3.2404 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{7/2}}{32 (3 x+2)^4}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{7/2}}{12 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{5/2}}{1344 (3 x+2)^3}-\frac{73205 \sqrt{1-2 x} (5 x+3)^{3/2}}{37632 (3 x+2)^2}-\frac{805255 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{8857805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(-805255*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (73205*Sqrt[1 - 2*x]*
(3 + 5*x)^(3/2))/(37632*(2 + 3*x)^2) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(134
4*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(6*(2 + 3*x)^6) + (11*(1 - 2*
x)^(3/2)*(3 + 5*x)^(7/2))/(12*(2 + 3*x)^5) + (121*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))
/(32*(2 + 3*x)^4) - (8857805*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175
616*Sqrt[7])

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Rubi [A]  time = 0.324629, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{7/2}}{32 (3 x+2)^4}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{7/2}}{12 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{5/2}}{1344 (3 x+2)^3}-\frac{73205 \sqrt{1-2 x} (5 x+3)^{3/2}}{37632 (3 x+2)^2}-\frac{805255 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{8857805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]

[Out]

(-805255*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (73205*Sqrt[1 - 2*x]*
(3 + 5*x)^(3/2))/(37632*(2 + 3*x)^2) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(134
4*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(6*(2 + 3*x)^6) + (11*(1 - 2*
x)^(3/2)*(3 + 5*x)^(7/2))/(12*(2 + 3*x)^5) + (121*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))
/(32*(2 + 3*x)^4) - (8857805*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175
616*Sqrt[7])

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Rubi in Sympy [A]  time = 24.6177, size = 190, normalized size = 0.91 \[ - \frac{6655 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{65856 \left (3 x + 2\right )^{3}} - \frac{605 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{4704 \left (3 x + 2\right )^{4}} - \frac{11 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{84 \left (3 x + 2\right )^{5}} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{6 \left (3 x + 2\right )^{6}} + \frac{73205 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{263424 \left (3 x + 2\right )^{2}} + \frac{805255 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{175616 \left (3 x + 2\right )} - \frac{8857805 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**7,x)

[Out]

-6655*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(65856*(3*x + 2)**3) - 605*(-2*x + 1)**(5/
2)*(5*x + 3)**(3/2)/(4704*(3*x + 2)**4) - 11*(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/
(84*(3*x + 2)**5) + (-2*x + 1)**(5/2)*(5*x + 3)**(7/2)/(6*(3*x + 2)**6) + 73205*
(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(263424*(3*x + 2)**2) + 805255*sqrt(-2*x + 1)*sq
rt(5*x + 3)/(175616*(3*x + 2)) - 8857805*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*
sqrt(5*x + 3)))/1229312

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Mathematica [A]  time = 0.184421, size = 92, normalized size = 0.44 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (568572155 x^5+1905431420 x^4+2573967504 x^3+1743189856 x^2+589734736 x+79536960\right )}{(3 x+2)^6}-26573415 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{7375872} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(79536960 + 589734736*x + 1743189856*x^2 + 2573
967504*x^3 + 1905431420*x^4 + 568572155*x^5))/(2 + 3*x)^6 - 26573415*Sqrt[7]*Arc
Tan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/7375872

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Maple [B]  time = 0.019, size = 346, normalized size = 1.7 \[{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 19372019535\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+24404657984\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(19372019535*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+77488078140*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^5+129146796900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/
(-10*x^2-x+3)^(1/2))*x^4+7960010170*x^5*(-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^3+26676039880*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+36035545056*x^3*(-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+24404657984*x^2*(-10*x^2-x+3)^(1/2)[phone]
60*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))/(-10*x^2-x+3)^(1/2)/(2+3*x)^6

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Maxima [A]  time = 1.6233, size = 408, normalized size = 1.95 \[ \frac{3304795}{19361664} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{196 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{4387 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{10976 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{81733 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{153664 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{660959 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{4302592 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{59208325}{12907776} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{113659535}{25815552} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{109715471 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{77446656 \,{\left (3 \, x + 2\right )}} + \frac{13542925}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{8857805}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{11932415}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

3304795/19361664*(-10*x^2 - x + 3)^(5/2) + 1/14*(-10*x^2 - x + 3)^(7/2)/(729*x^6
+ 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 37/196*(-10*x^2 - x
+ 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 4387/10976*(
-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 81733/153664*(
-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 660959/4302592*(-10*x^2 -
x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 59208325/12907776*(-10*x^2 - x + 3)^(3/2)*x +
113659535/25815552*(-10*x^2 - x + 3)^(3/2) - 109715471/77446656*(-10*x^2 - x + 3
)^(5/2)/(3*x + 2) + 13542925/614656*sqrt(-10*x^2 - x + 3)*x + 8857805/2458624*sq
rt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 11932415/1229312*sqrt(
-10*x^2 - x + 3)

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Fricas [A]  time = 0.230502, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (568572155 \, x^{5} + 1905431420 \, x^{4} + 2573967504 \, x^{3} + 1743189856 \, x^{2} + 589734736 \, x + 79536960\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 26573415 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7375872 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

1/7375872*sqrt(7)*(2*sqrt(7)*(568572155*x^5 + 1905431420*x^4 + 2573967504*x^3 +
1743189856*x^2 + 589734736*x + 79536960)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 26573415
*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*
sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(729*x^6 + 2916*x^5 + 4860*
x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)*(3+5*x)**(5/2)/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.657319, size = 676, normalized size = 3.23 \[ \frac{1771561}{4917248} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{8857805 \,{\left (3 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 4760 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 3104640 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 869299200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 104491520000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 5163110400000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{263424 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="giac")

[Out]

1771561/4917248*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)))) - 8857805/263424*(3*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)))^11 + 4760*
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)))^9 + 3104640*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)))^7 - 869299200*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)))^5 - 104491520000*sqrt(1
0)*((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 - 5163110400000*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))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s
qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^6