3.27.13 \(\int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4)} \, dx\) [2613]

3.27.13.1 Optimal result

Integrand size = 73, antiderivative size = 228 \[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c} \sqrt [3]{q+p x^4}}{-2 b \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{c} \sqrt [3]{q+p x^4}}\right )}{c^{2/3} \sqrt [3]{d}}-\frac {\log \left (b \sqrt [3]{d}+a \sqrt [3]{d} x+\sqrt [3]{c} \sqrt [3]{q+p x^4}\right )}{c^{2/3} \sqrt [3]{d}}+\frac {\log \left (b^2 d^{2/3}+2 a b d^{2/3} x+a^2 d^{2/3} x^2+\left (-b \sqrt [3]{c} \sqrt [3]{d}-a \sqrt [3]{c} \sqrt [3]{d} x\right ) \sqrt [3]{q+p x^4}+c^{2/3} \left (q+p x^4\right )^{2/3}\right )}{2 c^{2/3} \sqrt [3]{d}} \]

output

-3^(1/2)*arctan(3^(1/2)*c^(1/3)*(p*x^4+q)^(1/3)/(-2*b*d^(1/3)-2*a*d^(1/3)* 
x+c^(1/3)*(p*x^4+q)^(1/3)))/c^(2/3)/d^(1/3)-ln(b*d^(1/3)+a*d^(1/3)*x+c^(1/ 
3)*(p*x^4+q)^(1/3))/c^(2/3)/d^(1/3)+1/2*ln(b^2*d^(2/3)+2*a*b*d^(2/3)*x+a^2 
*d^(2/3)*x^2+(-b*c^(1/3)*d^(1/3)-a*c^(1/3)*d^(1/3)*x)*(p*x^4+q)^(1/3)+c^(2 
/3)*(p*x^4+q)^(2/3))/c^(2/3)/d^(1/3)
 

3.27.13.2 Mathematica [F]

\[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=\int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx \]

input

Integrate[(-3*a*q + 4*b*p*x^3 + a*p*x^4)/((q + p*x^4)^(1/3)*(b^3*d + c*q + 
 3*a*b^2*d*x + 3*a^2*b*d*x^2 + a^3*d*x^3 + c*p*x^4)),x]
 

output

Integrate[(-3*a*q + 4*b*p*x^3 + a*p*x^4)/((q + p*x^4)^(1/3)*(b^3*d + c*q + 
 3*a*b^2*d*x + 3*a^2*b*d*x^2 + a^3*d*x^3 + c*p*x^4)), x]
 

3.27.13.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a p x^4-3 a q+4 b p x^3}{\sqrt [3]{p x^4+q} \left (a^3 d x^3+3 a^2 b d x^2+3 a b^2 d x+b^3 d+c p x^4+c q\right )} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {a}{c \sqrt [3]{p x^4+q}}-\frac {x^3 \left (a^4 d-4 b c p\right )+3 a^3 b d x^2+3 a^2 b^2 d x+a \left (b^3 d+4 c q\right )}{c \sqrt [3]{p x^4+q} \left (a^3 d x^3+3 a^2 b d x^2+3 a b^2 d x+b^3 d+c p x^4+c q\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 a^3 b d \int \frac {x^2}{\sqrt [3]{p x^4+q} \left (c p x^4+a^3 d x^3+3 a^2 b d x^2+3 a b^2 d x+b^3 d+c q\right )}dx}{c}-\frac {3 a^2 b^2 d \int \frac {x}{\sqrt [3]{p x^4+q} \left (c p x^4+a^3 d x^3+3 a^2 b d x^2+3 a b^2 d x+b^3 d+c q\right )}dx}{c}-\frac {a \left (b^3 d+4 c q\right ) \int \frac {1}{\sqrt [3]{p x^4+q} \left (c p x^4+a^3 d x^3+3 a^2 b d x^2+3 a b^2 d x+b^3 d+c q\right )}dx}{c}-\frac {\left (a^4 d-4 b c p\right ) \int \frac {x^3}{\sqrt [3]{p x^4+q} \left (c p x^4+a^3 d x^3+3 a^2 b d x^2+3 a b^2 d x+b^3 d+c q\right )}dx}{c}+\frac {a x \sqrt [3]{\frac {p x^4}{q}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {5}{4},-\frac {p x^4}{q}\right )}{c \sqrt [3]{p x^4+q}}\)

