3.2387 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=157 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac{8}{27} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{247}{270} (5 x+3)^{3/2} \sqrt{1-2 x}+\frac{24251 \sqrt{5 x+3} \sqrt{1-2 x}}{3240}+\frac{326717 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9720 \sqrt{10}}+\frac{805}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(24251*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3240 - (247*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/2
70 - (8*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/
(3*(2 + 3*x)) + (326717*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(9720*Sqrt[10]) + (805
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi [A]  time = 0.368572, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{3 (3 x+2)}-\frac{8}{27} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{247}{270} (5 x+3)^{3/2} \sqrt{1-2 x}+\frac{24251 \sqrt{5 x+3} \sqrt{1-2 x}}{3240}+\frac{326717 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9720 \sqrt{10}}+\frac{805}{243} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(24251*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3240 - (247*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/2
70 - (8*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/27 - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/
(3*(2 + 3*x)) + (326717*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(9720*Sqrt[10]) + (805
*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/243

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Rubi in Sympy [A]  time = 37.9256, size = 141, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (3 x + 2\right )} - \frac{8 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{27} + \frac{247 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{108} + \frac{7949 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3240} + \frac{326717 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{97200} + \frac{805 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{243} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/(3*(3*x + 2)) - 8*(-2*x + 1)**(3/2)*(5*x + 3
)**(3/2)/27 + 247*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/108 + 7949*sqrt(-2*x + 1)*sqrt
(5*x + 3)/3240 + 326717*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/97200 + 805*sqr
t(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/243

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Mathematica [A]  time = 0.225735, size = 117, normalized size = 0.75 \[ \frac{\frac{60 \sqrt{1-2 x} \sqrt{5 x+3} \left (7200 x^3-13740 x^2+17277 x+21718\right )}{3 x+2}+322000 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )+326717 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{194400} \]

Antiderivative was successfully verified.

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

[Out]

((60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(21718 + 17277*x - 13740*x^2 + 7200*x^3))/(2 +
3*x) + 322000*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] + 32
6717*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/194400

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Maple [A]  time = 0.017, size = 180, normalized size = 1.2 \[ -{\frac{1}{388800+583200\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -432000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+966000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-980151\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+824400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+644000\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -653434\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1036620\,x\sqrt{-10\,{x}^{2}-x+3}-1303080\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/194400*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-432000*x^3*(-10*x^2-x+3)^(1/2)+966000*7^
(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-980151*10^(1/2)*arcsi
n(20/11*x+1/11)*x+824400*x^2*(-10*x^2-x+3)^(1/2)+644000*7^(1/2)*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-653434*10^(1/2)*arcsin(20/11*x+1/11)-1036620*
x*(-10*x^2-x+3)^(1/2)-1303080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.48877, size = 140, normalized size = 0.89 \[ -\frac{2}{27} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{247}{54} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{326717}{194400} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{805}{486} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{15359}{3240} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{9 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(-10*x^2 - x + 3)^(3/2) - 247/54*sqrt(-10*x^2 - x + 3)*x + 326717/194400*s
qrt(10)*arcsin(20/11*x + 1/11) - 805/486*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 2
0/11/abs(3*x + 2)) + 15359/3240*sqrt(-10*x^2 - x + 3) - 7/9*(-10*x^2 - x + 3)^(3
/2)/(3*x + 2)

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Fricas [A]  time = 0.232213, size = 158, normalized size = 1.01 \[ -\frac{\sqrt{10}{\left (32200 \, \sqrt{10} \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{10}{\left (7200 \, x^{3} - 13740 \, x^{2} + 17277 \, x + 21718\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 326717 \,{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{194400 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/194400*sqrt(10)*(32200*sqrt(10)*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x +
20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(10)*(7200*x^3 - 13740*x^2 + 17277*
x + 21718)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 326717*(3*x + 2)*arctan(1/20*sqrt(10)*
(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(3*x + 2)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.408503, size = 412, normalized size = 2.62 \[ -\frac{161}{972} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{5400} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 151 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4817 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{326717}{194400} \, \sqrt{10}{\left (\pi - 2 \, \arctan \left (\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1078 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{81 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-161/972*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(2
2)))) + 1/5400*(4*(8*sqrt(5)*(5*x + 3) - 151*sqrt(5))*(5*x + 3) + 4817*sqrt(5))*
sqrt(5*x + 3)*sqrt(-10*x + 5) + 326717/194400*sqrt(10)*(pi - 2*arctan(1/4*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)))) + 1078/81*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(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)