3.2441 \(\int \frac{(2+3 x)^4 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=135 \[ -\frac{3}{50} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3-\frac{987 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{4000}-\frac{21 \sqrt{1-2 x} (5 x+3)^{3/2} (92040 x+194923)}{640000}-\frac{97032047 \sqrt{1-2 x} \sqrt{5 x+3}}{2560000}+\frac{1067352517 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2560000 \sqrt{10}} \]

[Out]

(-97032047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 - (987*Sqrt[1 - 2*x]*(2 + 3*x)^2
*(3 + 5*x)^(3/2))/4000 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/50 - (21*
Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(194923 + 92040*x))/640000 + (1067352517*ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]])/(2560000*Sqrt[10])

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Rubi [A]  time = 0.201474, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3}{50} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3-\frac{987 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{4000}-\frac{21 \sqrt{1-2 x} (5 x+3)^{3/2} (92040 x+194923)}{640000}-\frac{97032047 \sqrt{1-2 x} \sqrt{5 x+3}}{2560000}+\frac{1067352517 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2560000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-97032047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 - (987*Sqrt[1 - 2*x]*(2 + 3*x)^2
*(3 + 5*x)^(3/2))/4000 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/50 - (21*
Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(194923 + 92040*x))/640000 + (1067352517*ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]])/(2560000*Sqrt[10])

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Rubi in Sympy [A]  time = 20.9834, size = 124, normalized size = 0.92 \[ - \frac{3 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac{3}{2}}}{50} - \frac{987 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac{3}{2}}}{4000} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}} \left (3624075 x + \frac{61400745}{8}\right )}{1200000} - \frac{97032047 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2560000} + \frac{1067352517 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{25600000} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-2*x + 1)*(3*x + 2)**3*(5*x + 3)**(3/2)/50 - 987*sqrt(-2*x + 1)*(3*x + 2
)**2*(5*x + 3)**(3/2)/4000 - sqrt(-2*x + 1)*(5*x + 3)**(3/2)*(3624075*x + 614007
45/8)/1200000 - 97032047*sqrt(-2*x + 1)*sqrt(5*x + 3)/2560000 + 1067352517*sqrt(
10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/25600000

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Mathematica [A]  time = 0.12566, size = 70, normalized size = 0.52 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (20736000 x^4+82339200 x^3+146144160 x^2+163168620 x+157419203\right )-1067352517 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25600000} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(157419203 + 163168620*x + 146144160*x^2 + 8233
9200*x^3 + 20736000*x^4) - 1067352517*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])
/25600000

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Maple [A]  time = 0.017, size = 121, normalized size = 0.9 \[{\frac{1}{51200000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -414720000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1646784000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-2922883200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3263372400\,x\sqrt{-10\,{x}^{2}-x+3}-3148384060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/51200000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-414720000*x^4*(-10*x^2-x+3)^(1/2)-16467
84000*x^3*(-10*x^2-x+3)^(1/2)-2922883200*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(
1/2)*arcsin(20/11*x+1/11)-3263372400*x*(-10*x^2-x+3)^(1/2)-3148384060*(-10*x^2-x
+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.5029, size = 122, normalized size = 0.9 \[ \frac{81}{100} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{25083}{8000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1067352517}{51200000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{180423}{32000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{8640723}{128000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{200720723}{2560000} \, \sqrt{-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81/100*(-10*x^2 - x + 3)^(3/2)*x^2 + 25083/8000*(-10*x^2 - x + 3)^(3/2)*x + 1067
352517/51200000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 180423/32000*(-10*x^2 -
x + 3)^(3/2) - 8640723/128000*sqrt(-10*x^2 - x + 3)*x - 200720723/2560000*sqrt(
-10*x^2 - x + 3)

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Fricas [A]  time = 0.222856, size = 97, normalized size = 0.72 \[ -\frac{1}{51200000} \, \sqrt{10}{\left (2 \, \sqrt{10}{\left (20736000 \, x^{4} + 82339200 \, x^{3} + 146144160 \, x^{2} + 163168620 \, x + 157419203\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 1067352517 \, \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/51200000*sqrt(10)*(2*sqrt(10)*(20736000*x^4 + 82339200*x^3 + 146144160*x^2 +
163168620*x + 157419203)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1067352517*arctan(1/20*s
qrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))

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Sympy [A]  time = 22.8166, size = 665, normalized size = 4.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + asi
n(sqrt(22)*sqrt(5*x + 3)/11)/2)/4, (x >= -3/5) & (x < 1/2)))/3125 + 24*sqrt(5)*P
iecewise((121*sqrt(2)*(sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/968 - s
qrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8
, (x >= -3/5) & (x < 1/2)))/3125 + 108*sqrt(5)*Piecewise((1331*sqrt(2)*(3*sqrt(2
)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/1936 + sqrt(2)*(-10*x + 5)**(3/2)*(5
*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22)
*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/3125 + 216*sqrt(5)*Piecewis
e((14641*sqrt(2)*(7*sqrt(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/3872 + 2*s
qrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/3993 + sqrt(2)*sqrt(-10*x + 5)*sqrt(5
*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqrt(
2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32
, (x >= -3/5) & (x < 1/2)))/3125 + 162*sqrt(5)*Piecewise((161051*sqrt(2)*(15*sqr
t(2)*(-20*x - 1)*sqrt(-10*x + 5)*sqrt(5*x + 3)/7744 - 2*sqrt(2)*(-10*x + 5)**(5/
2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(-10*x + 5)**(3/2)*(5*x + 3)**(3/2)/1331 +
5*sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x
+ 3)**2 - 5929)/3748096 - sqrt(2)*sqrt(-10*x + 5)*sqrt(5*x + 3)/22 + 63*asin(sq
rt(22)*sqrt(5*x + 3)/11)/256)/64, (x >= -3/5) & (x < 1/2)))/3125

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GIAC/XCAS [A]  time = 0.232899, size = 97, normalized size = 0.72 \[ -\frac{1}{128000000} \, \sqrt{5}{\left (2 \,{\left (12 \,{\left (24 \,{\left (12 \,{\left (240 \, x + 521\right )}{\left (5 \, x + 3\right )} + 29669\right )}{\left (5 \, x + 3\right )} + 4900505\right )}{\left (5 \, x + 3\right )} + 485160235\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 5336762585 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128000000*sqrt(5)*(2*(12*(24*(12*(240*x + 521)*(5*x + 3) + 29669)*(5*x + 3) +
4900505)*(5*x + 3) + 485160235)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 5336762585*sqrt
(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))