Integrand size = 43, antiderivative size = 399 \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt {q+p x^3}}{-\sqrt [4]{a} q+\sqrt [4]{b} x^2-\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}} \]
-1/4*(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^3+q)^(1/2)/(-a^(1/4)*q+b^(1/4)* x^2-a^(1/4)*p*x^3))/a^(3/8)/b^(5/8)+1/4*(2-2^(1/2))^(1/2)*arctan((2+2^(1/2 ))^(1/2)*a^(1/8)*b^(1/8)*x*(p*x^3+q)^(1/2)/(a^(1/4)*q-b^(1/4)*x^2+a^(1/4)* p*x^3))/a^(3/8)/b^(5/8)-1/4*(2+2^(1/2))^(1/2)*arctanh((2-2^(1/2))^(1/2)*a^ (1/8)*b^(1/8)*x*(p*x^3+q)^(1/2)/(a^(1/4)*q+b^(1/4)*x^2+a^(1/4)*p*x^3))/a^( 3/8)/b^(5/8)+1/4*(2-2^(1/2))^(1/2)*arctanh((2+2^(1/2))^(1/2)*a^(1/8)*b^(1/ 8)*x*(p*x^3+q)^(1/2)/(a^(1/4)*q+b^(1/4)*x^2+a^(1/4)*p*x^3))/a^(3/8)/b^(5/8 )
Time = 2.99 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.77 \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{-\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt [8]{a} \sqrt [8]{b} x \sqrt {-\left (\left (-2+\sqrt {2}\right ) \left (q+p x^3\right )\right )}}{-\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{b} x \sqrt {-\left (\left (-2+\sqrt {2}\right ) \left (q+p x^3\right )\right )}}{\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )}{4 a^{3/8} b^{5/8}} \]
(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p* x^3])/(-(b^(1/4)*x^2) + a^(1/4)*(q + p*x^3))] - Sqrt[2 + Sqrt[2]]*ArcTan[( a^(1/8)*b^(1/8)*x*Sqrt[-((-2 + Sqrt[2])*(q + p*x^3))])/(-(b^(1/4)*x^2) + a ^(1/4)*(q + p*x^3))] + Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8 )*b^(1/8)*x*Sqrt[q + p*x^3])/(b^(1/4)*x^2 + a^(1/4)*(q + p*x^3))] - Sqrt[2 + Sqrt[2]]*ArcTanh[(a^(1/8)*b^(1/8)*x*Sqrt[-((-2 + Sqrt[2])*(q + p*x^3))] )/(b^(1/4)*x^2 + a^(1/4)*(q + p*x^3))])/(4*a^(3/8)*b^(5/8))
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.
\(\displaystyle \int \frac {x^4 \left (p x^3-2 q\right ) \sqrt {p x^3+q}}{a \left (p x^3+q\right )^4+b x^8} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {p x^7 \sqrt {p x^3+q}}{a p^4 x^{12}+4 a p^3 q x^9+6 a p^2 q^2 x^6+4 a p q^3 x^3+a q^4+b x^8}-\frac {2 q x^4 \sqrt {p x^3+q}}{a p^4 x^{12}+4 a p^3 q x^9+6 a p^2 q^2 x^6+4 a p q^3 x^3+a q^4+b x^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle p \int \frac {x^7 \sqrt {p x^3+q}}{b x^8+a \left (p x^3+q\right )^4}dx-2 q \int \frac {x^4 \sqrt {p x^3+q}}{b x^8+a \left (p x^3+q\right )^4}dx\) |
3.30.98.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.99 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.10
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {p \,x^{3}+q}}{x}\right )}{\textit {\_R}^{5}}}{4 a}\) | \(40\) |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {p \,x^{3}+q}}{x}\right )}{\textit {\_R}^{5}}}{4 a}\) | \(40\) |
elliptic | \(\text {Expression too large to display}\) | \(1596\) |
Result contains complex when optimal does not.
Time = 6.40 (sec) , antiderivative size = 3667, normalized size of antiderivative = 9.19 \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\text {Too large to display} \]
(1/16*I + 1/16)*sqrt(2)*(-1/(a^3*b^5))^(1/8)*log((a*p^4*x^12 + 4*a*p^3*q*x ^9 + 6*a*p^2*q^2*x^6 - b*x^8 + 4*a*p*q^3*x^3 + a*q^4 - 2*(I*a^3*b^4*p^3*x^ 11 + 3*I*a^3*b^4*p^2*q*x^8 + 3*I*a^3*b^4*p*q^2*x^5 + I*a^3*b^4*q^3*x^2)*(- 1/(a^3*b^5))^(3/4) + ((I + 1)*sqrt(2)*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - s qrt(2)*((I - 1)*a^3*b^5*p*x^8 + (I - 1)*a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8 ) - sqrt(2)*((I - 1)*a^2*b^2*p^3*x^10 + (3*I - 3)*a^2*b^2*p^2*q*x^7 + (3*I - 3)*a^2*b^2*p*q^2*x^4 + (I - 1)*a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) - sq rt(2)*(-(I + 1)*a*b*p^2*x^9 - (2*I + 2)*a*b*p*q*x^6 - (I + 1)*a*b*q^2*x^3) *(-1/(a^3*b^5))^(1/8))*sqrt(p*x^3 + q) + 2*(a^2*b^3*p^2*x^10 + 2*a^2*b^3*p *q*x^7 + a^2*b^3*q^2*x^4)*sqrt(-1/(a^3*b^5)) - 2*(-I*a*b^2*p*x^9 - I*a*b^2 *q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^ 6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4)) - (1/16*I + 1/16)*sqrt(2)*(-1/(a^3*b^5 ))^(1/8)*log((a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 - b*x^8 + 4*a*p *q^3*x^3 + a*q^4 - 2*(I*a^3*b^4*p^3*x^11 + 3*I*a^3*b^4*p^2*q*x^8 + 3*I*a^3 *b^4*p*q^2*x^5 + I*a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) + (-(I + 1)*sqrt( 2)*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - sqrt(2)*(-(I - 1)*a^3*b^5*p*x^8 - (I - 1)*a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) - sqrt(2)*(-(I - 1)*a^2*b^2*p^3* x^10 - (3*I - 3)*a^2*b^2*p^2*q*x^7 - (3*I - 3)*a^2*b^2*p*q^2*x^4 - (I - 1) *a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) - sqrt(2)*((I + 1)*a*b*p^2*x^9 + (2*I + 2)*a*b*p*q*x^6 + (I + 1)*a*b*q^2*x^3)*(-1/(a^3*b^5))^(1/8))*sqrt(p*x...
\[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\int \frac {x^{4} \left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{4} x^{12} + 4 a p^{3} q x^{9} + 6 a p^{2} q^{2} x^{6} + 4 a p q^{3} x^{3} + a q^{4} + b x^{8}}\, dx \]
Integral(x**4*(p*x**3 - 2*q)*sqrt(p*x**3 + q)/(a*p**4*x**12 + 4*a*p**3*q*x **9 + 6*a*p**2*q**2*x**6 + 4*a*p*q**3*x**3 + a*q**4 + b*x**8), x)
\[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\int { \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )} x^{4}}{b x^{8} + {\left (p x^{3} + q\right )}^{4} a} \,d x } \]
Timed out. \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\text {Hanged} \]