3.2602 \(\int \frac{(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1099 (3 x+2)^2}{726 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} (8200665 x+4898747)}{798600 \sqrt{5 x+3}}+\frac{4887 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(-1099*(2 + 3*x)^2)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(33*(1 -
2*x)^(3/2)*Sqrt[3 + 5*x]) - (Sqrt[1 - 2*x]*(4898747 + 8200665*x))/(798600*Sqrt[
3 + 5*x]) + (4887*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi [A]  time = 0.201064, 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 \[ \frac{7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt{5 x+3}}-\frac{1099 (3 x+2)^2}{726 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{\sqrt{1-2 x} (8200665 x+4898747)}{798600 \sqrt{5 x+3}}+\frac{4887 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-1099*(2 + 3*x)^2)/(726*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(33*(1 -
2*x)^(3/2)*Sqrt[3 + 5*x]) - (Sqrt[1 - 2*x]*(4898747 + 8200665*x))/(798600*Sqrt[
3 + 5*x]) + (4887*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi in Sympy [A]  time = 18.7356, size = 105, normalized size = 0.93 \[ - \frac{\sqrt{- 2 x + 1} \left (\frac{8200665 x}{8} + \frac{4898747}{8}\right )}{99825 \sqrt{5 x + 3}} + \frac{4887 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2000} - \frac{1099 \left (3 x + 2\right )^{2}}{726 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{7 \left (3 x + 2\right )^{3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}} \]

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)**(3/2),x)

[Out]

-sqrt(-2*x + 1)*(8200665*x/8 + 4898747/8)/(99825*sqrt(5*x + 3)) + 4887*sqrt(10)*
asin(sqrt(22)*sqrt(5*x + 3)/11)/2000 - 1099*(3*x + 2)**2/(726*sqrt(-2*x + 1)*sqr
t(5*x + 3)) + 7*(3*x + 2)**3/(33*(-2*x + 1)**(3/2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.149278, size = 84, normalized size = 0.74 \[ -\frac{10 \sqrt{5 x+3} \left (6468660 x^3-40488772 x^2-12657123 x+8379147\right )-19513791 \sqrt{10-20 x} \left (10 x^2+x-3\right ) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{7986000 (1-2 x)^{3/2} (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[3 + 5*x]*(8379147 - 12657123*x - 40488772*x^2 + 6468660*x^3) - 1951379
1*Sqrt[10 - 20*x]*(-3 + x + 10*x^2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(7986000*(
1 - 2*x)^(3/2)*(3 + 5*x))

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Maple [A]  time = 0.021, size = 151, normalized size = 1.3 \[{\frac{1}{15972000\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 390275820\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-156110328\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-129373200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-136596537\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+809775440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-167582940\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15972000*(1-2*x)^(1/2)*(390275820*10^(1/2)*arcsin(20/11*x+1/11)*x^3-156110328*
10^(1/2)*arcsin(20/11*x+1/11)*x^2-129373200*x^3*(-10*x^2-x+3)^(1/2)-136596537*10
^(1/2)*arcsin(20/11*x+1/11)*x+809775440*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)-167582940*(-10*x^2-x+3)^(
1/2))/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49997, size = 128, normalized size = 1.13 \[ \frac{4887}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{81 \, x^{2}}{20 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{18627221 \, x}{798600 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3910543}{199650 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2401}{264 \,{\left (2 \, \sqrt{-10 \, x^{2} - x + 3} x - \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4887/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 81/20*x^2/sqrt(-10*x^2 - x +
3) - 18627221/798600*x/sqrt(-10*x^2 - x + 3) - 3910543/199650/sqrt(-10*x^2 - x +
3) - 2401/264/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.226737, size = 134, normalized size = 1.19 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (6468660 \, x^{3} - 40488772 \, x^{2} - 12657123 \, x + 8379147\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 19513791 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{15972000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15972000*sqrt(10)*(2*sqrt(10)*(6468660*x^3 - 40488772*x^2 - 12657123*x + 8379
147)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 19513791*(20*x^3 - 8*x^2 - 7*x + 3)*arctan(1
/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(20*x^3 - 8*x^2 - 7*x +
3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{4}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257955, size = 177, normalized size = 1.57 \[ \frac{4887}{2000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{332750 \, \sqrt{5 \, x + 3}} - \frac{{\left (4 \,{\left (323433 \, \sqrt{5}{\left (5 \, x + 3\right )} - 13033138 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 214579893 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{99825000 \,{\left (2 \, x - 1\right )}^{2}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{166375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4887/2000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/332750*sqrt(10)*(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/99825000*(4*(323433*sqrt(5)*(5
*x + 3) - 13033138*sqrt(5))*(5*x + 3) + 214579893*sqrt(5))*sqrt(5*x + 3)*sqrt(-1
0*x + 5)/(2*x - 1)^2 + 2/166375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))