3.2481 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{55 \sqrt{5 x+3}}-\frac{21}{550} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (3660 x+8987)}{88000}+\frac{143283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*Sqrt[3 + 5*x]) - (21*Sqrt[1 - 2*x]*(2 + 3*x)^
2*Sqrt[3 + 5*x])/550 - (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8987 + 3660*x))/88000 +
(143283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000*Sqrt[10])

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Rubi [A]  time = 0.187896, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{55 \sqrt{5 x+3}}-\frac{21}{550} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (3660 x+8987)}{88000}+\frac{143283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*Sqrt[3 + 5*x]) - (21*Sqrt[1 - 2*x]*(2 + 3*x)^
2*Sqrt[3 + 5*x])/550 - (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8987 + 3660*x))/88000 +
(143283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000*Sqrt[10])

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Rubi in Sympy [A]  time = 19.1658, size = 105, normalized size = 0.93 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{55 \sqrt{5 x + 3}} - \frac{21 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{550} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{288225 x}{2} + \frac{2830905}{8}\right )}{165000} + \frac{143283 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{80000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**3/(55*sqrt(5*x + 3)) - 21*sqrt(-2*x + 1)*(3*x + 2)*
*2*sqrt(5*x + 3)/550 - sqrt(-2*x + 1)*sqrt(5*x + 3)*(288225*x/2 + 2830905/8)/165
000 + 143283*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/80000

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Mathematica [A]  time = 0.164665, size = 65, normalized size = 0.58 \[ \frac{-\frac{10 \sqrt{1-2 x} \left (237600 x^3+849420 x^2+1477575 x+632101\right )}{\sqrt{5 x+3}}-1576113 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{880000} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*(632101 + 1477575*x + 849420*x^2 + 237600*x^3))/Sqrt[3 + 5*x
] - 1576113*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/880000

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Maple [A]  time = 0.02, size = 116, normalized size = 1. \[{\frac{1}{1760000} \left ( -4752000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7880565\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-16988400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4728339\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -29551500\,x\sqrt{-10\,{x}^{2}-x+3}-12642020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1760000*(-4752000*x^3*(-10*x^2-x+3)^(1/2)+7880565*10^(1/2)*arcsin(20/11*x+1/11
)*x-16988400*x^2*(-10*x^2-x+3)^(1/2)+4728339*10^(1/2)*arcsin(20/11*x+1/11)-29551
500*x*(-10*x^2-x+3)^(1/2)-12642020*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x
+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.49565, size = 111, normalized size = 0.98 \[ -\frac{27}{50} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{143283}{160000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3213}{2000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{95769}{40000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{6875 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/50*sqrt(-10*x^2 - x + 3)*x^2 + 143283/160000*sqrt(5)*sqrt(2)*arcsin(20/11*x
+ 1/11) - 3213/2000*sqrt(-10*x^2 - x + 3)*x - 95769/40000*sqrt(-10*x^2 - x + 3)
- 2/6875*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.224995, size = 107, normalized size = 0.95 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (237600 \, x^{3} + 849420 \, x^{2} + 1477575 \, x + 632101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1576113 \,{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1760000 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1760000*sqrt(10)*(2*sqrt(10)*(237600*x^3 + 849420*x^2 + 1477575*x + 632101)*s
qrt(5*x + 3)*sqrt(-2*x + 1) - 1576113*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)/
(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241646, size = 167, normalized size = 1.48 \[ -\frac{27}{200000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 71 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 2407 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{143283}{80000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{68750 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{34375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/200000*(4*(8*sqrt(5)*(5*x + 3) + 71*sqrt(5))*(5*x + 3) + 2407*sqrt(5))*sqrt(
5*x + 3)*sqrt(-10*x + 5) + 143283/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3)) - 1/68750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/3
4375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))