3.2309 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{5 x+3}}-\frac{638165 \sqrt{1-2 x}}{1176 (5 x+3)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (3 x+2) (5 x+3)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{13246251 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-638165*Sqrt[1 - 2*x])/(1176*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3
+ 5*x)^(3/2)) + (313*Sqrt[1 - 2*x])/(84*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (25441*S
qrt[1 - 2*x])/(392*(2 + 3*x)*(3 + 5*x)^(3/2)) + (63678595*Sqrt[1 - 2*x])/(12936*
Sqrt[3 + 5*x]) - (13246251*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*S
qrt[7])

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Rubi [A]  time = 0.391021, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{63678595 \sqrt{1-2 x}}{12936 \sqrt{5 x+3}}-\frac{638165 \sqrt{1-2 x}}{1176 (5 x+3)^{3/2}}+\frac{25441 \sqrt{1-2 x}}{392 (3 x+2) (5 x+3)^{3/2}}+\frac{313 \sqrt{1-2 x}}{84 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{13246251 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-638165*Sqrt[1 - 2*x])/(1176*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(3*(2 + 3*x)^3*(3
+ 5*x)^(3/2)) + (313*Sqrt[1 - 2*x])/(84*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (25441*S
qrt[1 - 2*x])/(392*(2 + 3*x)*(3 + 5*x)^(3/2)) + (63678595*Sqrt[1 - 2*x])/(12936*
Sqrt[3 + 5*x]) - (13246251*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*S
qrt[7])

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Rubi in Sympy [A]  time = 37.4506, size = 151, normalized size = 0.91 \[ \frac{63678595 \sqrt{- 2 x + 1}}{12936 \sqrt{5 x + 3}} - \frac{638165 \sqrt{- 2 x + 1}}{1176 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{25441 \sqrt{- 2 x + 1}}{392 \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{313 \sqrt{- 2 x + 1}}{84 \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{\sqrt{- 2 x + 1}}{3 \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{13246251 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2744} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

63678595*sqrt(-2*x + 1)/(12936*sqrt(5*x + 3)) - 638165*sqrt(-2*x + 1)/(1176*(5*x
+ 3)**(3/2)) + 25441*sqrt(-2*x + 1)/(392*(3*x + 2)*(5*x + 3)**(3/2)) + 313*sqrt
(-2*x + 1)/(84*(3*x + 2)**2*(5*x + 3)**(3/2)) + sqrt(-2*x + 1)/(3*(3*x + 2)**3*(
5*x + 3)**(3/2)) - 13246251*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)
))/2744

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Mathematica [A]  time = 0.113586, size = 87, normalized size = 0.52 \[ \frac{\sqrt{1-2 x} \left (8596610325 x^4+22161651840 x^3+21406565457 x^2+9181937962 x+1475586688\right )}{12936 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{13246251 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{784 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 - 2*x]*(1475586688 + 9181937962*x + 21406565457*x^2 + 22161651840*x^3 +
8596610325*x^4))/(12936*(2 + 3*x)^3*(3 + 5*x)^(3/2)) - (13246251*ArcTan[(-20 - 3
7*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/(784*Sqrt[7])

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Maple [B]  time = 0.022, size = 298, normalized size = 1.8 \[{\frac{1}{181104\, \left ( 2+3\,x \right ) ^{3}} \left ( 295060241025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+299691916398\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/181104*(295060241025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)
)*x^5+944192771280*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+1207779919929*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1
20352544550*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+310263125760*x^3*(-10*x^2-x+3)^(1/2)[phone]
2*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+299691916398*x^2*
(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(
1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51906, size = 324, normalized size = 1.95 \[ \frac{13246251}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{63678595 \, x}{6468 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{66486521}{12936 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{207835 \, x}{84 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{27 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{77}{4 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{24617}{72 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{2020657}{1512 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13246251/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 636785
95/6468*x/sqrt(-10*x^2 - x + 3) + 66486521/12936/sqrt(-10*x^2 - x + 3) + 207835/
84*x/(-10*x^2 - x + 3)^(3/2) + 49/27/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x
^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)
) + 77/4/(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)) + 24617/72/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3
)^(3/2)) - 2020657/1512/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.227515, size = 167, normalized size = 1.01 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (8596610325 \, x^{4} + 22161651840 \, x^{3} + 21406565457 \, x^{2} + 9181937962 \, x + 1475586688\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 437126283 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{181104 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/181104*sqrt(7)*(2*sqrt(7)*(8596610325*x^4 + 22161651840*x^3 + 21406565457*x^2
+ 9181937962*x + 1475586688)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 437126283*(675*x^5 +
2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37*x + 20)/(s
qrt(5*x + 3)*sqrt(-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)**(1/2)/(2+3*x)**4/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.458562, size = 587, normalized size = 3.54 \[ -\frac{1}{1811040} \, \sqrt{5}{\left (85750 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 437126283 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 271656000 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{2744280 \, \sqrt{2}{\left (22317 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 10704960 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{1323627200 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{5294508800 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1811040*sqrt(5)*(85750*sqrt(2)*((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 - 437126283*sqrt
(70)*sqrt(2)*(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)))) - 271656
000*sqrt(2)*((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))) - 2744280*sqrt(2)*(22317*((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)))^5 + 10704960*((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 + 1323627200*(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 5294508800*sqrt(5*x + 3)/(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22)))/(((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)))^2 + 280)^3)