3.2612 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{203 (3 x+2)^2}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x} (991010 x+627287)}{2196150 (5 x+3)^{3/2}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

[Out]

(-203*(2 + 3*x)^2)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^3)/(33*(1
- 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (Sqrt[1 - 2*x]*(627287 + 991010*x))/(2196150*(3
+ 5*x)^(3/2)) + (81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10])

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Rubi [A]  time = 0.205359, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{203 (3 x+2)^2}{242 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x} (991010 x+627287)}{2196150 (5 x+3)^{3/2}}+\frac{81 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-203*(2 + 3*x)^2)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^3)/(33*(1
- 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (Sqrt[1 - 2*x]*(627287 + 991010*x))/(2196150*(3
+ 5*x)^(3/2)) + (81*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt[10])

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Rubi in Sympy [A]  time = 19.3002, size = 107, normalized size = 0.95 \[ \frac{4 \sqrt{- 2 x + 1} \left (\frac{1486515 x}{4} + \frac{1881861}{8}\right )}{3294225 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{81 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{500} - \frac{203 \left (3 x + 2\right )^{2}}{242 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{7 \left (3 x + 2\right )^{3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(-2*x + 1)*(1486515*x/4 + 1881861/8)/(3294225*(5*x + 3)**(3/2)) + 81*sqrt(
10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/500 - 203*(3*x + 2)**2/(242*sqrt(-2*x + 1)*(
5*x + 3)**(3/2)) + 7*(3*x + 2)**3/(33*(-2*x + 1)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.247855, size = 65, normalized size = 0.58 \[ \frac{49702040 x^3+51334383 x^2+7883562 x-3014813}{2196150 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{81 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-3014813 + 7883562*x + 51334383*x^2 + 49702040*x^3)/(2196150*(1 - 2*x)^(3/2)*(3
+ 5*x)^(3/2)) - (81*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(50*Sqrt[10])

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Maple [A]  time = 0.022, size = 165, normalized size = 1.5 \[{\frac{1}{43923000\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 355776300\,\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) \sqrt{10}{x}^{4}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-209908017\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-21346578\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1026687660\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-60296260\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43923000*(1-2*x)^(1/2)*(355776300*arcsin(20/11*x+1/11)*10^(1/2)*x^4+71155260*1
0^(1/2)*arcsin(20/11*x+1/11)*x^3-209908017*10^(1/2)*arcsin(20/11*x+1/11)*x^2+994
040800*x^3*(-10*x^2-x+3)^(1/2)-21346578*10^(1/2)*arcsin(20/11*x+1/11)*x+10266876
60*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-
10*x^2-x+3)^(1/2)-60296260*(-10*x^2-x+3)^(1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(
3+5*x)^(3/2)

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Maxima [A]  time = 1.51541, size = 243, normalized size = 2.15 \[ \frac{27}{1464100} \, x{\left (\frac{7220 \, x}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{361}{\sqrt{-10 \, x^{2} - x + 3}} + \frac{21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} - \frac{81}{1000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{9747}{732050} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{1588351 \, x}{1098075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{108 \, x^{2}}{5 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{34823}{1098075 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{86854 \, x}{9075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{12682}{9075 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27/1464100*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x + 3)^(3/2)
+ 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^(3/2) - 87483/(-10*x^2 -
x + 3)^(3/2)) - 81/1000*sqrt(10)*arcsin(-20/11*x - 1/11) + 9747/732050*sqrt(-10
*x^2 - x + 3) - 1588351/1098075*x/sqrt(-10*x^2 - x + 3) + 108/5*x^2/(-10*x^2 - x
+ 3)^(3/2) - 34823/1098075/sqrt(-10*x^2 - x + 3) + 86854/9075*x/(-10*x^2 - x +
3)^(3/2) - 12682/9075/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 0.223779, size = 147, normalized size = 1.3 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (49702040 \, x^{3} + 51334383 \, x^{2} + 7883562 \, x - 3014813\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 3557763 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{43923000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/43923000*sqrt(10)*(2*sqrt(10)*(49702040*x^3 + 51334383*x^2 + 7883562*x - 30148
13)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 3557763*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)
*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(100*x^4 + 20*
x^3 - 59*x^2 - 6*x + 9)

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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((2+3*x)**4/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276901, size = 247, normalized size = 2.19 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{87846000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{81}{500} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{26620 \, \sqrt{5 \, x + 3}} + \frac{343 \,{\left (232 \, \sqrt{5}{\left (5 \, x + 3\right )} - 891 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2196150 \,{\left (2 \, x - 1\right )}^{2}} + \frac{{\left (\frac{825 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{5490375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/87846000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 81
/500*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/26620*sqrt(10)*(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 343/2196150*(232*sqrt(5)*(5*x + 3) - 8
91*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 1/5490375*(825*sqrt(10)*
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3