3.2398 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{2+3 x} \, dx\)

Optimal. Leaf size=172 \[ \frac{1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}+\frac{37}{360} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{4783 \sqrt{1-2 x} (5 x+3)^{5/2}}{32400}-\frac{14557 \sqrt{1-2 x} (5 x+3)^{3/2}}{28800}-\frac{1994287 \sqrt{1-2 x} \sqrt{5 x+3}}{3110400}+\frac{109715471 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9331200 \sqrt{10}}+\frac{98}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-1994287*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3110400 - (14557*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2))/28800 + (4783*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/32400 + (37*(1 - 2*x)^(3/2)*
(3 + 5*x)^(5/2))/360 + ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/15 + (109715471*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(9331200*Sqrt[10]) + (98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/729

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Rubi [A]  time = 0.446653, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{1}{15} (1-2 x)^{5/2} (5 x+3)^{5/2}+\frac{37}{360} (1-2 x)^{3/2} (5 x+3)^{5/2}+\frac{4783 \sqrt{1-2 x} (5 x+3)^{5/2}}{32400}-\frac{14557 \sqrt{1-2 x} (5 x+3)^{3/2}}{28800}-\frac{1994287 \sqrt{1-2 x} \sqrt{5 x+3}}{3110400}+\frac{109715471 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{9331200 \sqrt{10}}+\frac{98}{729} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1994287*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3110400 - (14557*Sqrt[1 - 2*x]*(3 + 5*x)^
(3/2))/28800 + (4783*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/32400 + (37*(1 - 2*x)^(3/2)*
(3 + 5*x)^(5/2))/360 + ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/15 + (109715471*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(9331200*Sqrt[10]) + (98*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/729

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Rubi in Sympy [A]  time = 44.8716, size = 158, normalized size = 0.92 \[ \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{15} - \frac{37 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{144} + \frac{2543 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{12960} + \frac{79439 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{259200} - \frac{1994287 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3110400} + \frac{109715471 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{93312000} + \frac{98 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{729} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*x + 1)**(5/2)*(5*x + 3)**(5/2)/15 - 37*(-2*x + 1)**(5/2)*(5*x + 3)**(3/2)/14
4 + 2543*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2)/12960 + 79439*sqrt(-2*x + 1)*(5*x +
3)**(3/2)/259200 - 1994287*sqrt(-2*x + 1)*sqrt(5*x + 3)/3110400 + 109715471*sqrt
(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/93312000 + 98*sqrt(7)*atan(sqrt(7)*sqrt(-2*
x + 1)/(7*sqrt(5*x + 3)))/729

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Mathematica [A]  time = 0.21048, size = 115, normalized size = 0.67 \[ \frac{60 \sqrt{1-2 x} \sqrt{5 x+3} \left (20736000 x^4-11836800 x^3-11943840 x^2+8506260 x+2165117\right )[phone] \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )[phone] \sqrt{10} \tan ^{-1}\left (\frac{20 x+1}{2 \sqrt{1-2 x} \sqrt{50 x+30}}\right )}{186624000} \]

Antiderivative was successfully verified.

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

[Out]

(60*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2165117 + 8506260*x - 11943840*x^2 - 11836800*x
^3 + 20736000*x^4) + 12544000*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt
[3 + 5*x])] + 109715471*Sqrt[10]*ArcTan[(1 + 20*x)/(2*Sqrt[1 - 2*x]*Sqrt[30 + 50
*x])])/186624000

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Maple [A]  time = 0.014, size = 149, normalized size = 0.9 \[ -{\frac{1}{186624000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -1244160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -109715471\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -510375600\,x\sqrt{-10\,{x}^{2}-x+3}-129907020\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/186624000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-1244160000*x^4*(-10*x^2-x+3)^(1/2)+71
0208000*x^3*(-10*x^2-x+3)^(1/2)[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))-109715471*10^(1/2)*arcsin(
20/11*x+1/11)-510375600*x*(-10*x^2-x+3)^(1/2)-129907020*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5108, size = 151, normalized size = 0.88 \[ \frac{1}{15} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{37}{72} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{787}{12960} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{79439}{51840} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{109715471}{186624000} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49}{729} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{865517}{3110400} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-10*x^2 - x + 3)^(5/2) + 37/72*(-10*x^2 - x + 3)^(3/2)*x - 787/12960*(-10*
x^2 - x + 3)^(3/2) + 79439/51840*sqrt(-10*x^2 - x + 3)*x + 109715471/186624000*s
qrt(10)*arcsin(20/11*x + 1/11) - 49/729*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20
/11/abs(3*x + 2)) + 865517/3110400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.235263, size = 142, normalized size = 0.83 \[ \frac{1}{186624000} \, \sqrt{10}{\left (6 \, \sqrt{10}{\left (20736000 \, x^{4} - 11836800 \, x^{3} - 11943840 \, x^{2} + 8506260 \, x + 2165117\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1254400 \, \sqrt{10} \sqrt{7} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right ) + 109715471 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/186624000*sqrt(10)*(6*sqrt(10)*(20736000*x^4 - 11836800*x^3 - 11943840*x^2 + 8
506260*x + 2165117)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1254400*sqrt(10)*sqrt(7)*arct
an(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))) + 109715471*arctan(1
/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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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)**(5/2)/(2+3*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.335548, size = 286, normalized size = 1.66 \[ -\frac{49}{7290} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{1}{77760000} \,{\left (12 \,{\left (8 \,{\left (36 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} - 713 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 112817 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 655065 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 9971435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{109715471}{186624000} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-49/7290*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)))) + 1/77760000*(12*(8*(36*(48*sqrt(5)*(5*x + 3) - 713*sqrt(5))*(5*x + 3) + 1
12817*sqrt(5))*(5*x + 3) - 655065*sqrt(5))*(5*x + 3) - 9971435*sqrt(5))*sqrt(5*x
+ 3)*sqrt(-10*x + 5) + 109715471/186624000*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))))