3.2015 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)^4} \, dx\)

Optimal. Leaf size=80 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)^3}+\frac{5 \sqrt{1-2 x} (1867 x+1205)}{9261 (3 x+2)^2}-\frac{78710 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}} \]

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x]*(1205 + 1867*x))
/(9261*(2 + 3*x)^2) - (78710*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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Rubi [A]  time = 0.113402, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)^3}+\frac{5 \sqrt{1-2 x} (1867 x+1205)}{9261 (3 x+2)^2}-\frac{78710 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9261 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x]*(1205 + 1867*x))
/(9261*(2 + 3*x)^2) - (78710*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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Rubi in Sympy [A]  time = 11.8951, size = 70, normalized size = 0.88 \[ \frac{\sqrt{- 2 x + 1} \left (56010 x + 36150\right )}{55566 \left (3 x + 2\right )^{2}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{63 \left (3 x + 2\right )^{3}} - \frac{78710 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{194481} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(56010*x + 36150)/(55566*(3*x + 2)**2) + sqrt(-2*x + 1)*(5*x + 3)
**2/(63*(3*x + 2)**3) - 78710*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/194481

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Mathematica [A]  time = 0.105045, size = 58, normalized size = 0.72 \[ \frac{\frac{21 \sqrt{1-2 x} \left (31680 x^2+41155 x+13373\right )}{(3 x+2)^3}-78710 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{194481} \]

Antiderivative was successfully verified.

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

[Out]

((21*Sqrt[1 - 2*x]*(13373 + 41155*x + 31680*x^2))/(2 + 3*x)^3 - 78710*Sqrt[21]*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/194481

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Maple [A]  time = 0.017, size = 57, normalized size = 0.7 \[ 54\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ( -{\frac{3520\, \left ( 1-2\,x \right ) ^{5/2}}{27783}}+{\frac{20810\, \left ( 1-2\,x \right ) ^{3/2}}{35721}}-{\frac{3418\,\sqrt{1-2\,x}}{5103}} \right ) }-{\frac{78710\,\sqrt{21}}{194481}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

54*(-3520/27783*(1-2*x)^(5/2)+20810/35721*(1-2*x)^(3/2)-3418/5103*(1-2*x)^(1/2))
/(-4-6*x)^3-78710/194481*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.49929, size = 124, normalized size = 1.55 \[ \frac{39355}{194481} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (15840 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 72835 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 83741 \, \sqrt{-2 \, x + 1}\right )}}{9261 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39355/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 4/9261*(15840*(-2*x + 1)^(5/2) - 72835*(-2*x + 1)^(3/2) + 83741*sqrt(-
2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 0.239747, size = 120, normalized size = 1.5 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (31680 \, x^{2} + 41155 \, x + 13373\right )} \sqrt{-2 \, x + 1} + 39355 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{194481 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/194481*sqrt(21)*(sqrt(21)*(31680*x^2 + 41155*x + 13373)*sqrt(-2*x + 1) + 39355
*(27*x^3 + 54*x^2 + 36*x + 8)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x
+ 2)))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**3/(2+3*x)**4/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.214403, size = 113, normalized size = 1.41 \[ \frac{39355}{194481} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{15840 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 72835 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 83741 \, \sqrt{-2 \, x + 1}}{18522 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

39355/194481*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) + 1/18522*(15840*(2*x - 1)^2*sqrt(-2*x + 1) - 72835*(-2*x + 1)^(
3/2) + 83741*sqrt(-2*x + 1))/(3*x + 2)^3