3.1973 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=208 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}+\frac{12068887 \sqrt{1-2 x}}{1323 (3 x+2) (5 x+3)}+\frac{924025 \sqrt{1-2 x}}{1512 (3 x+2)^2 (5 x+3)}+\frac{16549 \sqrt{1-2 x}}{270 (3 x+2)^3 (5 x+3)}+\frac{1379 \sqrt{1-2 x}}{180 (3 x+2)^4 (5 x+3)}-\frac{323422735 \sqrt{1-2 x}}{3528 (5 x+3)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-323422735*Sqrt[1 - 2*x])/(3528*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^
5*(3 + 5*x)) + (1379*Sqrt[1 - 2*x])/(180*(2 + 3*x)^4*(3 + 5*x)) + (16549*Sqrt[1
- 2*x])/(270*(2 + 3*x)^3*(3 + 5*x)) + (924025*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2*(
3 + 5*x)) + (12068887*Sqrt[1 - 2*x])/(1323*(2 + 3*x)*(3 + 5*x)) - (2231141147*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) + 111650*Sqrt[55]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.461337, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{7 (1-2 x)^{3/2}}{15 (3 x+2)^5 (5 x+3)}+\frac{12068887 \sqrt{1-2 x}}{1323 (3 x+2) (5 x+3)}+\frac{924025 \sqrt{1-2 x}}{1512 (3 x+2)^2 (5 x+3)}+\frac{16549 \sqrt{1-2 x}}{270 (3 x+2)^3 (5 x+3)}+\frac{1379 \sqrt{1-2 x}}{180 (3 x+2)^4 (5 x+3)}-\frac{323422735 \sqrt{1-2 x}}{3528 (5 x+3)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-323422735*Sqrt[1 - 2*x])/(3528*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^
5*(3 + 5*x)) + (1379*Sqrt[1 - 2*x])/(180*(2 + 3*x)^4*(3 + 5*x)) + (16549*Sqrt[1
- 2*x])/(270*(2 + 3*x)^3*(3 + 5*x)) + (924025*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^2*(
3 + 5*x)) + (12068887*Sqrt[1 - 2*x])/(1323*(2 + 3*x)*(3 + 5*x)) - (2231141147*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) + 111650*Sqrt[55]*ArcTanh[Sqrt[5/
11]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 49.0816, size = 180, normalized size = 0.87 \[ \frac{7 \left (- 2 x + 1\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{5} \left (5 x + 3\right )} - \frac{64684547 \sqrt{- 2 x + 1}}{1176 \left (3 x + 2\right )} - \frac{13928935 \sqrt{- 2 x + 1}}{1512 \left (3 x + 2\right ) \left (5 x + 3\right )} + \frac{924025 \sqrt{- 2 x + 1}}{1512 \left (3 x + 2\right )^{2} \left (5 x + 3\right )} + \frac{16549 \sqrt{- 2 x + 1}}{270 \left (3 x + 2\right )^{3} \left (5 x + 3\right )} + \frac{1379 \sqrt{- 2 x + 1}}{180 \left (3 x + 2\right )^{4} \left (5 x + 3\right )} - \frac{2231141147 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{12348} + 111650 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7*(-2*x + 1)**(3/2)/(15*(3*x + 2)**5*(5*x + 3)) - 64684547*sqrt(-2*x + 1)/(1176*
(3*x + 2)) - 13928935*sqrt(-2*x + 1)/(1512*(3*x + 2)*(5*x + 3)) + 924025*sqrt(-2
*x + 1)/(1512*(3*x + 2)**2*(5*x + 3)) + 16549*sqrt(-2*x + 1)/(270*(3*x + 2)**3*(
5*x + 3)) + 1379*sqrt(-2*x + 1)/(180*(3*x + 2)**4*(5*x + 3)) - 2231141147*sqrt(2
1)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/12348 + 111650*sqrt(55)*atanh(sqrt(55)*sqrt(
-2*x + 1)/11)

