3.1838 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=153 \[ \frac{34655 \sqrt{1-2 x}}{77 (5 x+3)}-\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)^2}+\frac{139 \sqrt{1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac{43467}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{66325}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^
2) + (139*Sqrt[1 - 2*x])/(14*(2 + 3*x)*(3 + 5*x)^2) + (34655*Sqrt[1 - 2*x])/(77*
(3 + 5*x)) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (66325*Sqrt[
5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.301441, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{34655 \sqrt{1-2 x}}{77 (5 x+3)}-\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)^2}+\frac{139 \sqrt{1-2 x}}{14 (3 x+2) (5 x+3)^2}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^2}+\frac{43467}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{66325}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1045*Sqrt[1 - 2*x])/(14*(3 + 5*x)^2) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^
2) + (139*Sqrt[1 - 2*x])/(14*(2 + 3*x)*(3 + 5*x)^2) + (34655*Sqrt[1 - 2*x])/(77*
(3 + 5*x)) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (66325*Sqrt[
5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi in Sympy [A]  time = 35.4562, size = 131, normalized size = 0.86 \[ \frac{34655 \sqrt{- 2 x + 1}}{77 \left (5 x + 3\right )} - \frac{1045 \sqrt{- 2 x + 1}}{14 \left (5 x + 3\right )^{2}} + \frac{139 \sqrt{- 2 x + 1}}{14 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{\sqrt{- 2 x + 1}}{2 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{43467 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} - \frac{66325 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{121} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

34655*sqrt(-2*x + 1)/(77*(5*x + 3)) - 1045*sqrt(-2*x + 1)/(14*(5*x + 3)**2) + 13
9*sqrt(-2*x + 1)/(14*(3*x + 2)*(5*x + 3)**2) + sqrt(-2*x + 1)/(2*(3*x + 2)**2*(5
*x + 3)**2) + 43467*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49 - 66325*sqrt(55
)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/121

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Mathematica [A]  time = 0.151565, size = 101, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (3118950 x^3+5926515 x^2+3748007 x+788875\right )}{154 (3 x+2)^2 (5 x+3)^2}+\frac{43467}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{66325}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(788875 + 3748007*x + 5926515*x^2 + 3118950*x^3))/(154*(2 + 3*x)^
2*(3 + 5*x)^2) + (43467*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (66325*S
qrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]  time = 0.022, size = 94, normalized size = 0.6 \[ -972\,{\frac{1}{ \left ( -4-6\,x \right ) ^{2}} \left ({\frac{209\, \left ( 1-2\,x \right ) ^{3/2}}{252}}-{\frac{211\,\sqrt{1-2\,x}}{108}} \right ) }+{\frac{43467\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+2500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{199\, \left ( 1-2\,x \right ) ^{3/2}}{220}}+{\frac{197\,\sqrt{1-2\,x}}{100}} \right ) }-{\frac{66325\,\sqrt{55}}{121}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-972*(209/252*(1-2*x)^(3/2)-211/108*(1-2*x)^(1/2))/(-4-6*x)^2+43467/49*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2500*(-199/220*(1-2*x)^(3/2)+197/100*(1-2*x)
^(1/2))/(-6-10*x)^2-66325/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.49512, size = 197, normalized size = 1.29 \[ \frac{66325}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{43467}{98} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1559475 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 10604940 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 24027469 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 18137504 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

66325/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 43467/98*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-
2*x + 1))) - 2/77*(1559475*(-2*x + 1)^(7/2) - 10604940*(-2*x + 1)^(5/2) + 240274
69*(-2*x + 1)^(3/2) - 18137504*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)
^3 + 6934*(2*x - 1)^2 + 20944*x - 4543)

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Fricas [A]  time = 0.224679, size = 240, normalized size = 1.57 \[ \frac{\sqrt{11} \sqrt{7}{\left (464275 \, \sqrt{7} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 478137 \, \sqrt{11} \sqrt{3}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (3118950 \, x^{3} + 5926515 \, x^{2} + 3748007 \, x + 788875\right )} \sqrt{-2 \, x + 1}\right )}}{11858 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11858*sqrt(11)*sqrt(7)*(464275*sqrt(7)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 +
228*x + 36)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*x + 3)) + 47
8137*sqrt(11)*sqrt(3)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(7)*(3
*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(11)*sqrt(7)*(3118950*x^3 +
5926515*x^2 + 3748007*x + 788875)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2
+ 228*x + 36)

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Sympy [A]  time = 100.473, size = 614, normalized size = 4.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3708*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/4 + log(sqrt(21)*s
qrt(-2*x + 1)/7 + 1)/4 - 1/(4*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) - 1/(4*(sqrt(21)*
sqrt(-2*x + 1)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 504*Piecewise((sqrt(21)
*(3*log(sqrt(21)*sqrt(-2*x + 1)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(-2*x + 1)/7 + 1)
/16 + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(-2*x + 1)/7
+ 1)**2) + 3/(16*(sqrt(21)*sqrt(-2*x + 1)/7 - 1)) - 1/(16*(sqrt(21)*sqrt(-2*x +
1)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3))) + 10100*Piecewise((sqrt(55)*(-log
(sqrt(55)*sqrt(-2*x + 1)/11 - 1)/4 + log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/4 - 1/(
4*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)))/60
5, (x <= 1/2) & (x > -3/5))) + 2200*Piecewise((sqrt(55)*(3*log(sqrt(55)*sqrt(-2*
x + 1)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(-2*x + 1)/11 + 1)/16 + 3/(16*(sqrt(55)*s
qrt(-2*x + 1)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 + 1)**2) + 3/(16*(sqr
t(55)*sqrt(-2*x + 1)/11 - 1)) - 1/(16*(sqrt(55)*sqrt(-2*x + 1)/11 - 1)**2))/6655
, (x <= 1/2) & (x > -3/5))) - 18360*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(-2*
x + 1)/7)/21, -2*x + 1 > 7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/21, -
2*x + 1 < 7/3)) + 30600*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/5
5, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 <
11/5))

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GIAC/XCAS [A]  time = 0.251543, size = 200, normalized size = 1.31 \[ \frac{66325}{242} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{43467}{98} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{2 \,{\left (1559475 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 10604940 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 24027469 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 18137504 \, \sqrt{-2 \, x + 1}\right )}}{77 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

66325/242*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 43467/98*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sq
rt(21) + 3*sqrt(-2*x + 1))) + 2/77*(1559475*(2*x - 1)^3*sqrt(-2*x + 1) + 1060494
0*(2*x - 1)^2*sqrt(-2*x + 1) - 24027469*(-2*x + 1)^(3/2) + 18137504*sqrt(-2*x +
1))/(15*(2*x - 1)^2 + 136*x + 9)^2