3.1784 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=110 \[ \frac{(1-2 x)^{3/2}}{84 (3 x+2)^4}+\frac{15 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{5 \sqrt{1-2 x}}{392 (3 x+2)^2}-\frac{5 \sqrt{1-2 x}}{28 (3 x+2)^3}+\frac{5 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372} \]

[Out]

(1 - 2*x)^(3/2)/(84*(2 + 3*x)^4) - (5*Sqrt[1 - 2*x])/(28*(2 + 3*x)^3) + (5*Sqrt[
1 - 2*x])/(392*(2 + 3*x)^2) + (15*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (5*Sqrt[3/7]
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1372

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Rubi [A]  time = 0.103278, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(1-2 x)^{3/2}}{84 (3 x+2)^4}+\frac{15 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{5 \sqrt{1-2 x}}{392 (3 x+2)^2}-\frac{5 \sqrt{1-2 x}}{28 (3 x+2)^3}+\frac{5 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372} \]

Antiderivative was successfully verified.

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

[Out]

(1 - 2*x)^(3/2)/(84*(2 + 3*x)^4) - (5*Sqrt[1 - 2*x])/(28*(2 + 3*x)^3) + (5*Sqrt[
1 - 2*x])/(392*(2 + 3*x)^2) + (15*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (5*Sqrt[3/7]
*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1372

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Rubi in Sympy [A]  time = 11.0725, size = 94, normalized size = 0.85 \[ \frac{\left (- 2 x + 1\right )^{\frac{3}{2}}}{84 \left (3 x + 2\right )^{4}} + \frac{15 \sqrt{- 2 x + 1}}{2744 \left (3 x + 2\right )} + \frac{5 \sqrt{- 2 x + 1}}{392 \left (3 x + 2\right )^{2}} - \frac{5 \sqrt{- 2 x + 1}}{28 \left (3 x + 2\right )^{3}} + \frac{5 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{9604} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(3/2)/(84*(3*x + 2)**4) + 15*sqrt(-2*x + 1)/(2744*(3*x + 2)) + 5*sqr
t(-2*x + 1)/(392*(3*x + 2)**2) - 5*sqrt(-2*x + 1)/(28*(3*x + 2)**3) + 5*sqrt(21)
*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/9604

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Mathematica [A]  time = 0.103055, size = 63, normalized size = 0.57 \[ \frac{\frac{7 \sqrt{1-2 x} \left (1215 x^3+3375 x^2-1726 x-2062\right )}{(3 x+2)^4}+30 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{57624} \]

Antiderivative was successfully verified.

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

[Out]

((7*Sqrt[1 - 2*x]*(-2062 - 1726*x + 3375*x^2 + 1215*x^3))/(2 + 3*x)^4 + 30*Sqrt[
21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/57624

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Maple [A]  time = 0.016, size = 66, normalized size = 0.6 \[ -1296\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{5\, \left ( 1-2\,x \right ) ^{7/2}}{21952}}-{\frac{55\, \left ( 1-2\,x \right ) ^{5/2}}{28224}}+{\frac{209\, \left ( 1-2\,x \right ) ^{3/2}}{108864}}+{\frac{5\,\sqrt{1-2\,x}}{1728}} \right ) }+{\frac{5\,\sqrt{21}}{9604}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1296*(5/21952*(1-2*x)^(7/2)-55/28224*(1-2*x)^(5/2)+209/108864*(1-2*x)^(3/2)+5/1
728*(1-2*x)^(1/2))/(-4-6*x)^4+5/9604*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2
)

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Maxima [A]  time = 1.4952, size = 149, normalized size = 1.35 \[ -\frac{5}{19208} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{1215 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 10395 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 10241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 15435 \, \sqrt{-2 \, x + 1}}{4116 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/19208*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1
))) - 1/4116*(1215*(-2*x + 1)^(7/2) - 10395*(-2*x + 1)^(5/2) + 10241*(-2*x + 1)^
(3/2) + 15435*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)
^2 + 8232*x - 1715)

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Fricas [A]  time = 0.211163, size = 149, normalized size = 1.35 \[ \frac{\sqrt{7}{\left (15 \, \sqrt{3}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (1215 \, x^{3} + 3375 \, x^{2} - 1726 \, x - 2062\right )} \sqrt{-2 \, x + 1}\right )}}{57624 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/57624*sqrt(7)*(15*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(7
)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(7)*(1215*x^3 + 3375*x^
2 - 1726*x - 2062)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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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)*(1-2*x)**(1/2)/(2+3*x)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215421, size = 135, normalized size = 1.23 \[ -\frac{5}{19208} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1215 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 10395 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 10241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 15435 \, \sqrt{-2 \, x + 1}}{65856 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/19208*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/65856*(1215*(2*x - 1)^3*sqrt(-2*x + 1) + 10395*(2*x - 1)^2*sqrt(
-2*x + 1) - 10241*(-2*x + 1)^(3/2) - 15435*sqrt(-2*x + 1))/(3*x + 2)^4