3.838 \(\int x^3 (2+3 x)^{3/2} \sqrt{1+4 x} \, dx\)

Optimal. Leaf size=146 \[ \frac{1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}+\frac{(4103-7968 x) (4 x+1)^{3/2} (3 x+2)^{5/2}}{829440}-\frac{8543 \sqrt{4 x+1} (3 x+2)^{5/2}}{995328}+\frac{42715 \sqrt{4 x+1} (3 x+2)^{3/2}}{15925248}+\frac{213575 \sqrt{4 x+1} \sqrt{3 x+2}}{42467328}+\frac{1067875 \sinh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{4 x+1}\right )}{84934656 \sqrt{3}} \]

[Out]

(213575*Sqrt[2 + 3*x]*Sqrt[1 + 4*x])/42467328 + (42715*(2 + 3*x)^(3/2)*Sqrt[1 +
4*x])/15925248 - (8543*(2 + 3*x)^(5/2)*Sqrt[1 + 4*x])/995328 + ((4103 - 7968*x)*
(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/829440 + (x^2*(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/
72 + (1067875*ArcSinh[Sqrt[3/5]*Sqrt[1 + 4*x]])/(84934656*Sqrt[3])

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Rubi [A]  time = 0.15733, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}+\frac{(4103-7968 x) (4 x+1)^{3/2} (3 x+2)^{5/2}}{829440}-\frac{8543 \sqrt{4 x+1} (3 x+2)^{5/2}}{995328}+\frac{42715 \sqrt{4 x+1} (3 x+2)^{3/2}}{15925248}+\frac{213575 \sqrt{4 x+1} \sqrt{3 x+2}}{42467328}+\frac{1067875 \sinh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{4 x+1}\right )}{84934656 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(213575*Sqrt[2 + 3*x]*Sqrt[1 + 4*x])/42467328 + (42715*(2 + 3*x)^(3/2)*Sqrt[1 +
4*x])/15925248 - (8543*(2 + 3*x)^(5/2)*Sqrt[1 + 4*x])/995328 + ((4103 - 7968*x)*
(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/829440 + (x^2*(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/
72 + (1067875*ArcSinh[Sqrt[3/5]*Sqrt[1 + 4*x]])/(84934656*Sqrt[3])

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Rubi in Sympy [A]  time = 12.739, size = 133, normalized size = 0.91 \[ \frac{x^{2} \left (3 x + 2\right )^{\frac{5}{2}} \left (4 x + 1\right )^{\frac{3}{2}}}{72} + \frac{\left (- 1992 x + \frac{4103}{4}\right ) \left (3 x + 2\right )^{\frac{5}{2}} \left (4 x + 1\right )^{\frac{3}{2}}}{207360} - \frac{8543 \left (3 x + 2\right )^{\frac{3}{2}} \left (4 x + 1\right )^{\frac{3}{2}}}{1327104} - \frac{42715 \sqrt{3 x + 2} \left (4 x + 1\right )^{\frac{3}{2}}}{7077888} - \frac{213575 \sqrt{3 x + 2} \sqrt{4 x + 1}}{42467328} + \frac{1067875 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{15} \sqrt{4 x + 1}}{5} \right )}}{254803968} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(3*x + 2)**(5/2)*(4*x + 1)**(3/2)/72 + (-1992*x + 4103/4)*(3*x + 2)**(5/2)*
(4*x + 1)**(3/2)/207360 - 8543*(3*x + 2)**(3/2)*(4*x + 1)**(3/2)/1327104 - 42715
*sqrt(3*x + 2)*(4*x + 1)**(3/2)/7077888 - 213575*sqrt(3*x + 2)*sqrt(4*x + 1)/424
67328 + 1067875*sqrt(3)*asinh(sqrt(15)*sqrt(4*x + 1)/5)/254803968

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Mathematica [A]  time = 0.111896, size = 79, normalized size = 0.54 \[ \frac{6 \sqrt{3 x+2} \sqrt{4 x+1} \left (106168320 x^5+94666752 x^4+4119552 x^3-1849728 x^2+1089592 x-881613\right )+5339375 \sqrt{3} \log \left (2 \sqrt{3 x+2}+\sqrt{12 x+3}\right )}{1274019840} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[2 + 3*x]*Sqrt[1 + 4*x]*(-881613 + 1089592*x - 1849728*x^2 + 4119552*x^3
+ 94666752*x^4 + 106168320*x^5) + 5339375*Sqrt[3]*Log[2*Sqrt[2 + 3*x] + Sqrt[3 +
12*x]])/1274019840

