3.2546 \(\int \frac{(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=84 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{\sqrt{1-2 x} (50985 x+30443)}{12100 \sqrt{5 x+3}}-\frac{999 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (Sqrt[1 - 2*x]*(30443 + 50985
*x))/(12100*Sqrt[3 + 5*x]) - (999*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[10
])

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Rubi [A]  time = 0.122271, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{7 (3 x+2)^2}{11 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{\sqrt{1-2 x} (50985 x+30443)}{12100 \sqrt{5 x+3}}-\frac{999 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{100 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(7*(2 + 3*x)^2)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (Sqrt[1 - 2*x]*(30443 + 50985
*x))/(12100*Sqrt[3 + 5*x]) - (999*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(100*Sqrt[10
])

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Rubi in Sympy [A]  time = 11.8214, size = 78, normalized size = 0.93 \[ \frac{\sqrt{- 2 x + 1} \left (\frac{50985 x}{4} + \frac{30443}{4}\right )}{3025 \sqrt{5 x + 3}} - \frac{999 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{1000} + \frac{7 \left (3 x + 2\right )^{2}}{11 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-2*x + 1)*(50985*x/4 + 30443/4)/(3025*sqrt(5*x + 3)) - 999*sqrt(10)*asin(sq
rt(22)*sqrt(5*x + 3)/11)/1000 + 7*(3*x + 2)**2/(11*sqrt(-2*x + 1)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.13241, size = 70, normalized size = 0.83 \[ \frac{-326700 x^2+824990 x+120879 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+612430}{121000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

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

[Out]

(612430 + 824990*x - 326700*x^2 + 120879*Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*ArcSin[Sq
rt[5/11]*Sqrt[1 - 2*x]])/(121000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.02, size = 120, normalized size = 1.4 \[ -{\frac{1}{-242000+484000\,x}\sqrt{1-2\,x} \left ( 1208790\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+120879\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-653400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-362637\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1649980\,x\sqrt{-10\,{x}^{2}-x+3}+1224860\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/242000*(1-2*x)^(1/2)*(1208790*10^(1/2)*arcsin(20/11*x+1/11)*x^2+120879*10^(1/
2)*arcsin(20/11*x+1/11)*x-653400*x^2*(-10*x^2-x+3)^(1/2)-362637*10^(1/2)*arcsin(
20/11*x+1/11)+1649980*x*(-10*x^2-x+3)^(1/2)+1224860*(-10*x^2-x+3)^(1/2))/(-1+2*x
)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49957, size = 78, normalized size = 0.93 \[ -\frac{27 \, x^{2}}{10 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{999}{2000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{82499 \, x}{12100 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{61243}{12100 \, \sqrt{-10 \, x^{2} - x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/10*x^2/sqrt(-10*x^2 - x + 3) + 999/2000*sqrt(10)*arcsin(-20/11*x - 1/11) + 8
2499/12100*x/sqrt(-10*x^2 - x + 3) + 61243/12100/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 0.236266, size = 108, normalized size = 1.29 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (32670 \, x^{2} - 82499 \, x - 61243\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 120879 \,{\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{242000 \,{\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/242000*sqrt(10)*(2*sqrt(10)*(32670*x^2 - 82499*x - 61243)*sqrt(5*x + 3)*sqrt(-
2*x + 1) - 120879*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3
)*sqrt(-2*x + 1))))/(10*x^2 + x - 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{3}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.250225, size = 159, normalized size = 1.89 \[ -\frac{999}{1000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6534 \, \sqrt{5}{\left (5 \, x + 3\right )} - 121687 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{302500 \,{\left (2 \, x - 1\right )}} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{30250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{15125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-999/1000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/302500*(6534*sqrt(5)*
(5*x + 3) - 121687*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 1/30250*sq
rt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/15125*sqrt(10)*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))