3.2437 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (3 x+2) (5 x+3)^{3/2}}-\frac{13145 (1-2 x)^{3/2}}{84 (5 x+3)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{5 x+3}}-\frac{13145}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-13145*(1 - 2*x)^(3/2))/(84*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(7/2))/(14*(2 + 3*x
)^2*(3 + 5*x)^(3/2)) + (239*(1 - 2*x)^(5/2))/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (1
3145*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) - (13145*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqr
t[7]*Sqrt[3 + 5*x])])/4

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Rubi [A]  time = 0.212926, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (3 x+2) (5 x+3)^{3/2}}-\frac{13145 (1-2 x)^{3/2}}{84 (5 x+3)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{5 x+3}}-\frac{13145}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-13145*(1 - 2*x)^(3/2))/(84*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(7/2))/(14*(2 + 3*x
)^2*(3 + 5*x)^(3/2)) + (239*(1 - 2*x)^(5/2))/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (1
3145*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) - (13145*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqr
t[7]*Sqrt[3 + 5*x])])/4

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Rubi in Sympy [A]  time = 17.0358, size = 138, normalized size = 1.01 \[ - \frac{10 \left (- 2 x + 1\right )^{\frac{7}{2}}}{33 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{478 \left (- 2 x + 1\right )^{\frac{5}{2}}}{33 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{8365 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{66 \left (3 x + 2\right )^{2}} + \frac{8365 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{4 \left (3 x + 2\right )} - \frac{13145 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*(-2*x + 1)**(7/2)/(33*(3*x + 2)**2*(5*x + 3)**(3/2)) + 478*(-2*x + 1)**(5/2)
/(33*(3*x + 2)**2*sqrt(5*x + 3)) + 8365*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(66*(3*x
+ 2)**2) + 8365*sqrt(-2*x + 1)*sqrt(5*x + 3)/(4*(3*x + 2)) - 13145*sqrt(7)*atan
(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/4

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Mathematica [A]  time = 0.0929637, size = 82, normalized size = 0.6 \[ \frac{\sqrt{1-2 x} \left (1809585 x^3+3458634 x^2+2200321 x+465916\right )}{12 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{13145}{8} \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(465916 + 2200321*x + 3458634*x^2 + 1809585*x^3))/(12*(2 + 3*x)^2
*(3 + 5*x)^(3/2)) - (13145*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3
+ 5*x])])/8

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Maple [B]  time = 0.022, size = 250, normalized size = 1.8 \[{\frac{1}{24\, \left ( 2+3\,x \right ) ^{2}} \left ( 8872875\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+3619170\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+8991180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+6917268\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1419660\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4400642\,x\sqrt{-10\,{x}^{2}-x+3}+931832\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(8872875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+224
77950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+21334335*7^
(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3619170*x^3*(-10*x^
2-x+3)^(1/2)+8991180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x+6917268*x^2*(-10*x^2-x+3)^(1/2)+1419660*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))+4400642*x*(-10*x^2-x+3)^(1/2)+931832*(-10*x^2-x+3)^(1/2))*(
1-2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51186, size = 232, normalized size = 1.69 \[ \frac{13145}{8} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{40213 \, x}{6 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{69977}{20 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{454757 \, x}{270 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{162 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{25039}{108 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{1473541}{1620 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13145/8*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 40213/6*x/sq
rt(-10*x^2 - x + 3) + 69977/20/sqrt(-10*x^2 - x + 3) + 454757/270*x/(-10*x^2 - x
+ 3)^(3/2) + 2401/162/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/
2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 25039/108/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(
-10*x^2 - x + 3)^(3/2)) - 1473541/1620/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.227032, size = 143, normalized size = 1.04 \[ \frac{39435 \, \sqrt{7}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 2 \,{\left (1809585 \, x^{3} + 3458634 \, x^{2} + 2200321 \, x + 465916\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{24 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(39435*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1/14*sqrt(
7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 2*(1809585*x^3 + 3458634*x^2 +
2200321*x + 465916)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 +
228*x + 36)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.375035, size = 509, normalized size = 3.72 \[ -\frac{11}{240} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{2629}{16} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1133}{10} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} + \frac{77 \,{\left (437 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 103880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{2 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/240*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 2629/16*sqrt(70)*sqrt(10)*(pi +
2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/
(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 1133/10*sqrt(10)*((sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10
*x + 5) - sqrt(22))) + 77/2*(437*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/
sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 103880
*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2