input

Int[(-3*a*q + 4*b*p*x^3 + a*p*x^4)/((q + p*x^4)^(1/3)*(b^3*d + c*q + 3*a*b 
^2*d*x + 3*a^2*b*d*x^2 + a^3*d*x^3 + c*p*x^4)),x]
 

output

$Aborted
 

3.27.13.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.27.13.4 Maple [F]

\[\int \frac {p a \,x^{4}+4 b p \,x^{3}-3 a q}{\left (p \,x^{4}+q \right )^{\frac {1}{3}} \left (a^{3} d \,x^{3}+3 a^{2} b d \,x^{2}+c p \,x^{4}+3 a \,b^{2} d x +b^{3} d +c q \right )}d x\]

input

int((a*p*x^4+4*b*p*x^3-3*a*q)/(p*x^4+q)^(1/3)/(a^3*d*x^3+3*a^2*b*d*x^2+c*p 
*x^4+3*a*b^2*d*x+b^3*d+c*q),x)
 

output

int((a*p*x^4+4*b*p*x^3-3*a*q)/(p*x^4+q)^(1/3)/(a^3*d*x^3+3*a^2*b*d*x^2+c*p 
*x^4+3*a*b^2*d*x+b^3*d+c*q),x)
 

3.27.13.5 Fricas [F(-1)]

Timed out. \[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=\text {Timed out} \]

input

integrate((a*p*x^4+4*b*p*x^3-3*a*q)/(p*x^4+q)^(1/3)/(a^3*d*x^3+3*a^2*b*d*x 
^2+c*p*x^4+3*a*b^2*d*x+b^3*d+c*q),x, algorithm="fricas")
 

output

Timed out
 

3.27.13.6 Sympy [F(-1)]

Timed out. \[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=\text {Timed out} \]

input

integrate((a*p*x**4+4*b*p*x**3-3*a*q)/(p*x**4+q)**(1/3)/(a**3*d*x**3+3*a** 
2*b*d*x**2+c*p*x**4+3*a*b**2*d*x+b**3*d+c*q),x)
 

output

Timed out
 

3.27.13.7 Maxima [F]

\[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=\int { \frac {a p x^{4} + 4 \, b p x^{3} - 3 \, a q}{{\left (a^{3} d x^{3} + 3 \, a^{2} b d x^{2} + c p x^{4} + 3 \, a b^{2} d x + b^{3} d + c q\right )} {\left (p x^{4} + q\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate((a*p*x^4+4*b*p*x^3-3*a*q)/(p*x^4+q)^(1/3)/(a^3*d*x^3+3*a^2*b*d*x 
^2+c*p*x^4+3*a*b^2*d*x+b^3*d+c*q),x, algorithm="maxima")
 

output

integrate((a*p*x^4 + 4*b*p*x^3 - 3*a*q)/((a^3*d*x^3 + 3*a^2*b*d*x^2 + c*p* 
x^4 + 3*a*b^2*d*x + b^3*d + c*q)*(p*x^4 + q)^(1/3)), x)
 

3.27.13.8 Giac [F(-1)]

Timed out. \[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=\text {Timed out} \]

input

integrate((a*p*x^4+4*b*p*x^3-3*a*q)/(p*x^4+q)^(1/3)/(a^3*d*x^3+3*a^2*b*d*x 
^2+c*p*x^4+3*a*b^2*d*x+b^3*d+c*q),x, algorithm="giac")
 

output

Timed out
 

3.27.13.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-3 a q+4 b p x^3+a p x^4}{\sqrt [3]{q+p x^4} \left (b^3 d+c q+3 a b^2 d x+3 a^2 b d x^2+a^3 d x^3+c p x^4\right )} \, dx=\int \frac {a\,p\,x^4+4\,b\,p\,x^3-3\,a\,q}{{\left (p\,x^4+q\right )}^{1/3}\,\left (d\,a^3\,x^3+3\,d\,a^2\,b\,x^2+3\,d\,a\,b^2\,x+d\,b^3+c\,p\,x^4+c\,q\right )} \,d x \]

input

int((a*p*x^4 - 3*a*q + 4*b*p*x^3)/((q + p*x^4)^(1/3)*(c*q + b^3*d + c*p*x^ 
4 + a^3*d*x^3 + 3*a*b^2*d*x + 3*a^2*b*d*x^2)),x)
 

output

int((a*p*x^4 - 3*a*q + 4*b*p*x^3)/((q + p*x^4)^(1/3)*(c*q + b^3*d + c*p*x^ 
4 + a^3*d*x^3 + 3*a*b^2*d*x + 3*a^2*b*d*x^2)), x)