3.2470 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}+\frac{1948963 \sqrt{1-2 x} \sqrt{5 x+3}}{8297856 (3 x+2)}-\frac{12371 \sqrt{1-2 x} \sqrt{5 x+3}}{592704 (3 x+2)^2}-\frac{14831 \sqrt{1-2 x} \sqrt{5 x+3}}{105840 (3 x+2)^3}+\frac{437 \sqrt{1-2 x} \sqrt{5 x+3}}{17640 (3 x+2)^4}-\frac{933031 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]

[Out]

(437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(17640*(2 + 3*x)^4) - (14831*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(105840*(2 + 3*x)^3) - (12371*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(592704*(
2 + 3*x)^2) + (1948963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8297856*(2 + 3*x)) + (Sqrt[
1 - 2*x]*(3 + 5*x)^(3/2))/(105*(2 + 3*x)^5) - (933031*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(307328*Sqrt[7])

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Rubi [A]  time = 0.370489, antiderivative size = 180, 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 \[ \frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^5}+\frac{1948963 \sqrt{1-2 x} \sqrt{5 x+3}}{8297856 (3 x+2)}-\frac{12371 \sqrt{1-2 x} \sqrt{5 x+3}}{592704 (3 x+2)^2}-\frac{14831 \sqrt{1-2 x} \sqrt{5 x+3}}{105840 (3 x+2)^3}+\frac{437 \sqrt{1-2 x} \sqrt{5 x+3}}{17640 (3 x+2)^4}-\frac{933031 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

(437*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(17640*(2 + 3*x)^4) - (14831*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(105840*(2 + 3*x)^3) - (12371*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(592704*(
2 + 3*x)^2) + (1948963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8297856*(2 + 3*x)) + (Sqrt[
1 - 2*x]*(3 + 5*x)^(3/2))/(105*(2 + 3*x)^5) - (933031*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(307328*Sqrt[7])

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Rubi in Sympy [A]  time = 36.1264, size = 163, normalized size = 0.91 \[ \frac{1948963 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{8297856 \left (3 x + 2\right )} - \frac{12371 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{592704 \left (3 x + 2\right )^{2}} - \frac{14831 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{105840 \left (3 x + 2\right )^{3}} + \frac{437 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{17640 \left (3 x + 2\right )^{4}} + \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{105 \left (3 x + 2\right )^{5}} - \frac{933031 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{2151296} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(5/2)/(2+3*x)**6/(1-2*x)**(1/2),x)

[Out]

1948963*sqrt(-2*x + 1)*sqrt(5*x + 3)/(8297856*(3*x + 2)) - 12371*sqrt(-2*x + 1)*
sqrt(5*x + 3)/(592704*(3*x + 2)**2) - 14831*sqrt(-2*x + 1)*sqrt(5*x + 3)/(105840
*(3*x + 2)**3) + 437*sqrt(-2*x + 1)*sqrt(5*x + 3)/(17640*(3*x + 2)**4) + sqrt(-2
*x + 1)*(5*x + 3)**(3/2)/(105*(3*x + 2)**5) - 933031*sqrt(7)*atan(sqrt(7)*sqrt(-
2*x + 1)/(7*sqrt(5*x + 3)))/2151296

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Mathematica [A]  time = 0.161246, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (87703335 x^4+231277650 x^3+222865988 x^2+93291272 x+14330592\right )}{(3 x+2)^5}-13995465 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{64538880} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(14330592 + 93291272*x + 222865988*x^2 + 231277
650*x^3 + 87703335*x^4))/(2 + 3*x)^5 - 13995465*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*S
qrt[7 - 14*x]*Sqrt[3 + 5*x])])/64538880

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Maple [B]  time = 0.023, size = 298, normalized size = 1.7 \[{\frac{1}{64538880\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3400897995\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3120123832\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(5/2)/(2+3*x)^6/(1-2*x)^(1/2),x)

[Out]

1/64538880*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(3400897995*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+11336326650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+15115102200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^3+1227846690*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3237887100*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
[phone]*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))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.52156, size = 248, normalized size = 1.38 \[ \frac{933031}{4302592} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{315 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{239 \, \sqrt{-10 \, x^{2} - x + 3}}{5880 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{14831 \, \sqrt{-10 \, x^{2} - x + 3}}{105840 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{12371 \, \sqrt{-10 \, x^{2} - x + 3}}{592704 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1948963 \, \sqrt{-10 \, x^{2} - x + 3}}{8297856 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^6*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

933031/4302592*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/315
*sqrt(-10*x^2 - x + 3)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 2
39/5880*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) - 14831/1
05840*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) - 12371/592704*sqrt(-10
*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 1948963/8297856*sqrt(-10*x^2 - x + 3)/(3*x +
2)

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Fricas [A]  time = 0.229376, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (87703335 \, x^{4} + 231277650 \, x^{3} + 222865988 \, x^{2} + 93291272 \, x + 14330592\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 13995465 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{64538880 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^6*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

1/64538880*sqrt(7)*(2*sqrt(7)*(87703335*x^4 + 231277650*x^3 + 222865988*x^2 + 93
291272*x + 14330592)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 13995465*(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3
)*sqrt(-2*x + 1))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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((3+5*x)**(5/2)/(2+3*x)**6/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.517146, size = 594, normalized size = 3.3 \[ \frac{933031}{43025920} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (2103 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 2747920 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 1406935040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 74141312000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 10228753920000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{460992 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^6*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

933031/43025920*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)))) - 1331/460992*(2103*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 + 274792
0*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 + 1406935040*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^5 - 74141312000*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 - 102287539200
00*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)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5