3.2109 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=127 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^3}{1210 (5 x+3)^2}-\frac{1344 \sqrt{1-2 x} (3 x+2)^2}{33275 (5 x+3)}+\frac{441 \sqrt{1-2 x} (1125 x+3344)}{332750}-\frac{4557 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375 \sqrt{55}} \]

[Out]

(-71*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(1210*(3 + 5*x)^2) + (7*(2 + 3*x)^4)/(11*Sqrt[1
- 2*x]*(3 + 5*x)^2) - (1344*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(33275*(3 + 5*x)) + (441*
Sqrt[1 - 2*x]*(3344 + 1125*x))/332750 - (4557*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/(166375*Sqrt[55])

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Rubi [A]  time = 0.224547, antiderivative size = 127, 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 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^2}-\frac{71 \sqrt{1-2 x} (3 x+2)^3}{1210 (5 x+3)^2}-\frac{1344 \sqrt{1-2 x} (3 x+2)^2}{33275 (5 x+3)}+\frac{441 \sqrt{1-2 x} (1125 x+3344)}{332750}-\frac{4557 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{166375 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-71*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(1210*(3 + 5*x)^2) + (7*(2 + 3*x)^4)/(11*Sqrt[1
- 2*x]*(3 + 5*x)^2) - (1344*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(33275*(3 + 5*x)) + (441*
Sqrt[1 - 2*x]*(3344 + 1125*x))/332750 - (4557*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])
/(166375*Sqrt[55])

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Rubi in Sympy [A]  time = 25.2885, size = 112, normalized size = 0.88 \[ - \frac{71 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{1210 \left (5 x + 3\right )^{2}} - \frac{1344 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{33275 \left (5 x + 3\right )} + \frac{\sqrt{- 2 x + 1} \left (7441875 x + 22120560\right )}{4991250} - \frac{4557 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{9150625} + \frac{7 \left (3 x + 2\right )^{4}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-71*sqrt(-2*x + 1)*(3*x + 2)**3/(1210*(5*x + 3)**2) - 1344*sqrt(-2*x + 1)*(3*x +
2)**2/(33275*(5*x + 3)) + sqrt(-2*x + 1)*(7441875*x + 22120560)/4991250 - 4557*
sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/9150625 + 7*(3*x + 2)**4/(11*sqrt(-2*
x + 1)*(5*x + 3)**2)

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Mathematica [A]  time = 0.191047, size = 68, normalized size = 0.54 \[ \frac{-\frac{55 \left (5390550 x^4+42046290 x^3-764310 x^2-41668993 x-16342856\right )}{\sqrt{1-2 x} (5 x+3)^2}-9114 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{18301250} \]

Antiderivative was successfully verified.

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

[Out]

((-55*(-16342856 - 41668993*x - 764310*x^2 + 42046290*x^3 + 5390550*x^4))/(Sqrt[
1 - 2*x]*(3 + 5*x)^2) - 9114*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1830125
0

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Maple [A]  time = 0.02, size = 75, normalized size = 0.6 \[ -{\frac{81}{500} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1539}{625}\sqrt{1-2\,x}}+{\frac{16807}{5324}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{4}{33275\, \left ( -6-10\,x \right ) ^{2}} \left ({\frac{337}{20} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3729}{100}\sqrt{1-2\,x}} \right ) }-{\frac{4557\,\sqrt{55}}{9150625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/500*(1-2*x)^(3/2)+1539/625*(1-2*x)^(1/2)+16807/5324/(1-2*x)^(1/2)+4/33275*(3
37/20*(1-2*x)^(3/2)-3729/100*(1-2*x)^(1/2))/(-6-10*x)^2-4557/9150625*arctanh(1/1
1*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51364, size = 136, normalized size = 1.07 \[ -\frac{81}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4557}{18301250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1539}{625} \, \sqrt{-2 \, x + 1} + \frac{262616115 \,{\left (2 \, x - 1\right )}^{2} + 2310992332 \, x + 115533209}{3327500 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/500*(-2*x + 1)^(3/2) + 4557/18301250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1539/625*sqrt(-2*x + 1) + 1/3327500*(26261
6115*(2*x - 1)^2 + 2310992332*x + 115533209)/(25*(-2*x + 1)^(5/2) - 110*(-2*x +
1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 0.249558, size = 131, normalized size = 1.03 \[ \frac{\sqrt{55}{\left (4557 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) - \sqrt{55}{\left (5390550 \, x^{4} + 42046290 \, x^{3} - 764310 \, x^{2} - 41668993 \, x - 16342856\right )}\right )}}{18301250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18301250*sqrt(55)*(4557*(25*x^2 + 30*x + 9)*sqrt(-2*x + 1)*log((sqrt(55)*(5*x
- 8) + 55*sqrt(-2*x + 1))/(5*x + 3)) - sqrt(55)*(5390550*x^4 + 42046290*x^3 - 76
4310*x^2 - 41668993*x - 16342856))/((25*x^2 + 30*x + 9)*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((2+3*x)**5/(1-2*x)**(3/2)/(3+5*x)**3,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.221981, size = 128, normalized size = 1.01 \[ -\frac{81}{500} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{4557}{18301250} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1539}{625} \, \sqrt{-2 \, x + 1} + \frac{16807}{5324 \, \sqrt{-2 \, x + 1}} + \frac{1685 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 3729 \, \sqrt{-2 \, x + 1}}{3327500 \,{\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/500*(-2*x + 1)^(3/2) + 4557/18301250*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sq
rt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 1539/625*sqrt(-2*x + 1) + 16807/5
324/sqrt(-2*x + 1) + 1/3327500*(1685*(-2*x + 1)^(3/2) - 3729*sqrt(-2*x + 1))/(5*
x + 3)^2