3.24.93 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (b x^6+a (q+p x^3)^3)}{x^{11}} \, dx\) [2393]

3.24.93.1 Optimal result

Integrand size = 59, antiderivative size = 192 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (6 a q^4+24 a p q^3 x^3-4 a p q^3 x^4+15 b q x^6+36 a p^2 q^2 x^6-8 a p^2 q^2 x^7-16 a p^2 q^2 x^8+15 b p x^9+24 a p^3 q x^9-4 a p^3 q x^{10}+6 a p^4 x^{12}\right )}{30 x^{10}}+2 b p q \log (x)-b p q \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \]

output

1/30*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(6*a*p^4*x^12-4*a*p^3*q*x^10+ 
24*a*p^3*q*x^9-16*a*p^2*q^2*x^8-8*a*p^2*q^2*x^7+36*a*p^2*q^2*x^6+15*b*p*x^ 
9-4*a*p*q^3*x^4+24*a*p*q^3*x^3+15*b*q*x^6+6*a*q^4)/x^10+2*b*p*q*ln(x)-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))
 

3.24.93.2 Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\frac {\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (15 b x^6 \left (q+p x^3\right )+2 a \left (3 q^4-2 p q^3 (-6+x) x^3-2 p^3 q (-6+x) x^9+3 p^4 x^{12}-2 p^2 q^2 x^6 \left (-9+2 x+4 x^2\right )\right )\right )}{30 x^{10}}+2 b p q \log (x)-b p q \log \left (q+p x^3+\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}\right ) \]

input

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 
^6 + a*(q + p*x^3)^3))/x^11,x]
 

output

(Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]*(15*b*x^6*(q + p*x^3) + 2*a*(3*q 
^4 - 2*p*q^3*(-6 + x)*x^3 - 2*p^3*q*(-6 + x)*x^9 + 3*p^4*x^12 - 2*p^2*q^2* 
x^6*(-9 + 2*x + 4*x^2))))/(30*x^10) + 2*b*p*q*Log[x] - b*p*q*Log[q + p*x^3 
 + Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]]
 

3.24.93.3 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.

\(\displaystyle \int \frac {\left (p x^3-2 q\right ) \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a \left (p x^3+q\right )^3+b x^6\right )}{x^{11}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {q \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (3 a p^2 q+2 b\right )}{x^5}+\frac {p \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p^2 q+b\right )}{x^2}-\frac {2 a q^4 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{11}}-\frac {5 a p q^3 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^8}+a p^4 x \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -q \left (3 a p^2 q+2 b\right ) \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^5}dx+p \left (a p^2 q+b\right ) \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^2}dx-2 a q^4 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{11}}dx-5 a p q^3 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^8}dx+a p^4 \int x \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}dx\)

input

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^6 + a 
*(q + p*x^3)^3))/x^11,x]
 

output

$Aborted
 

3.24.93.3.1 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] 
]
 

3.24.93.4 Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {\left (6 a \,p^{4} x^{12}-4 a \,p^{3} q \,x^{10}+\left (24 q a \,p^{3}+15 b p \right ) x^{9}-16 a \,p^{2} q^{2} x^{8}-8 a \,p^{2} q^{2} x^{7}+\left (36 q^{2} a \,p^{2}+15 q b \right ) x^{6}-4 a p \,q^{3} x^{4}+24 a p \,q^{3} x^{3}+6 a \,q^{4}\right ) \sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}-30 b p q \ln \left (\frac {p \,x^{3}+\sqrt {\frac {p^{2} x^{6}-2 p q \,x^{3} \left (-1+x \right )+q^{2}}{x^{2}}}\, x +q}{x^{2}}\right ) x^{9}}{30 x^{9}}\)	\(188\)

input

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^6+a*(p*x^3+q) 
^3)/x^11,x,method=_RETURNVERBOSE)
 

output

1/30*((6*a*p^4*x^12-4*a*p^3*q*x^10+(24*a*p^3*q+15*b*p)*x^9-16*a*p^2*q^2*x^ 
8-8*a*p^2*q^2*x^7+(36*a*p^2*q^2+15*b*q)*x^6-4*a*p*q^3*x^4+24*a*p*q^3*x^3+6 
*a*q^4)*((p^2*x^6-2*p*q*x^3*(-1+x)+q^2)/x^2)^(1/2)-30*b*p*q*ln((p*x^3+((p^ 
2*x^6-2*p*q*x^3*(-1+x)+q^2)/x^2)^(1/2)*x+q)/x^2)*x^9)/x^9
 

3.24.93.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\text {Timed out} \]

input

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^6+a*(p* 
x^3+q)^3)/x^11,x, algorithm="fricas")
 

output

Timed out
 

3.24.93.6 Sympy [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + q^{2}} \left (a p^{3} x^{9} + 3 a p^{2} q x^{6} + 3 a p q^{2} x^{3} + a q^{3} + b x^{6}\right )}{x^{11}}\, dx \]

input

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* 
*6+a*(p*x**3+q)**3)/x**11,x)
 

output

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**3*x**9 + 3*a*p**2*q*x**6 + 3*a*p*q**2*x**3 + a*q**3 + b*x**6)/x**11, 
x)
 

3.24.93.7 Maxima [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\int { \frac {\sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (b x^{6} + {\left (p x^{3} + q\right )}^{3} a\right )} {\left (p x^{3} - 2 \, q\right )}}{x^{11}} \,d x } \]

input

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^6+a*(p* 
x^3+q)^3)/x^11,x, algorithm="maxima")
 

output

integrate(sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(b*x^6 + (p*x^3 + q) 
^3*a)*(p*x^3 - 2*q)/x^11, x)
 

3.24.93.8 Giac [F]

\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\int { \frac {\sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (b x^{6} + {\left (p x^{3} + q\right )}^{3} a\right )} {\left (p x^{3} - 2 \, q\right )}}{x^{11}} \,d x } \]

input

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^6+a*(p* 
x^3+q)^3)/x^11,x, algorithm="giac")
 

output

integrate(sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(b*x^6 + (p*x^3 + q) 
^3*a)*(p*x^3 - 2*q)/x^11, x)
 

3.24.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx=\int -\frac {\left (a\,{\left (p\,x^3+q\right )}^3+b\,x^6\right )\,\left (2\,q-p\,x^3\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}}{x^{11}} \,d x \]

input

int(-((a*(q + p*x^3)^3 + b*x^6)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 
 2*p*q*x^4)^(1/2))/x^11,x)
 

output

int(-((a*(q + p*x^3)^3 + b*x^6)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 
 2*p*q*x^4)^(1/2))/x^11, x)