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Mathematica [A]  time = 0.20682, size = 105, normalized size = 0.5 \[ -\frac{\sqrt{1-2 x} \left (130986207675 x^5+432275892930 x^4+570477768855 x^3+376323861626 x^2+124085884254 x+16360698684\right )}{5880 (3 x+2)^5 (5 x+3)}-\frac{2231141147 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{588 \sqrt{21}}+111650 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*(16360698684 + 124085884254*x + 376323861626*x^2 + 570477768855*
x^3 + 432275892930*x^4 + 130986207675*x^5))/(5880*(2 + 3*x)^5*(3 + 5*x)) - (2231
141147*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(588*Sqrt[21]) + 111650*Sqrt[55]*ArcTan
h[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.022, size = 109, normalized size = 0.5 \[ 972\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ({\frac{54012347\, \left ( 1-2\,x \right ) ^{9/2}}{7056}}-{\frac{46563587\, \left ( 1-2\,x \right ) ^{7/2}}{648}}+{\frac{307361449\, \left ( 1-2\,x \right ) ^{5/2}}{1215}}-{\frac{2308578797\, \left ( 1-2\,x \right ) ^{3/2}}{5832}}+{\frac{2709545797\,\sqrt{1-2\,x}}{11664}} \right ) }-{\frac{2231141147\,\sqrt{21}}{12348}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+6050\,{\frac{\sqrt{1-2\,x}}{-6/5-2\,x}}+111650\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

972*(54012347/7056*(1-2*x)^(9/2)-46563587/648*(1-2*x)^(7/2)[phone]/1215*(1-2*
x)^(5/2)-2308578797/5832*(1-2*x)^(3/2)[phone]/11664*(1-2*x)^(1/2))/(-4-6*x)^
5-2231141147/12348*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+6050*(1-2*x)^(1/
2)/(-6/5-2*x)+111650*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51451, size = 246, normalized size = 1.18 \[ -55825 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2231141147}{24696} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{130986207675 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 1519482824235 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 7049980295610 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 16353496911178 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 18965427342155 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 8796956467915 \, \sqrt{-2 \, x + 1}}{2940 \,{\left (1215 \,{\left (2 \, x - 1\right )}^{6} + 16848 \,{\left (2 \, x - 1\right )}^{5} + 97335 \,{\left (2 \, x - 1\right )}^{4} + 299880 \,{\left (2 \, x - 1\right )}^{3} + 519645 \,{\left (2 \, x - 1\right )}^{2} + 960400 \, x - 295323\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-55825*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))
) + 2231141147/24696*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*s
qrt(-2*x + 1))) + 1/2940*(130986207675*(-2*x + 1)^(11/2) - 1519482824235*(-2*x +
1)^(9/2) + 7049980295610*(-2*x + 1)^(7/2) - 16353496911178*(-2*x + 1)^(5/2) + 1
8965427342155*(-2*x + 1)^(3/2) - 8796956467915*sqrt(-2*x + 1))/(1215*(2*x - 1)^6
+ 16848*(2*x - 1)^5 + 97335*(2*x - 1)^4 + 299880*(2*x - 1)^3 + 519645*(2*x - 1)
^2 + 960400*x - 295323)

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Fricas [A]  time = 0.221, size = 269, normalized size = 1.29 \[ \frac{\sqrt{21}{\left (328251000 \, \sqrt{55} \sqrt{21}{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - \sqrt{21}{\left (130986207675 \, x^{5} + 432275892930 \, x^{4} + 570477768855 \, x^{3} + 376323861626 \, x^{2} + 124085884254 \, x + 16360698684\right )} \sqrt{-2 \, x + 1} + 11155705735 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{123480 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/123480*sqrt(21)*(328251000*sqrt(55)*sqrt(21)*(1215*x^6 + 4779*x^5 + 7830*x^4 +
6840*x^3 + 3360*x^2 + 880*x + 96)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) - sqrt(21)*(130986207675*x^5 + 432275892930*x^4 + 570477768855*x^3 + 37632
3861626*x^2 + 124085884254*x + 16360698684)*sqrt(-2*x + 1) + 11155705735*(1215*x
^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log((sqrt(21)*(3*x
- 5) + 21*sqrt(-2*x + 1))/(3*x + 2)))/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3
+ 3360*x^2 + 880*x + 96)

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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)**6/(3+5*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221103, size = 231, normalized size = 1.11 \[ -55825 \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2231141147}{24696} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{15125 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{21875000535 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 205345418670 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 722914128048 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1131203610530 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 663838720265 \, \sqrt{-2 \, x + 1}}{94080 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-55825*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-
2*x + 1))) + 2231141147/24696*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1)
)/(sqrt(21) + 3*sqrt(-2*x + 1))) - 15125*sqrt(-2*x + 1)/(5*x + 3) - 1/94080*(218
75000535*(2*x - 1)^4*sqrt(-2*x + 1) + 205345418670*(2*x - 1)^3*sqrt(-2*x + 1) +
722914128048*(2*x - 1)^2*sqrt(-2*x + 1) - 1131203610530*(-2*x + 1)^(3/2) + 66383
8720265*sqrt(-2*x + 1))/(3*x + 2)^5