3.1910 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=178 \[ \frac{117955 \sqrt{1-2 x}}{14 (5 x+3)}-\frac{176065 \sqrt{1-2 x}}{126 (5 x+3)^2}+\frac{1301 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac{28 \sqrt{1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-176065*Sqrt[1 - 2*x])/(126*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3
+ 5*x)^2) + (28*Sqrt[1 - 2*x])/(3*(2 + 3*x)^2*(3 + 5*x)^2) + (1301*Sqrt[1 - 2*x]
)/(7*(2 + 3*x)*(3 + 5*x)^2) + (117955*Sqrt[1 - 2*x])/(14*(3 + 5*x)) + (813716*Sq
rt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.388923, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{117955 \sqrt{1-2 x}}{14 (5 x+3)}-\frac{176065 \sqrt{1-2 x}}{126 (5 x+3)^2}+\frac{1301 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac{28 \sqrt{1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-176065*Sqrt[1 - 2*x])/(126*(3 + 5*x)^2) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3
+ 5*x)^2) + (28*Sqrt[1 - 2*x])/(3*(2 + 3*x)^2*(3 + 5*x)^2) + (1301*Sqrt[1 - 2*x]
)/(7*(2 + 3*x)*(3 + 5*x)^2) + (117955*Sqrt[1 - 2*x])/(14*(3 + 5*x)) + (813716*Sq
rt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/1
1]*Sqrt[1 - 2*x]]

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Rubi in Sympy [A]  time = 41.2455, size = 162, normalized size = 0.91 \[ \frac{70773 \sqrt{- 2 x + 1}}{14 \left (3 x + 2\right )} + \frac{2540 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right ) \left (5 x + 3\right )} - \frac{1685 \sqrt{- 2 x + 1}}{18 \left (3 x + 2\right ) \left (5 x + 3\right )^{2}} + \frac{28 \sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{2}} + \frac{7 \sqrt{- 2 x + 1}}{9 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{2}} + \frac{813716 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{49} - \frac{112875 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

70773*sqrt(-2*x + 1)/(14*(3*x + 2)) + 2540*sqrt(-2*x + 1)/(3*(3*x + 2)*(5*x + 3)
) - 1685*sqrt(-2*x + 1)/(18*(3*x + 2)*(5*x + 3)**2) + 28*sqrt(-2*x + 1)/(3*(3*x
+ 2)**2*(5*x + 3)**2) + 7*sqrt(-2*x + 1)/(9*(3*x + 2)**3*(5*x + 3)**2) + 813716*
sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/49 - 112875*sqrt(55)*atanh(sqrt(55)*sq
rt(-2*x + 1)/11)/11

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Mathematica [A]  time = 0.185318, size = 104, normalized size = 0.58 \[ \frac{\sqrt{1-2 x} \left (15923925 x^4+40874010 x^3+39307638 x^2+16784696 x+2685098\right )}{14 (3 x+2)^3 (5 x+3)^2}+\frac{813716}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-112875 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(2685098 + 16784696*x + 39307638*x^2 + 40874010*x^3 + 15923925*x^
4))/(14*(2 + 3*x)^3*(3 + 5*x)^2) + (813716*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/7 - 112875*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.02, size = 103, normalized size = 0.6 \[ -324\,{\frac{1}{ \left ( -4-6\,x \right ) ^{3}} \left ({\frac{3544\, \left ( 1-2\,x \right ) ^{5/2}}{21}}-{\frac{21418\, \left ( 1-2\,x \right ) ^{3/2}}{27}}+{\frac{25172\,\sqrt{1-2\,x}}{27}} \right ) }+{\frac{813716\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+2500\,{\frac{1}{ \left ( -6-10\,x \right ) ^{2}} \left ( -{\frac{269\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{2937\,\sqrt{1-2\,x}}{100}} \right ) }-{\frac{112875\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-324*(3544/21*(1-2*x)^(5/2)-21418/27*(1-2*x)^(3/2)+25172/27*(1-2*x)^(1/2))/(-4-6
*x)^3+813716/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2500*(-269/20*(1-2*
x)^(3/2)+2937/100*(1-2*x)^(1/2))/(-6-10*x)^2-112875/11*arctanh(1/11*55^(1/2)*(1-
2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.50358, size = 221, normalized size = 1.24 \[ \frac{112875}{22} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{406858}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{15923925 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 145443720 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 498018162 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 757678432 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 432141633 \, \sqrt{-2 \, x + 1}}{7 \,{\left (675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

112875/22*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x +
1))) - 406858/49*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) + 1/7*(15923925*(-2*x + 1)^(9/2) - 145443720*(-2*x + 1)^(7/2) + 4980
18162*(-2*x + 1)^(5/2) - 757678432*(-2*x + 1)^(3/2) + 432141633*sqrt(-2*x + 1))/
(675*(2*x - 1)^5 + 7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 18
2182*x - 49588)

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Fricas [A]  time = 0.239876, size = 267, normalized size = 1.5 \[ \frac{\sqrt{11} \sqrt{7}{\left (790125 \, \sqrt{7} \sqrt{5}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{11}{\left (5 \, x - 8\right )} + 11 \, \sqrt{5} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right ) + 813716 \, \sqrt{11} \sqrt{3}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} - 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{11} \sqrt{7}{\left (15923925 \, x^{4} + 40874010 \, x^{3} + 39307638 \, x^{2} + 16784696 \, x + 2685098\right )} \sqrt{-2 \, x + 1}\right )}}{1078 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1078*sqrt(11)*sqrt(7)*(790125*sqrt(7)*sqrt(5)*(675*x^5 + 2160*x^4 + 2763*x^3 +
1766*x^2 + 564*x + 72)*log((sqrt(11)*(5*x - 8) + 11*sqrt(5)*sqrt(-2*x + 1))/(5*
x + 3)) + 813716*sqrt(11)*sqrt(3)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 56
4*x + 72)*log((sqrt(7)*(3*x - 5) - 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)) + sqrt(1
1)*sqrt(7)*(15923925*x^4 + 40874010*x^3 + 39307638*x^2 + 16784696*x + 2685098)*s
qrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236531, size = 204, normalized size = 1.15 \[ \frac{112875}{22} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{406858}{49} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25 \,{\left (1345 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2937 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} + \frac{3 \,{\left (15948 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 74963 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 88102 \, \sqrt{-2 \, x + 1}\right )}}{7 \,{\left (3 \, x + 2\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

112875/22*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 406858/49*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(s
qrt(21) + 3*sqrt(-2*x + 1))) - 25/4*(1345*(-2*x + 1)^(3/2) - 2937*sqrt(-2*x + 1)
)/(5*x + 3)^2 + 3/7*(15948*(2*x - 1)^2*sqrt(-2*x + 1) - 74963*(-2*x + 1)^(3/2) +
88102*sqrt(-2*x + 1))/(3*x + 2)^3