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Maple [A]  time = 0.019, size = 157, normalized size = 1.1 \[{\frac{1}{2548039680}\sqrt{2+3\,x}\sqrt{4\,x+1} \left ( 1274019840\,{x}^{5}\sqrt{12\,{x}^{2}+11\,x+2}[phone]\,{x}^{4}\sqrt{12\,{x}^{2}+11\,x+2}[phone]\,{x}^{3}\sqrt{12\,{x}^{2}+11\,x+2}-22196736\,{x}^{2}\sqrt{12\,{x}^{2}+11\,x+2}+5339375\,\ln \left ({\frac{11\,\sqrt{3}}{12}}+2\,\sqrt{3}x+\sqrt{12\,{x}^{2}+11\,x+2} \right ) \sqrt{3}[phone]\,\sqrt{12\,{x}^{2}+11\,x+2}x-10579356\,\sqrt{12\,{x}^{2}+11\,x+2} \right ){\frac{1}{\sqrt{12\,{x}^{2}+11\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2548039680*(2+3*x)^(1/2)*(4*x+1)^(1/2)*(1274019840*x^5*(12*x^2+11*x+2)^(1/2)+1
136001024*x^4*(12*x^2+11*x+2)^(1/2)[phone]*x^3*(12*x^2+11*x+2)^(1/2)-22196736*
x^2*(12*x^2+11*x+2)^(1/2)+5339375*ln(11/12*3^(1/2)+2*3^(1/2)*x+(12*x^2+11*x+2)^(
1/2))*3^(1/2)[phone]*(12*x^2+11*x+2)^(1/2)*x-10579356*(12*x^2+11*x+2)^(1/2))/(
12*x^2+11*x+2)^(1/2)

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Maxima [A]  time = 1.49966, size = 163, normalized size = 1.12 \[ \frac{1}{24} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} x^{3} - \frac{1}{960} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} x^{2} - \frac{403}{92160} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} x + \frac{22933}{6635520} \,{\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac{3}{2}} - \frac{42715}{1769472} \, \sqrt{12 \, x^{2} + 11 \, x + 2} x + \frac{1067875}{509607936} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{12 \, x^{2} + 11 \, x + 2} + 24 \, x + 11\right ) - \frac{469865}{42467328} \, \sqrt{12 \, x^{2} + 11 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(12*x^2 + 11*x + 2)^(3/2)*x^3 - 1/960*(12*x^2 + 11*x + 2)^(3/2)*x^2 - 403/9
2160*(12*x^2 + 11*x + 2)^(3/2)*x + 22933/6635520*(12*x^2 + 11*x + 2)^(3/2) - 427
15/1769472*sqrt(12*x^2 + 11*x + 2)*x + 1067875/509607936*sqrt(3)*log(4*sqrt(3)*s
qrt(12*x^2 + 11*x + 2) + 24*x + 11) - 469865/42467328*sqrt(12*x^2 + 11*x + 2)

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Fricas [A]  time = 0.25113, size = 120, normalized size = 0.82 \[ \frac{1}{5096079360} \, \sqrt{3}{\left (8 \, \sqrt{3}{\left (106168320 \, x^{5} + 94666752 \, x^{4} + 4119552 \, x^{3} - 1849728 \, x^{2} + 1089592 \, x - 881613\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} + 5339375 \, \log \left (24 \,{\left (24 \, x + 11\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} + \sqrt{3}{\left (1152 \, x^{2} + 1056 \, x + 217\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5096079360*sqrt(3)*(8*sqrt(3)*(106168320*x^5 + 94666752*x^4 + 4119552*x^3 - 18
49728*x^2 + 1089592*x - 881613)*sqrt(4*x + 1)*sqrt(3*x + 2) + 5339375*log(24*(24
*x + 11)*sqrt(4*x + 1)*sqrt(3*x + 2) + sqrt(3)*(1152*x^2 + 1056*x + 217)))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.2248, size = 181, normalized size = 1.24 \[ \frac{1}{23592960} \,{\left (2 \,{\left (12 \,{\left (2 \,{\left (8 \,{\left (120 \, x - 109\right )}{\left (4 \, x + 1\right )} + 1845\right )}{\left (4 \, x + 1\right )} - 1415\right )}{\left (4 \, x + 1\right )} - 62545\right )}{\left (4 \, x + 1\right )} + 427925\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} + \frac{1}{6635520} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (96 \, x - 61\right )}{\left (4 \, x + 1\right )} + 1535\right )}{\left (4 \, x + 1\right )} + 13465\right )}{\left (4 \, x + 1\right )} - 153725\right )} \sqrt{4 \, x + 1} \sqrt{3 \, x + 2} - \frac{1067875}{254803968} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} \sqrt{4 \, x + 1} + 2 \, \sqrt{3 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/23592960*(2*(12*(2*(8*(120*x - 109)*(4*x + 1) + 1845)*(4*x + 1) - 1415)*(4*x +
1) - 62545)*(4*x + 1) + 427925)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1/6635520*(2*(12*
(18*(96*x - 61)*(4*x + 1) + 1535)*(4*x + 1) + 13465)*(4*x + 1) - 153725)*sqrt(4*
x + 1)*sqrt(3*x + 2) - 1067875/254803968*sqrt(3)*ln(-sqrt(3)*sqrt(4*x + 1) + 2*s
qrt(3*x + 2))