3.2267 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}+\frac{1852307 \sqrt{1-2 x} \sqrt{5 x+3}}{1185408 (3 x+2)}+\frac{17981 \sqrt{1-2 x} \sqrt{5 x+3}}{84672 (3 x+2)^2}+\frac{641 \sqrt{1-2 x} \sqrt{5 x+3}}{15120 (3 x+2)^3}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{2520 (3 x+2)^4}-\frac{783959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2520*(2 + 3*x)^4) + (641*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(15120*(2 + 3*x)^3) + (17981*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 +
3*x)^2) + (1852307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1185408*(2 + 3*x)) - (Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(15*(2 + 3*x)^5) - (783959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.367994, 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 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}+\frac{1852307 \sqrt{1-2 x} \sqrt{5 x+3}}{1185408 (3 x+2)}+\frac{17981 \sqrt{1-2 x} \sqrt{5 x+3}}{84672 (3 x+2)^2}+\frac{641 \sqrt{1-2 x} \sqrt{5 x+3}}{15120 (3 x+2)^3}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{2520 (3 x+2)^4}-\frac{783959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2520*(2 + 3*x)^4) + (641*Sqrt[1 - 2*x]*Sqrt[
3 + 5*x])/(15120*(2 + 3*x)^3) + (17981*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 +
3*x)^2) + (1852307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1185408*(2 + 3*x)) - (Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(15*(2 + 3*x)^5) - (783959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*S
qrt[3 + 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 35.8922, size = 163, normalized size = 0.91 \[ \frac{1852307 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1185408 \left (3 x + 2\right )} + \frac{17981 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84672 \left (3 x + 2\right )^{2}} + \frac{641 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15120 \left (3 x + 2\right )^{3}} - \frac{107 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2520 \left (3 x + 2\right )^{4}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{15 \left (3 x + 2\right )^{5}} - \frac{783959 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1852307*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1185408*(3*x + 2)) + 17981*sqrt(-2*x + 1)*
sqrt(5*x + 3)/(84672*(3*x + 2)**2) + 641*sqrt(-2*x + 1)*sqrt(5*x + 3)/(15120*(3*
x + 2)**3) - 107*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2520*(3*x + 2)**4) - sqrt(-2*x +
1)*(5*x + 3)**(3/2)/(15*(3*x + 2)**5) - 783959*sqrt(7)*atan(sqrt(7)*sqrt(-2*x +
1)/(7*sqrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.107858, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (83353815 x^4+226052850 x^3+230080132 x^2+103856008 x+17507808\right )}{(3 x+2)^5}-11759385 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{9219840} \]

Antiderivative was successfully verified.

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

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(17507808 + 103856008*x + 230080132*x^2 + 22605
2850*x^3 + 83353815*x^4))/(2 + 3*x)^5 - 11759385*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*
Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/9219840

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Maple [B]  time = 0.018, size = 298, normalized size = 1.7 \[{\frac{1}{9219840\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2857530555\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3221121848\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9219840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2857530555*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+9525101850*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x^4+12700135800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-1
0*x^2-x+3)^(1/2))*x^3+1166953410*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3164739900*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+32
21121848*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 = 1.5123, size = 267, normalized size = 1.48 \[ \frac{783959}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{32395}{32928} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{13 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{280 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{545 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2352 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{19437 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{239723 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

783959/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 32395/
32928*sqrt(-10*x^2 - x + 3) - 1/35*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32) + 13/280*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*
x^3 + 216*x^2 + 96*x + 16) + 545/2352*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 +
36*x + 8) + 19437/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 239723/131
712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.225077, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (83353815 \, x^{4} + 226052850 \, x^{3} + 230080132 \, x^{2} + 103856008 \, x + 17507808\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 11759385 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{9219840 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9219840*sqrt(7)*(2*sqrt(7)*(83353815*x^4 + 226052850*x^3 + 230080132*x^2 + 103
856008*x + 17507808)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 11759385*(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)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.465361, size = 594, normalized size = 3.3 \[ \frac{783959}{6146560} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1331 \,{\left (1767 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} + 2308880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 925245440 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 177804928000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 10860971520000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{65856 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

783959/6146560*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)))) - 1331/65856*(1767*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 2308880*
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 - 925245440*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)))^5 - 177804928000*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 - 10860971520000
*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))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5