Problem number 850

5.13 Problem number 850

$\int x^{3} (2 + 3 x)^{3 / 2} \sqrt{1 + 4 x} d x$

Optimal antiderivative $\frac{(4103 - 7968 x) {(2 + 3 x)}^{\frac{5}{2}} {(1 + 4 x)}^{\frac{3}{2}}}{829440} + \frac{x^{2} {(2 + 3 x)}^{\frac{5}{2}} {(1 + 4 x)}^{\frac{3}{2}}}{72} + \frac{1067875 arcsinh (\frac{\sqrt{15} \sqrt{1 + 4 x}}{5}) \sqrt{3}}{254803968} + \frac{42715 {(2 + 3 x)}^{\frac{3}{2}} \sqrt{1 + 4 x}}{15925248} - \frac{8543 {(2 + 3 x)}^{\frac{5}{2}} \sqrt{1 + 4 x}}{995328} + \frac{213575 \sqrt{2 + 3 x} \sqrt{1 + 4 x}}{42467328}$

command

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

Sympy 1.10.1 under Python 3.10.4 output

$Timed out$

Sympy 1.8 under Python 3.8.8 output

$\frac{3 {(4 x + 1)}^{\frac{13}{2}}}{4096 \sqrt{12 x + 8}} + \frac{7 {(4 x + 1)}^{\frac{11}{2}}}{40960 \sqrt{12 x + 8}} - \frac{869 {(4 x + 1)}^{\frac{9}{2}}}{196608 \sqrt{12 x + 8}} + \frac{2027 {(4 x + 1)}^{\frac{7}{2}}}{1179648 \sqrt{12 x + 8}} + \frac{119135 {(4 x + 1)}^{\frac{5}{2}}}{14155776 \sqrt{12 x + 8}} - \frac{904775 {(4 x + 1)}^{\frac{3}{2}}}{84934656 \sqrt{12 x + 8}} - \frac{1067875 \sqrt{4 x + 1}}{84934656 \sqrt{12 x + 8}} + \frac{1067875 \sqrt{3} asinh (\frac{\sqrt{15} \sqrt{4 x + 1}}{5})}{254803968}$

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