3.2342 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{297 \sqrt{1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac{9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac{1089 \sqrt{1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac{11979 \sqrt{1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac{395307 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{4348377 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(-395307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (11979*Sqrt[1 - 2*x]*
(3 + 5*x)^(3/2))/(12544*(2 + 3*x)^2) - (1089*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(224
0*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 + 3*x)^6) + (9*(1 - 2*
x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (297*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))
/(160*(2 + 3*x)^4) - (4348377*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(17
5616*Sqrt[7])

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Rubi [A]  time = 0.320546, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{297 \sqrt{1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac{9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac{1089 \sqrt{1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac{11979 \sqrt{1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac{395307 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{4348377 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-395307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (11979*Sqrt[1 - 2*x]*
(3 + 5*x)^(3/2))/(12544*(2 + 3*x)^2) - (1089*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(224
0*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 + 3*x)^6) + (9*(1 - 2*
x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (297*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))
/(160*(2 + 3*x)^4) - (4348377*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(17
5616*Sqrt[7])

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Rubi in Sympy [A]  time = 24.1518, size = 190, normalized size = 0.91 \[ - \frac{99 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{1568 \left (3 x + 2\right )^{4}} - \frac{9 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{140 \left (3 x + 2\right )^{5}} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{7}{2}}}{14 \left (3 x + 2\right )^{6}} - \frac{35937 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{87808 \left (3 x + 2\right )^{2}} + \frac{1089 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3136 \left (3 x + 2\right )^{3}} + \frac{395307 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{175616 \left (3 x + 2\right )} - \frac{4348377 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-99*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/(1568*(3*x + 2)**4) - 9*(-2*x + 1)**(5/2)
*(5*x + 3)**(5/2)/(140*(3*x + 2)**5) + (-2*x + 1)**(5/2)*(5*x + 3)**(7/2)/(14*(3
*x + 2)**6) - 35937*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(87808*(3*x + 2)**2) + 1089*
(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/(3136*(3*x + 2)**3) + 395307*sqrt(-2*x + 1)*s
qrt(5*x + 3)/(175616*(3*x + 2)) - 4348377*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7
*sqrt(5*x + 3)))/1229312

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Mathematica [A]  time = 0.153273, size = 92, normalized size = 0.44 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (460633945 x^5+1555340180 x^4+2108117296 x^3+1428134688 x^2+482263920 x+64829376\right )}{(3 x+2)^6}-21741885 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{12293120} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(64829376 + 482263920*x + 1428134688*x^2 + 2108
117296*x^3 + 1555340180*x^4 + 460633945*x^5))/(2 + 3*x)^6 - 21741885*Sqrt[7]*Arc
Tan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/12293120

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Maple [B]  time = 0.02, size = 346, normalized size = 1.7 \[{\frac{1}{12293120\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 15849834165\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+19993885632\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12293120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(15849834165*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+63399336660*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^5+105665561100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^4+6448875230*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+21774762520*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+29513642144*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+19993885632*x^2*(-10*x^2-x+3)^(1/2)[phone]
40*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 = 1.52244, size = 369, normalized size = 1.77 \[ \frac{272085}{307328} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{42 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{23 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{420 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{297 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1568 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{10989 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{489753 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{6648345}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{4348377}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{5857731}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{645909 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1229312 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

272085/307328*(-10*x^2 - x + 3)^(3/2) - 1/42*(-10*x^2 - x + 3)^(5/2)/(729*x^6 +
2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 23/420*(-10*x^2 - x +
3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 297/1568*(-10*x
^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 10989/21952*(-10*x^
2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 489753/614656*(-10*x^2 - x + 3)^
(5/2)/(9*x^2 + 12*x + 4) + 6648345/614656*sqrt(-10*x^2 - x + 3)*x + 4348377/2458
624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 5857731/1229312*
sqrt(-10*x^2 - x + 3) + 645909/1229312*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 0.228727, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (460633945 \, x^{5} + 1555340180 \, x^{4} + 2108117296 \, x^{3} + 1428134688 \, x^{2} + 482263920 \, x + 64829376\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 21741885 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{12293120 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12293120*sqrt(7)*(2*sqrt(7)*(460633945*x^5 + 1555340180*x^4 + 2108117296*x^3 +
1428134688*x^2 + 482263920*x + 64829376)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 2174188
5*(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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.602692, size = 676, normalized size = 3.23 \[ \frac{4348377}{24586240} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{161051 \,{\left (27 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} + 42840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 27941760 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 6539187200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 940423680000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 46467993600000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{87808 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4348377/24586240*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)))) - 161051/87808*(27*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 + 42840
*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 + 27941760*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) - sqr
t(22)))^7 - 6539187200*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 - 940423680000*sqr
t(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqr
t(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 46467993600000*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) - s
qrt(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)^6