*sqrt(-2*x + 1)*(3*x + 2)) - 1/(9*sqrt(-2*x + 1)*(3*x + 2)**2) + 1/
(63*sqrt(-2*x + 1)*(3*x + 2)**3)

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Mathematica [A]  time = 0.129925, size = 63, normalized size = 0.58 \[ \frac{90 x^3+145 x^2+57 x+1}{49 \sqrt{1-2 x} (3 x+2)^3}-\frac{10 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(1 + 57*x + 145*x^2 + 90*x^3)/(49*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (10*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21])

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Maple [A]  time = 0.019, size = 66, normalized size = 0.6 \[{\frac{88}{2401}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{216}{2401\, \left ( -4-6\,x \right ) ^{3}} \left ({\frac{113}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{1351}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{7007}{108}\sqrt{1-2\,x}} \right ) }-{\frac{10\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x)

[Out]

88/2401/(1-2*x)^(1/2)+216/2401*(113/12*(1-2*x)^(5/2)-1351/27*(1-2*x)^(3/2)+7007/
108*(1-2*x)^(1/2))/(-4-6*x)^3-10/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/
2)

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Maxima [A]  time = 1.5195, size = 136, normalized size = 1.26 \[ \frac{5}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 280 \,{\left (2 \, x - 1\right )}^{2} + 1078 \, x - 231\right )}}{49 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)/((3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

5/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))
) - 2/49*(45*(2*x - 1)^3 + 280*(2*x - 1)^2 + 1078*x - 231)/(27*(-2*x + 1)^(7/2)
- 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 343*sqrt(-2*x + 1))

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Fricas [A]  time = 0.224417, size = 136, normalized size = 1.26 \[ \frac{\sqrt{21}{\left (5 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (90 \, x^{3} + 145 \, x^{2} + 57 \, x + 1\right )}\right )}}{1029 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)/((3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/1029*sqrt(21)*(5*(27*x^3 + 54*x^2 + 36*x + 8)*sqrt(-2*x + 1)*log((sqrt(21)*(3*
x - 5) + 21*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(21)*(90*x^3 + 145*x^2 + 57*x + 1))
/((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(-2*x + 1))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)/(1-2*x)**(3/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.220105, size = 126, normalized size = 1.17 \[ \frac{5}{1029} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{88}{2401 \, \sqrt{-2 \, x + 1}} - \frac{1017 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 5404 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 7007 \, \sqrt{-2 \, x + 1}}{9604 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)/((3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

5/1029*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2
*x + 1))) + 88/2401/sqrt(-2*x + 1) - 1/9604*(1017*(2*x - 1)^2*sqrt(-2*x + 1) - 5
404*(-2*x + 1)^(3/2) + 7007*sqrt(-2*x + 1))/(3*x + 2)^3