3.2128 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=150 \[ -\frac{5805 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{1935 \sqrt{1-2 x}}{19208 (3 x+2)^2}-\frac{387 \sqrt{1-2 x}}{1372 (3 x+2)^3}+\frac{387}{686 \sqrt{1-2 x} (3 x+2)^3}+\frac{43}{294 (1-2 x)^{3/2} (3 x+2)^3}+\frac{1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228} \]

[Out]

1/(84*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + 43/(294*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 387/
(686*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (387*Sqrt[1 - 2*x])/(1372*(2 + 3*x)^3) - (1935
*Sqrt[1 - 2*x])/(19208*(2 + 3*x)^2) - (5805*Sqrt[1 - 2*x])/(134456*(2 + 3*x)) -
(1935*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/67228

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Rubi [A]  time = 0.163779, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5805 \sqrt{1-2 x}}{134456 (3 x+2)}-\frac{1935 \sqrt{1-2 x}}{19208 (3 x+2)^2}-\frac{387 \sqrt{1-2 x}}{1372 (3 x+2)^3}+\frac{387}{686 \sqrt{1-2 x} (3 x+2)^3}+\frac{43}{294 (1-2 x)^{3/2} (3 x+2)^3}+\frac{1}{84 (1-2 x)^{3/2} (3 x+2)^4}-\frac{1935 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{67228} \]

Antiderivative was successfully verified.

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

[Out]

1/(84*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + 43/(294*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 387/
(686*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (387*Sqrt[1 - 2*x])/(1372*(2 + 3*x)^3) - (1935
*Sqrt[1 - 2*x])/(19208*(2 + 3*x)^2) - (5805*Sqrt[1 - 2*x])/(134456*(2 + 3*x)) -
(1935*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/67228

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Rubi in Sympy [A]  time = 14.7344, size = 133, normalized size = 0.89 \[ - \frac{5805 \sqrt{- 2 x + 1}}{134456 \left (3 x + 2\right )} - \frac{1935 \sqrt{- 2 x + 1}}{19208 \left (3 x + 2\right )^{2}} - \frac{1935 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{470596} + \frac{129}{686 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{43}{686 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{2}} - \frac{43}{588 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{3}} + \frac{1}{84 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5805*sqrt(-2*x + 1)/(134456*(3*x + 2)) - 1935*sqrt(-2*x + 1)/(19208*(3*x + 2)**
2) - 1935*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/470596 + 129/(686*sqrt(-2*x
+ 1)*(3*x + 2)**2) + 43/(686*(-2*x + 1)**(3/2)*(3*x + 2)**2) - 43/(588*(-2*x + 1
)**(3/2)*(3*x + 2)**3) + 1/(84*(-2*x + 1)**(3/2)*(3*x + 2)**4)

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Mathematica [A]  time = 0.182707, size = 73, normalized size = 0.49 \[ \frac{-\frac{7 \left (1880820 x^5+3343680 x^4+1069281 x^3-1034451 x^2-611202 x-48490\right )}{(1-2 x)^{3/2} (3 x+2)^4}-11610 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2823576} \]

Antiderivative was successfully verified.

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

[Out]

((-7*(-48490 - 611202*x - 1034451*x^2 + 1069281*x^3 + 3343680*x^4 + 1880820*x^5)
)/((1 - 2*x)^(3/2)*(2 + 3*x)^4) - 11610*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]
])/2823576

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Maple [A]  time = 0.022, size = 84, normalized size = 0.6 \[{\frac{176}{50421} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2080}{117649}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{3888}{117649\, \left ( -4-6\,x \right ) ^{4}} \left ({\frac{5225}{192} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{119623}{576} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{921935}{1728} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2378705}{5184}\sqrt{1-2\,x}} \right ) }-{\frac{1935\,\sqrt{21}}{470596}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

176/50421/(1-2*x)^(3/2)+2080/117649/(1-2*x)^(1/2)+3888/117649*(5225/192*(1-2*x)^
(7/2)-119623/576*(1-2*x)^(5/2)+921935/1728*(1-2*x)^(3/2)-2378705/5184*(1-2*x)^(1
/2))/(-4-6*x)^4-1935/470596*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50868, size = 173, normalized size = 1.15 \[ \frac{1935}{941192} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{470205 \,{\left (2 \, x - 1\right )}^{5} + 4022865 \,{\left (2 \, x - 1\right )}^{4} + 12458691 \,{\left (2 \, x - 1\right )}^{3} + 15872031 \,{\left (2 \, x - 1\right )}^{2} + 11327232 \, x - 7353920}{201684 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 2401 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1935/941192*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) - 1/201684*(470205*(2*x - 1)^5 + 4022865*(2*x - 1)^4 + 12458691*(2*x - 1)
^3 + 15872031*(2*x - 1)^2 + 11327232*x - 7353920)/(81*(-2*x + 1)^(11/2) - 756*(-
2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(-2*x + 1)^(5/2) + 2401*(-2*x + 1)
^(3/2))

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Fricas [A]  time = 0.219016, size = 185, normalized size = 1.23 \[ \frac{\sqrt{7}{\left (5805 \, \sqrt{3}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (1880820 \, x^{5} + 3343680 \, x^{4} + 1069281 \, x^{3} - 1034451 \, x^{2} - 611202 \, x - 48490\right )}\right )}}{2823576 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2823576*sqrt(7)*(5805*sqrt(3)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 1
6)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2))
+ sqrt(7)*(1880820*x^5 + 3343680*x^4 + 1069281*x^3 - 1034451*x^2 - 611202*x - 48
490))/((162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*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((3+5*x)/(1-2*x)**(5/2)/(2+3*x)**5,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.230776, size = 163, normalized size = 1.09 \[ \frac{1935}{941192} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{16 \,{\left (780 \, x - 467\right )}}{352947 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{3 \,{\left (141075 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 1076607 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 2765805 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2378705 \, \sqrt{-2 \, x + 1}\right )}}{7529536 \,{\left (3 \, x + 2\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1935/941192*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) + 16/352947*(780*x - 467)/((2*x - 1)*sqrt(-2*x + 1)) - 3/7529536*
(141075*(2*x - 1)^3*sqrt(-2*x + 1) + 1076607*(2*x - 1)^2*sqrt(-2*x + 1) - 276580
5*(-2*x + 1)^(3/2) + 2378705*sqrt(-2*x + 1))/(3*x + 2)^4