\(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (b x^8+a (q+p x^3)^4)}{x^{13}} \, dx\) [2733]

Optimal result

Integrand size = 59, antiderivative size = 252 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (2 a q^5+10 a p q^4 x^3-a p q^4 x^4+20 a p^2 q^3 x^6-3 a p^2 q^3 x^7+6 b q x^8-3 a p^2 q^3 x^8+20 a p^3 q^2 x^9-3 a p^3 q^2 x^{10}+6 b p x^{11}-3 a p^3 q^2 x^{11}+10 a p^4 q x^{12}-a p^4 q x^{13}+2 a p^5 x^{15}\right )}{12 x^{12}}+\left (2 b p q+a p^3 q^3\right ) \log (x)+\frac {1}{2} \left (-2 b p q-a p^3 q^3\right ) \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \]

[Out]

1/12*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(2*a*p^5*x^15-a*p^4*q*x^13+10*a*p^4*q*x^12-3*a*p^3*q^2*x^11-3*a*p
^3*q^2*x^10+20*a*p^3*q^2*x^9-3*a*p^2*q^3*x^8-3*a*p^2*q^3*x^7+6*b*p*x^11+20*a*p^2*q^3*x^6-a*p*q^4*x^4+6*b*q*x^8
+10*a*p*q^4*x^3+2*a*q^5)/x^12+(a*p^3*q^3+2*b*p*q)*ln(x)+1/2*(-a*p^3*q^3-2*b*p*q)*ln(q+p*x^3+(p^2*x^6-2*p*q*x^4
+2*p*q*x^3+q^2)^(1/2))

Rubi [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx \]

[In]

Int[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^8 + a*(q + p*x^3)^4))/x^13,x]

[Out]

-2*a*q^5*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^13, x] - 7*a*p*q^4*Defer[Int][Sqrt[q^2 + 2*p
*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^10, x] - 8*a*p^2*q^3*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^
7, x] - 2*b*q*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^5, x] - 2*a*p^3*q^2*Defer[Int][Sqrt[q^2
+ 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^4, x] + b*p*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^2,
x] + 2*a*p^4*q*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x, x] + a*p^5*Defer[Int][x^2*Sqrt[q^2 +
2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 a q^5 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{13}}-\frac {7 a p q^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{10}}-\frac {8 a p^2 q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7}-\frac {2 b q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}-\frac {2 a p^3 q^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^4}+\frac {b p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+\frac {2 a p^4 q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x}+a p^5 x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \, dx \\ & = (b p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx+\left (a p^5\right ) \int x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-(2 b q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx+\left (2 a p^4 q\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx-\left (2 a p^3 q^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^4} \, dx-\left (8 a p^2 q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7} \, dx-\left (7 a p q^4\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{10}} \, dx-\left (2 a q^5\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{13}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.69 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\frac {\left (q+p x^3\right ) \sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (6 b x^8+a \left (2 q^4-p q^3 (-8+x) x^3-p^3 q (-8+x) x^9+2 p^4 x^{12}+p^2 q^2 x^6 \left (12-2 x-3 x^2\right )\right )\right )}{12 x^{12}}+\left (2 b p q+a p^3 q^3\right ) \log (x)-\frac {1}{2} p q \left (2 b+a p^2 q^2\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}\right ) \]

[In]

Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^8 + a*(q + p*x^3)^4))/x^13,x]

[Out]

((q + p*x^3)*Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]*(6*b*x^8 + a*(2*q^4 - p*q^3*(-8 + x)*x^3 - p^3*q*(-8 + x
)*x^9 + 2*p^4*x^12 + p^2*q^2*x^6*(12 - 2*x - 3*x^2))))/(12*x^12) + (2*b*p*q + a*p^3*q^3)*Log[x] - (p*q*(2*b +
a*p^2*q^2)*Log[q + p*x^3 + Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]])/2

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(\frac {-3 p q \,x^{11} \left (a \,p^{2} q^{2}+2 b \right ) \ln \left (\frac {p \,x^{3}+\sqrt {\frac {p^{2} x^{6}-2 q \,x^{3} \left (-1+x \right ) p +q^{2}}{x^{2}}}\, x +q}{x^{2}}\right )+\left (a \,p^{4} x^{12}-\frac {a \,p^{3} q \,x^{10}}{2}+4 a \,p^{3} q \,x^{9}+3 \left (-\frac {a \,p^{2} q^{2}}{2}+b \right ) x^{8}-a \,p^{2} q^{2} x^{7}+6 a \,p^{2} q^{2} x^{6}-\frac {a p \,q^{3} x^{4}}{2}+4 a p \,q^{3} x^{3}+a \,q^{4}\right ) \sqrt {\frac {p^{2} x^{6}-2 q \,x^{3} \left (-1+x \right ) p +q^{2}}{x^{2}}}\, \left (p \,x^{3}+q \right )}{6 x^{11}}\)	\(196\)

[In]

int((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^8+a*(p*x^3+q)^4)/x^13,x,method=_RETURNVERBOSE)

[Out]

1/6*(-3*p*q*x^11*(a*p^2*q^2+2*b)*ln((p*x^3+((p^2*x^6-2*q*x^3*(-1+x)*p+q^2)/x^2)^(1/2)*x+q)/x^2)+(a*p^4*x^12-1/
2*a*p^3*q*x^10+4*a*p^3*q*x^9+3*(-1/2*a*p^2*q^2+b)*x^8-a*p^2*q^2*x^7+6*a*p^2*q^2*x^6-1/2*a*p*q^3*x^4+4*a*p*q^3*
x^3+a*q^4)*((p^2*x^6-2*q*x^3*(-1+x)*p+q^2)/x^2)^(1/2)*(p*x^3+q))/x^11

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\text {Timed out} \]

[In]

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^8+a*(p*x^3+q)^4)/x^13,x, algorithm="fricas"
)

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + q^{2}} \left (a p^{4} x^{12} + 4 a p^{3} q x^{9} + 6 a p^{2} q^{2} x^{6} + 4 a p q^{3} x^{3} + a q^{4} + b x^{8}\right )}{x^{13}}\, dx \]

[In]

integrate((p*x**3-2*q)*(p**2*x**6-2*p*q*x**4+2*p*q*x**3+q**2)**(1/2)*(b*x**8+a*(p*x**3+q)**4)/x**13,x)

[Out]

Integral((p*x**3 - 2*q)*sqrt(p**2*x**6 - 2*p*q*x**4 + 2*p*q*x**3 + q**2)*(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**13, x)

Maxima [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\int { \frac {{\left (b x^{8} + {\left (p x^{3} + q\right )}^{4} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{13}} \,d x } \]

[In]

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^8+a*(p*x^3+q)^4)/x^13,x, algorithm="maxima"
)

[Out]

integrate((b*x^8 + (p*x^3 + q)^4*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^13, x)

Giac [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\int { \frac {{\left (b x^{8} + {\left (p x^{3} + q\right )}^{4} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{13}} \,d x } \]

[In]

integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^8+a*(p*x^3+q)^4)/x^13,x, algorithm="giac")

[Out]

integrate((b*x^8 + (p*x^3 + q)^4*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^13, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx=\int -\frac {\left (a\,{\left (p\,x^3+q\right )}^4+b\,x^8\right )\,\left (2\,q-p\,x^3\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}}{x^{13}} \,d x \]

[In]

int(-((a*(q + p*x^3)^4 + b*x^8)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2))/x^13,x)

[Out]

int(-((a*(q + p*x^3)^4 + b*x^8)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2))/x^13, x)