3.1963 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=141 \[ -\frac{(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac{32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac{2794 \sqrt{1-2 x}}{78125}-\frac{2794 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

[Out]

(2794*Sqrt[1 - 2*x])/78125 + (254*(1 - 2*x)^(3/2))/46875 - (32*(1 - 2*x)^(5/2)*(
2 + 3*x)^2)/4125 + (39*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/275 - ((1 - 2*x)^(5/2)*(2 +
3*x)^4)/(5*(3 + 5*x)) - ((1 - 2*x)^(5/2)*(1347116 + 1110975*x))/3609375 - (2794*
Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi [A]  time = 0.250919, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(1-2 x)^{5/2} (3 x+2)^4}{5 (5 x+3)}+\frac{39}{275} (1-2 x)^{5/2} (3 x+2)^3-\frac{32 (1-2 x)^{5/2} (3 x+2)^2}{4125}+\frac{254 (1-2 x)^{3/2}}{46875}-\frac{(1-2 x)^{5/2} (1110975 x+1347116)}{3609375}+\frac{2794 \sqrt{1-2 x}}{78125}-\frac{2794 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{78125} \]

Antiderivative was successfully verified.

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

[Out]

(2794*Sqrt[1 - 2*x])/78125 + (254*(1 - 2*x)^(3/2))/46875 - (32*(1 - 2*x)^(5/2)*(
2 + 3*x)^2)/4125 + (39*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/275 - ((1 - 2*x)^(5/2)*(2 +
3*x)^4)/(5*(3 + 5*x)) - ((1 - 2*x)^(5/2)*(1347116 + 1110975*x))/3609375 - (2794*
Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/78125

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Rubi in Sympy [A]  time = 28.9059, size = 121, normalized size = 0.86 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{4}}{5 \left (5 x + 3\right )} + \frac{39 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{275} - \frac{32 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2}}{4125} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (3332925 x + 4041348\right )}{10828125} + \frac{254 \left (- 2 x + 1\right )^{\frac{3}{2}}}{46875} + \frac{2794 \sqrt{- 2 x + 1}}{78125} - \frac{2794 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{390625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(3*x + 2)**4/(5*(5*x + 3)) + 39*(-2*x + 1)**(5/2)*(3*x + 2)**
3/275 - 32*(-2*x + 1)**(5/2)*(3*x + 2)**2/4125 - (-2*x + 1)**(5/2)*(3332925*x +
4041348)/10828125 + 254*(-2*x + 1)**(3/2)/46875 + 2794*sqrt(-2*x + 1)/78125 - 27
94*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/390625

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Mathematica [A]  time = 0.142793, size = 78, normalized size = 0.55 \[ \frac{\frac{5 \sqrt{1-2 x} \left (212625000 x^6+237037500 x^5-173598750 x^4-214071975 x^3+85482115 x^2+50081215 x-15982128\right )}{5 x+3}-645414 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{90234375} \]

Antiderivative was successfully verified.

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

[Out]

((5*Sqrt[1 - 2*x]*(-15982128 + 50081215*x + 85482115*x^2 - 214071975*x^3 - 17359
8750*x^4 + 237037500*x^5 + 212625000*x^6))/(3 + 5*x) - 645414*Sqrt[55]*ArcTanh[S
qrt[5/11]*Sqrt[1 - 2*x]])/90234375

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Maple [A]  time = 0.017, size = 90, normalized size = 0.6 \[ -{\frac{81}{1100} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}+{\frac{111}{250} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{12393}{17500} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{24}{15625} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{52}{9375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2816}{78125}\sqrt{1-2\,x}}+{\frac{242}{390625}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{2794\,\sqrt{55}}{390625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/1100*(1-2*x)^(11/2)+111/250*(1-2*x)^(9/2)-12393/17500*(1-2*x)^(7/2)+24/15625
*(1-2*x)^(5/2)+52/9375*(1-2*x)^(3/2)+2816/78125*(1-2*x)^(1/2)+242/390625*(1-2*x)
^(1/2)/(-6/5-2*x)-2794/390625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51153, size = 144, normalized size = 1.02 \[ -\frac{81}{1100} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{111}{250} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{12393}{17500} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{24}{15625} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{52}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1397}{390625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2816}{78125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81/1100*(-2*x + 1)^(11/2) + 111/250*(-2*x + 1)^(9/2) - 12393/17500*(-2*x + 1)^(
7/2) + 24/15625*(-2*x + 1)^(5/2) + 52/9375*(-2*x + 1)^(3/2) + 1397/390625*sqrt(5
5)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2816/7812
5*sqrt(-2*x + 1) - 121/78125*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.214952, size = 128, normalized size = 0.91 \[ \frac{\sqrt{5}{\left (322707 \, \sqrt{11}{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + \sqrt{5}{\left (212625000 \, x^{6} + 237037500 \, x^{5} - 173598750 \, x^{4} - 214071975 \, x^{3} + 85482115 \, x^{2} + 50081215 \, x - 15982128\right )} \sqrt{-2 \, x + 1}\right )}}{90234375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90234375*sqrt(5)*(322707*sqrt(11)*(5*x + 3)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11
)*sqrt(-2*x + 1))/(5*x + 3)) + sqrt(5)*(212625000*x^6 + 237037500*x^5 - 17359875
0*x^4 - 214071975*x^3 + 85482115*x^2 + 50081215*x - 15982128)*sqrt(-2*x + 1))/(5
*x + 3)

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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((1-2*x)**(5/2)*(2+3*x)**4/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216255, size = 186, normalized size = 1.32 \[ \frac{81}{1100} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{111}{250} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{12393}{17500} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{24}{15625} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{52}{9375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1397}{390625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2816}{78125} \, \sqrt{-2 \, x + 1} - \frac{121 \, \sqrt{-2 \, x + 1}}{78125 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/1100*(2*x - 1)^5*sqrt(-2*x + 1) + 111/250*(2*x - 1)^4*sqrt(-2*x + 1) + 12393/
17500*(2*x - 1)^3*sqrt(-2*x + 1) + 24/15625*(2*x - 1)^2*sqrt(-2*x + 1) + 52/9375
*(-2*x + 1)^(3/2) + 1397/390625*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x +
1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 2816/78125*sqrt(-2*x + 1) - 121/78125*sqrt
(-2*x + 1)/(5*x + 3)