3.2006 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=130 \[ -\frac{351 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{117 \sqrt{1-2 x}}{2744 (3 x+2)^2}-\frac{117 \sqrt{1-2 x}}{980 (3 x+2)^3}+\frac{341 \sqrt{1-2 x}}{8820 (3 x+2)^4}-\frac{\sqrt{1-2 x}}{315 (3 x+2)^5}-\frac{117 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604} \]

[Out]

-Sqrt[1 - 2*x]/(315*(2 + 3*x)^5) + (341*Sqrt[1 - 2*x])/(8820*(2 + 3*x)^4) - (117
*Sqrt[1 - 2*x])/(980*(2 + 3*x)^3) - (117*Sqrt[1 - 2*x])/(2744*(2 + 3*x)^2) - (35
1*Sqrt[1 - 2*x])/(19208*(2 + 3*x)) - (117*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/9604

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Rubi [A]  time = 0.15353, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{351 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{117 \sqrt{1-2 x}}{2744 (3 x+2)^2}-\frac{117 \sqrt{1-2 x}}{980 (3 x+2)^3}+\frac{341 \sqrt{1-2 x}}{8820 (3 x+2)^4}-\frac{\sqrt{1-2 x}}{315 (3 x+2)^5}-\frac{117 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^2/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

-Sqrt[1 - 2*x]/(315*(2 + 3*x)^5) + (341*Sqrt[1 - 2*x])/(8820*(2 + 3*x)^4) - (117
*Sqrt[1 - 2*x])/(980*(2 + 3*x)^3) - (117*Sqrt[1 - 2*x])/(2744*(2 + 3*x)^2) - (35
1*Sqrt[1 - 2*x])/(19208*(2 + 3*x)) - (117*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/9604

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Rubi in Sympy [A]  time = 13.6707, size = 112, normalized size = 0.86 \[ - \frac{351 \sqrt{- 2 x + 1}}{19208 \left (3 x + 2\right )} - \frac{117 \sqrt{- 2 x + 1}}{2744 \left (3 x + 2\right )^{2}} - \frac{117 \sqrt{- 2 x + 1}}{980 \left (3 x + 2\right )^{3}} + \frac{341 \sqrt{- 2 x + 1}}{8820 \left (3 x + 2\right )^{4}} - \frac{\sqrt{- 2 x + 1}}{315 \left (3 x + 2\right )^{5}} - \frac{117 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{67228} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**2/(2+3*x)**6/(1-2*x)**(1/2),x)

[Out]

-351*sqrt(-2*x + 1)/(19208*(3*x + 2)) - 117*sqrt(-2*x + 1)/(2744*(3*x + 2)**2) -
117*sqrt(-2*x + 1)/(980*(3*x + 2)**3) + 341*sqrt(-2*x + 1)/(8820*(3*x + 2)**4)
- sqrt(-2*x + 1)/(315*(3*x + 2)**5) - 117*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)
/7)/67228

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Mathematica [A]  time = 0.127829, size = 68, normalized size = 0.52 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (426465 x^4+1468935 x^3+2110212 x^2+1327058 x+298748\right )}{(3 x+2)^5}-10530 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{6050520} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^2/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

((-21*Sqrt[1 - 2*x]*(298748 + 1327058*x + 2110212*x^2 + 1468935*x^3 + 426465*x^4
))/(2 + 3*x)^5 - 10530*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/6050520

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Maple [A]  time = 0.017, size = 75, normalized size = 0.6 \[ -3888\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{117\, \left ( 1-2\,x \right ) ^{9/2}}{153664}}+{\frac{13\, \left ( 1-2\,x \right ) ^{7/2}}{1568}}-{\frac{26\, \left ( 1-2\,x \right ) ^{5/2}}{735}}+{\frac{77587\, \left ( 1-2\,x \right ) ^{3/2}}{1143072}}-{\frac{5287\,\sqrt{1-2\,x}}{108864}} \right ) }-{\frac{117\,\sqrt{21}}{67228}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^2/(2+3*x)^6/(1-2*x)^(1/2),x)

[Out]

-3888*(-117/153664*(1-2*x)^(9/2)+13/1568*(1-2*x)^(7/2)-26/735*(1-2*x)^(5/2)+7758
7/1143072*(1-2*x)^(3/2)-5287/108864*(1-2*x)^(1/2))/(-4-6*x)^5-117/67228*arctanh(
1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.56263, size = 173, normalized size = 1.33 \[ \frac{117}{134456} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{426465 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4643730 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 19813248 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 38017630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 27201615 \, \sqrt{-2 \, x + 1}}{144060 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2/((3*x + 2)^6*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

117/134456*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 1/144060*(426465*(-2*x + 1)^(9/2) - 4643730*(-2*x + 1)^(7/2) + 19813248*
(-2*x + 1)^(5/2) - 38017630*(-2*x + 1)^(3/2) + 27201615*sqrt(-2*x + 1))/(243*(2*
x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x -
19208)

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Fricas [A]  time = 0.234513, size = 170, normalized size = 1.31 \[ \frac{\sqrt{7}{\left (1755 \, \sqrt{3}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) - \sqrt{7}{\left (426465 \, x^{4} + 1468935 \, x^{3} + 2110212 \, x^{2} + 1327058 \, x + 298748\right )} \sqrt{-2 \, x + 1}\right )}}{2016840 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2/((3*x + 2)^6*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

1/2016840*sqrt(7)*(1755*sqrt(3)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) - sqrt(7)*(4
26465*x^4 + 1468935*x^3 + 2110212*x^2 + 1327058*x + 298748)*sqrt(-2*x + 1))/(243
*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**6/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.225632, size = 157, normalized size = 1.21 \[ \frac{117}{134456} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{426465 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 4643730 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 19813248 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 38017630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 27201615 \, \sqrt{-2 \, x + 1}}{4609920 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2/((3*x + 2)^6*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

117/134456*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) - 1/4609920*(426465*(2*x - 1)^4*sqrt(-2*x + 1) + 4643730*(2*x - 1)
^3*sqrt(-2*x + 1) + 19813248*(2*x - 1)^2*sqrt(-2*x + 1) - 38017630*(-2*x + 1)^(3
/2) + 27201615*sqrt(-2*x + 1))/(3*x + 2)^5