3.2256 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{463266973 \sqrt{1-2 x} \sqrt{5 x+3}}{11063808 (3 x+2)}+\frac{4429459 \sqrt{1-2 x} \sqrt{5 x+3}}{790272 (3 x+2)^2}+\frac{126799 \sqrt{1-2 x} \sqrt{5 x+3}}{141120 (3 x+2)^3}+\frac{10921 \sqrt{1-2 x} \sqrt{5 x+3}}{70560 (3 x+2)^4}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{1260 (3 x+2)^5}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{18 (3 x+2)^6}-\frac{588912203 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(1260*(2 + 3*x)^5) + (10921*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(70560*(2 + 3*x)^4)
+ (126799*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(141120*(2 + 3*x)^3) + (4429459*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(790272*(2 + 3*x)^2) + (463266973*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(11063808*(2 + 3*x)) - (588912203*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(1229312*Sqrt[7])

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Rubi [A]  time = 0.450174, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{463266973 \sqrt{1-2 x} \sqrt{5 x+3}}{11063808 (3 x+2)}+\frac{4429459 \sqrt{1-2 x} \sqrt{5 x+3}}{790272 (3 x+2)^2}+\frac{126799 \sqrt{1-2 x} \sqrt{5 x+3}}{141120 (3 x+2)^3}+\frac{10921 \sqrt{1-2 x} \sqrt{5 x+3}}{70560 (3 x+2)^4}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{1260 (3 x+2)^5}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{18 (3 x+2)^6}-\frac{588912203 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1229312 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(1260*(2 + 3*x)^5) + (10921*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(70560*(2 + 3*x)^4)
+ (126799*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(141120*(2 + 3*x)^3) + (4429459*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(790272*(2 + 3*x)^2) + (463266973*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(11063808*(2 + 3*x)) - (588912203*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(1229312*Sqrt[7])

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Rubi in Sympy [A]  time = 43.5276, size = 190, normalized size = 0.91 \[ \frac{463266973 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{11063808 \left (3 x + 2\right )} + \frac{4429459 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{790272 \left (3 x + 2\right )^{2}} + \frac{126799 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{141120 \left (3 x + 2\right )^{3}} + \frac{10921 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{70560 \left (3 x + 2\right )^{4}} + \frac{37 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1260 \left (3 x + 2\right )^{5}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{18 \left (3 x + 2\right )^{6}} - \frac{588912203 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{8605184} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

463266973*sqrt(-2*x + 1)*sqrt(5*x + 3)/(11063808*(3*x + 2)) + 4429459*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(790272*(3*x + 2)**2) + 126799*sqrt(-2*x + 1)*sqrt(5*x + 3)/(
141120*(3*x + 2)**3) + 10921*sqrt(-2*x + 1)*sqrt(5*x + 3)/(70560*(3*x + 2)**4) +
37*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1260*(3*x + 2)**5) - sqrt(-2*x + 1)*sqrt(5*x +
3)/(18*(3*x + 2)**6) - 588912203*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*
x + 3)))/8605184

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Mathematica [A]  time = 0.12971, size = 92, normalized size = 0.44 \[ \frac{\frac{126 \sqrt{1-2 x} \sqrt{5 x+3} \left (62541041355 x^5+211260697020 x^4+285550790544 x^3+193055073632 x^2+65287037520 x+8835086144\right )}{(3 x+2)^6}-26501049135 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{774466560} \]

Antiderivative was successfully verified.

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

[Out]

((126*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8835086144 + 65287037520*x + 193055073632*x^2
+ 285550790544*x^3 + 211260697020*x^4 + 62541041355*x^5))/(2 + 3*x)^6 - 2650104
9135*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/774466560

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Maple [B]  time = 0.023, size = 346, normalized size = 1.7 \[{\frac{1}{86051840\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2146584979935\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2702771030848\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/86051840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2146584979935*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+8586339919740*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+14310566532900*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+875574578970*x^5*(-10*x^2-x+3)^(1/2)[phone]
800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2957649758280
*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^2+3997711067616*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+2702771030848*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)^6

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Maxima [A]  time = 1.52078, size = 329, normalized size = 1.57 \[ \frac{588912203}{17210368} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{24335215}{921984} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{333 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{980 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{11721 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{7840 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{137455 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{14601129 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{180080591 \, \sqrt{-10 \, x^{2} - x + 3}}{3687936 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

588912203/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2
4335215/921984*sqrt(-10*x^2 - x + 3) + 1/14*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 2
916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 333/980*(-10*x^2 - x +
3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 11721/7840*(-10
*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 137455/21952*(-10
*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 14601129/614656*(-10*x^2 - x
+ 3)^(3/2)/(9*x^2 + 12*x + 4) - 180080591/3687936*sqrt(-10*x^2 - x + 3)/(3*x + 2
)

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Fricas [A]  time = 0.23489, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (62541041355 \, x^{5} + 211260697020 \, x^{4} + 285550790544 \, x^{3} + 193055073632 \, x^{2} + 65287037520 \, x + 8835086144\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 2944561015 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{86051840 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/86051840*sqrt(7)*(2*sqrt(7)*(62541041355*x^5 + 211260697020*x^4 + 285550790544
*x^3 + 193055073632*x^2 + 65287037520*x + 8835086144)*sqrt(5*x + 3)*sqrt(-2*x +
1) + 2944561015*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 6
4)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(729*x^6 + 2
916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-2*x + 1)*sqrt(5*x + 3)/(3*x + 2)**7, x)

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GIAC/XCAS [A]  time = 0.615711, size = 660, normalized size = 3.16 \[ \frac{121}{172103680} \, \sqrt{5}{\left (4867043 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{280 \, \sqrt{2}{\left (4867043 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} - 12766158440 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 6076175020160 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 1409555377484800 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 169516778170880000 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - \frac{8376360110182400000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{33505440440729600000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/172103680*sqrt(5)*(4867043*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*s
qrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))) - 280*sqrt(2)*(4867043*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11
- 12766158440*((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 - 6076175020160*((sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22)))^7 - 1409555377484800*((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 - 169516778170880000
*((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 - 8376360110182400000*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3) + 33505440440729600000*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)^6)