3.2005 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ -\frac{635 \sqrt{1-2 x}}{8232 (3 x+2)}-\frac{635 \sqrt{1-2 x}}{3528 (3 x+2)^2}+\frac{13 \sqrt{1-2 x}}{252 (3 x+2)^3}-\frac{\sqrt{1-2 x}}{252 (3 x+2)^4}-\frac{635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4116 \sqrt{21}} \]

[Out]

-Sqrt[1 - 2*x]/(252*(2 + 3*x)^4) + (13*Sqrt[1 - 2*x])/(252*(2 + 3*x)^3) - (635*S
qrt[1 - 2*x])/(3528*(2 + 3*x)^2) - (635*Sqrt[1 - 2*x])/(8232*(2 + 3*x)) - (635*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(4116*Sqrt[21])

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Rubi [A]  time = 0.131988, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{635 \sqrt{1-2 x}}{8232 (3 x+2)}-\frac{635 \sqrt{1-2 x}}{3528 (3 x+2)^2}+\frac{13 \sqrt{1-2 x}}{252 (3 x+2)^3}-\frac{\sqrt{1-2 x}}{252 (3 x+2)^4}-\frac{635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4116 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 - 2*x]/(252*(2 + 3*x)^4) + (13*Sqrt[1 - 2*x])/(252*(2 + 3*x)^3) - (635*S
qrt[1 - 2*x])/(3528*(2 + 3*x)^2) - (635*Sqrt[1 - 2*x])/(8232*(2 + 3*x)) - (635*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(4116*Sqrt[21])

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Rubi in Sympy [A]  time = 11.7493, size = 94, normalized size = 0.87 \[ - \frac{635 \sqrt{- 2 x + 1}}{8232 \left (3 x + 2\right )} - \frac{635 \sqrt{- 2 x + 1}}{3528 \left (3 x + 2\right )^{2}} + \frac{13 \sqrt{- 2 x + 1}}{252 \left (3 x + 2\right )^{3}} - \frac{\sqrt{- 2 x + 1}}{252 \left (3 x + 2\right )^{4}} - \frac{635 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{86436} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-635*sqrt(-2*x + 1)/(8232*(3*x + 2)) - 635*sqrt(-2*x + 1)/(3528*(3*x + 2)**2) +
13*sqrt(-2*x + 1)/(252*(3*x + 2)**3) - sqrt(-2*x + 1)/(252*(3*x + 2)**4) - 635*s
qrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/86436

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Mathematica [A]  time = 0.126633, size = 63, normalized size = 0.58 \[ -\frac{\sqrt{1-2 x} \left (17145 x^3+47625 x^2+39366 x+10190\right )}{8232 (3 x+2)^4}-\frac{635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4116 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(10190 + 39366*x + 47625*x^2 + 17145*x^3))/(8232*(2 + 3*x)^4) -
(635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(4116*Sqrt[21])

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Maple [A]  time = 0.017, size = 66, normalized size = 0.6 \[ 648\,{\frac{1}{ \left ( -4-6\,x \right ) ^{4}} \left ({\frac{635\, \left ( 1-2\,x \right ) ^{7/2}}{98784}}-{\frac{6985\, \left ( 1-2\,x \right ) ^{5/2}}{127008}}+{\frac{2717\, \left ( 1-2\,x \right ) ^{3/2}}{18144}}-{\frac{7171\,\sqrt{1-2\,x}}{54432}} \right ) }-{\frac{635\,\sqrt{21}}{86436}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

648*(635/98784*(1-2*x)^(7/2)-6985/127008*(1-2*x)^(5/2)+2717/18144*(1-2*x)^(3/2)-
7171/54432*(1-2*x)^(1/2))/(-4-6*x)^4-635/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2
))*21^(1/2)

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Maxima [A]  time = 1.50669, size = 149, normalized size = 1.38 \[ \frac{635}{172872} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17145 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 146685 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 399399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 351379 \, \sqrt{-2 \, x + 1}}{4116 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

635/172872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) + 1/4116*(17145*(-2*x + 1)^(7/2) - 146685*(-2*x + 1)^(5/2) + 399399*(-2*x
+ 1)^(3/2) - 351379*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 = 0.232563, size = 140, normalized size = 1.3 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (17145 \, x^{3} + 47625 \, x^{2} + 39366 \, x + 10190\right )} \sqrt{-2 \, x + 1} - 635 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{172872 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/172872*sqrt(21)*(sqrt(21)*(17145*x^3 + 47625*x^2 + 39366*x + 10190)*sqrt(-2*x
+ 1) - 635*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(21)*(3*x - 5) + 2
1*sqrt(-2*x + 1))/(3*x + 2)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.230436, size = 135, normalized size = 1.25 \[ \frac{635}{172872} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{17145 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 146685 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 399399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 351379 \, \sqrt{-2 \, x + 1}}{65856 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

635/172872*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 1/65856*(17145*(2*x - 1)^3*sqrt(-2*x + 1) + 146685*(2*x - 1)^2*s
qrt(-2*x + 1) - 399399*(-2*x + 1)^(3/2) + 351379*sqrt(-2*x + 1))/(3*x + 2)^4