3.2378 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{925 \sqrt{5 x+3} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{8}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{32765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{648 \sqrt{7}} \]

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^3) + (5*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/(12*(2 + 3*x)^2) + (925*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(216*(2 + 3*x)) - (8*S
qrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (32765*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(648*Sqrt[7])

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Rubi [A]  time = 0.313915, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^2}+\frac{925 \sqrt{5 x+3} \sqrt{1-2 x}}{216 (3 x+2)}-\frac{8}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{32765 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{648 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^4,x]

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(9*(2 + 3*x)^3) + (5*(1 - 2*x)^(3/2)*Sqrt[3 + 5
*x])/(12*(2 + 3*x)^2) + (925*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(216*(2 + 3*x)) - (8*S
qrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (32765*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(648*Sqrt[7])

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Rubi in Sympy [A]  time = 29.7673, size = 134, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{9 \left (3 x + 2\right )^{3}} + \frac{5 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{12 \left (3 x + 2\right )^{2}} + \frac{925 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{216 \left (3 x + 2\right )} - \frac{8 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{81} - \frac{32765 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{4536} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**4,x)

[Out]

-(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(9*(3*x + 2)**3) + 5*(-2*x + 1)**(3/2)*sqrt(5*x
+ 3)/(12*(3*x + 2)**2) + 925*sqrt(-2*x + 1)*sqrt(5*x + 3)/(216*(3*x + 2)) - 8*s
qrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/81 - 32765*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/4536

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Mathematica [A]  time = 0.19442, size = 112, normalized size = 0.75 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (7689 x^2+11106 x+3856\right )}{(3 x+2)^3}-32765 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-448 \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{9072} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^4,x]

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3856 + 11106*x + 7689*x^2))/(2 + 3*x)^3 - 3276
5*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])] - 448*Sqrt[10]*A
rcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50*x])])/9072

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Maple [B]  time = 0.017, size = 253, normalized size = 1.7 \[{\frac{1}{9072\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 884655\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-12096\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+1769310\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-24192\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+1179540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-16128\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+322938\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+262120\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -3584\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +466452\,x\sqrt{-10\,{x}^{2}-x+3}+161952\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^4,x)

[Out]

1/9072*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(884655*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x^3-12096*10^(1/2)*arcsin(20/11*x+1/11)*x^3+1769310*7^(1/2
)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-24192*10^(1/2)*arcsin(2
0/11*x+1/11)*x^2+1179540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x-16128*10^(1/2)*arcsin(20/11*x+1/11)*x+322938*x^2*(-10*x^2-x+3)^(1/2)+26212
0*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-3584*10^(1/2)*arcsi
n(20/11*x+1/11)+466452*x*(-10*x^2-x+3)^(1/2)+161952*(-10*x^2-x+3)^(1/2))/(-10*x^
2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.51618, size = 178, normalized size = 1.19 \[ -\frac{4}{81} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{32765}{9072} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{145}{54} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{9 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{29 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{12 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1105 \, \sqrt{-10 \, x^{2} - x + 3}}{216 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^4,x, algorithm="maxima")

[Out]

-4/81*sqrt(10)*arcsin(20/11*x + 1/11) + 32765/9072*sqrt(7)*arcsin(37/11*x/abs(3*
x + 2) + 20/11/abs(3*x + 2)) + 145/54*sqrt(-10*x^2 - x + 3) + 7/9*(-10*x^2 - x +
3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 29/12*(-10*x^2 - x + 3)^(3/2)/(9*x^2 +
12*x + 4) - 1105/216*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.236692, size = 192, normalized size = 1.29 \[ -\frac{\sqrt{7}{\left (64 \, \sqrt{10} \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) - 6 \, \sqrt{7}{\left (7689 \, x^{2} + 11106 \, x + 3856\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 32765 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{9072 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^4,x, algorithm="fricas")

[Out]

-1/9072*sqrt(7)*(64*sqrt(10)*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sq
rt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) - 6*sqrt(7)*(7689*x^2 + 11106*
x + 3856)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 32765*(27*x^3 + 54*x^2 + 36*x + 8)*arct
an(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(27*x^3 + 54*x^2 +
36*x + 8)

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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)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.400948, size = 520, normalized size = 3.49 \[ \frac{6553}{18144} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{4}{81} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (989 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 795200 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 72520000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{108 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^4,x, algorithm="giac")

[Out]

6553/18144*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)))) - 4/81*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)))) - 11/108
*(989*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 - 795200*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)))^3 - 72520000*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(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^2 + 280)^3