Integrand size = 43, antiderivative size = 457 \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt {q+p x^4}}{-\sqrt [4]{a} q+\sqrt [4]{b} x^2-\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^4}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{a} q}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [8]{a} p x^4}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\frac {\sqrt [8]{a} q}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}+\frac {\sqrt [8]{a} p x^4}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}} \] Output:
-1/8*(2+2^(1/2))^(1/2)*arctan((2^(1/2)/(2-2^(1/2))^(1/2)*a^(1/8)*b^(1/8)-2 *a^(1/8)*b^(1/8)/(2-2^(1/2))^(1/2))*x*(p*x^4+q)^(1/2)/(-a^(1/4)*q+b^(1/4)* x^2-a^(1/4)*p*x^4))/a^(3/8)/b^(5/8)+1/8*(2-2^(1/2))^(1/2)*arctan((2+2^(1/2 ))^(1/2)*a^(1/8)*b^(1/8)*x*(p*x^4+q)^(1/2)/(a^(1/4)*q-b^(1/4)*x^2+a^(1/4)* p*x^4))/a^(3/8)/b^(5/8)-1/8*(2+2^(1/2))^(1/2)*arctanh((a^(1/8)*q/(2-2^(1/2 ))^(1/2)/b^(1/8)+b^(1/8)*x^2/(2-2^(1/2))^(1/2)/a^(1/8)+a^(1/8)*p*x^4/(2-2^ (1/2))^(1/2)/b^(1/8))/x/(p*x^4+q)^(1/2))/a^(3/8)/b^(5/8)+1/8*(2-2^(1/2))^( 1/2)*arctanh((a^(1/8)*q/(2+2^(1/2))^(1/2)/b^(1/8)+b^(1/8)*x^2/(2+2^(1/2))^ (1/2)/a^(1/8)+a^(1/8)*p*x^4/(2+2^(1/2))^(1/2)/b^(1/8))/x/(p*x^4+q)^(1/2))/ a^(3/8)/b^(5/8)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 17.68 (sec) , antiderivative size = 15065, normalized size of antiderivative = 32.96 \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Result too large to show} \] Input:
Integrate[(x^4*(-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^8 + a*(q + p*x^4)^4),x]
Output:
Result too large to show
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.
| \(\displaystyle \int \frac {x^4 \left (p x^4-q\right ) \sqrt {p x^4+q}}{a \left (p x^4+q\right )^4+b x^8} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {p x^8 \sqrt {p x^4+q}}{b x^8 \left (\frac {6 a p^2 q^2}{b}+1\right )+a p^4 x^{16}+4 a p^3 q x^{12}+4 a p q^3 x^4+a q^4}+\frac {q x^4 \sqrt {p x^4+q}}{-b x^8 \left (\frac {6 a p^2 q^2}{b}+1\right )-a p^4 x^{16}-4 a p^3 q x^{12}-4 a p q^3 x^4-a q^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle q \int \frac {x^4 \sqrt {p x^4+q}}{-b x^8-a \left (p x^4+q\right )^4}dx+p \int \frac {x^8 \sqrt {p x^4+q}}{b x^8+a \left (p x^4+q\right )^4}dx\) |
Input:
Int[(x^4*(-q + p*x^4)*Sqrt[q + p*x^4])/(b*x^8 + a*(q + p*x^4)^4),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.83 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.09
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {p \,x^{4}+q}}{x}\right )}{\textit {\_R}^{5}}}{8 a}\) | \(40\) |
| pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {p \,x^{4}+q}}{x}\right )}{\textit {\_R}^{5}}}{8 a}\) | \(40\) |
| elliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right ) \sqrt {2}}{64 a}\) | \(47\) |
Input:
int(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x,method=_RETURNVE RBOSE)
Output:
1/8*sum(ln((-_R*x+(p*x^4+q)^(1/2))/x)/_R^5,_R=RootOf(_Z^8*a+b))/a
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Timed out} \] Input:
integrate(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x, algorithm ="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Timed out} \] Input:
integrate(x**4*(p*x**4-q)*(p*x**4+q)**(1/2)/(b*x**8+a*(p*x**4+q)**4),x)
Output:
Timed out
\[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\int { \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )} x^{4}}{b x^{8} + {\left (p x^{4} + q\right )}^{4} a} \,d x } \] Input:
integrate(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x, algorithm ="maxima")
Output:
integrate(sqrt(p*x^4 + q)*(p*x^4 - q)*x^4/(b*x^8 + (p*x^4 + q)^4*a), x)
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\text {Timed out} \] Input:
integrate(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x, algorithm ="giac")
Output:
Timed out
Timed out. \[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=-\int \frac {x^4\,\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{a\,{\left (p\,x^4+q\right )}^4+b\,x^8} \,d x \] Input:
int(-(x^4*(q + p*x^4)^(1/2)*(q - p*x^4))/(a*(q + p*x^4)^4 + b*x^8),x)
Output:
-int((x^4*(q + p*x^4)^(1/2)*(q - p*x^4))/(a*(q + p*x^4)^4 + b*x^8), x)
\[ \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx=\left (\int \frac {\sqrt {p \,x^{4}+q}\, x^{8}}{a \,p^{4} x^{16}+4 a \,p^{3} q \,x^{12}+6 a \,p^{2} q^{2} x^{8}+4 a p \,q^{3} x^{4}+b \,x^{8}+a \,q^{4}}d x \right ) p -\left (\int \frac {\sqrt {p \,x^{4}+q}\, x^{4}}{a \,p^{4} x^{16}+4 a \,p^{3} q \,x^{12}+6 a \,p^{2} q^{2} x^{8}+4 a p \,q^{3} x^{4}+b \,x^{8}+a \,q^{4}}d x \right ) q \] Input:
int(x^4*(p*x^4-q)*(p*x^4+q)^(1/2)/(b*x^8+a*(p*x^4+q)^4),x)
Output:
int((sqrt(p*x**4 + q)*x**8)/(a*p**4*x**16 + 4*a*p**3*q*x**12 + 6*a*p**2*q* *2*x**8 + 4*a*p*q**3*x**4 + a*q**4 + b*x**8),x)*p - int((sqrt(p*x**4 + q)* x**4)/(a*p**4*x**16 + 4*a*p**3*q*x**12 + 6*a*p**2*q**2*x**8 + 4*a*p*q**3*x **4 + a*q**4 + b*x**8),x)*q