\(\int \frac {x^4 (-2 q+p x^3) \sqrt {q+p x^3}}{b x^8+a (q+p x^3)^4} \, dx\) [2998]

Optimal result

Integrand size = 43, antiderivative size = 399 \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt {q+p x^3}}{-\sqrt [4]{a} q+\sqrt [4]{b} x^2-\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}} \]

[Out]

-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)

Rubi [F]

\[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx \]

[In]

Int[(x^4*(-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4),x]

[Out]

-2*q*Defer[Int][(x^4*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4), x] + p*Defer[Int][(x^7*Sqrt[q + p*x^3])/(b*x^
8 + a*(q + p*x^3)^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 q x^4 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}}+\frac {p x^7 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}}\right ) \, dx \\ & = p \int \frac {x^7 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}} \, dx-(2 q) \int \frac {x^4 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}} \, dx \\ & = p \int \frac {x^7 \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx-(2 q) \int \frac {x^4 \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.99 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.77 \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{-\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt [8]{a} \sqrt [8]{b} x \sqrt {-\left (\left (-2+\sqrt {2}\right ) \left (q+p x^3\right )\right )}}{-\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{b} x \sqrt {-\left (\left (-2+\sqrt {2}\right ) \left (q+p x^3\right )\right )}}{\sqrt [4]{b} x^2+\sqrt [4]{a} \left (q+p x^3\right )}\right )}{4 a^{3/8} b^{5/8}} \]

[In]

Integrate[(x^4*(-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4),x]

[Out]

(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))

Maple [C] (verified)

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	\(\text {Expression too large to display}\)	\(1596\)

[In]

int(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(ln((-_R*x+(p*x^3+q)^(1/2))/x)/_R^5,_R=RootOf(_Z^8*a+b))/a

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.40 (sec) , antiderivative size = 3667, normalized size of antiderivative = 9.19 \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\text {Too large to display} \]

[In]

integrate(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x, algorithm="fricas")

[Out]

(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^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^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^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) - sqr
t(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))
- 1/8*(-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*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) + 2*(
a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - (a^3*b^5*p*x^8 + a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) + (a^2*b^2*p^3*x^10 +
3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q^2*x^4 + a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) - (a*b*p^2*x^9 + 2*a*b*p*q*x^6
+ 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*(a*b^2*p*x^9 + 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)) + 1/8*(-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*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^2*x
^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) - 2*(a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - (a^3*b^5*p*x^8 + a^3*b^5*q*
x^5)*(-1/(a^3*b^5))^(7/8) + (a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q^2*x^4 + a^2*b^2*q^3*x)*(-1
/(a^3*b^5))^(3/8) - (a*b*p^2*x^9 + 2*a*b*p*q*x^6 + 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*(a*b^2*p*x^9 + 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)) + 1/8*I*(-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*(a^3*b^4*
p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) - 2*(I*a^2*b^4*x^
7*(-1/(a^3*b^5))^(5/8) + (I*a^3*b^5*p*x^8 + I*a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) + (-I*a^2*b^2*p^3*x^10 - 3*I
*a^2*b^2*p^2*q*x^7 - 3*I*a^2*b^2*p*q^2*x^4 - I*a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) + (-I*a*b*p^2*x^9 - 2*I*a*b
*p*q*x^6 - I*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*(a*b^2*p*x^9 + 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)) - 1/8*I*(-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*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3
*b^4*p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) - 2*(-I*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) + (-I*a^3*b^5*
p*x^8 - I*a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) + (I*a^2*b^2*p^3*x^10 + 3*I*a^2*b^2*p^2*q*x^7 + 3*I*a^2*b^2*p*q^
2*x^4 + I*a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) + (I*a*b*p^2*x^9 + 2*I*a*b*p*q*x^6 + I*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*
(a*b^2*p*x^9 + 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))

Sympy [F]

\[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\int \frac {x^{4} \left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{4} x^{12} + 4 a p^{3} q x^{9} + 6 a p^{2} q^{2} x^{6} + 4 a p q^{3} x^{3} + a q^{4} + b x^{8}}\, dx \]

[In]

integrate(x**4*(p*x**3-2*q)*(p*x**3+q)**(1/2)/(b*x**8+a*(p*x**3+q)**4),x)

[Out]

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)

Maxima [F]

\[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\int { \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )} x^{4}}{b x^{8} + {\left (p x^{3} + q\right )}^{4} a} \,d x } \]

[In]

integrate(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x, algorithm="maxima")

[Out]

integrate(sqrt(p*x^3 + q)*(p*x^3 - 2*q)*x^4/(b*x^8 + (p*x^3 + q)^4*a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\text {Timed out} \]

[In]

integrate(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx=\text {Hanged} \]

[In]

int(-(x^4*(q + p*x^3)^(1/2)*(2*q - p*x^3))/(a*(q + p*x^3)^4 + b*x^8),x)

[Out]

\text{Hanged}