\(\int \frac {(-q+2 p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} (b x^3+a (q+p x^3)^3)}{x^6} \, dx\) [2389]

Optimal result

Integrand size = 60, antiderivative size = 191 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, dx=\frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (6 a q^4-4 a p q^3 x^2+15 b q x^3+24 a p q^3 x^3-16 a p^2 q^2 x^4-8 a p^2 q^2 x^5+15 b p x^6+36 a p^2 q^2 x^6-4 a p^3 q x^8+24 a p^3 q x^9+6 a p^4 x^{12}\right )}{30 x^5}+b p q \log (x)-b p q \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \]

[Out]

1/30*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(6*a*p^4*x^12+24*a*p^3*q*x^9-4*a*p^3*q*x^8+36*a*p^2*q^2*x^6-8*a*p
^2*q^2*x^5-16*a*p^2*q^2*x^4+24*a*p*q^3*x^3+15*b*p*x^6-4*a*p*q^3*x^2+6*a*q^4+15*b*q*x^3)/x^5+b*p*q*ln(x)-b*p*q*
ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2))

Rubi [F]

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

[In]

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

[Out]

p*(2*b + 3*a*p*q^2)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] - a*q^4*Defer[Int][Sqrt[q^2 - 2
*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]/x^6, x] - q*(b + a*p*q^2)*Defer[Int][Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^
6]/x^3, x] + 5*a*p^3*q*Defer[Int][x^3*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x] + 2*a*p^4*Defer[Int][x^6
*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6], x]

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, dx=\frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (15 b x^3 \left (q+p x^3\right )+2 a \left (3 q^4+3 p^4 x^{12}+2 p q^3 x^2 (-1+6 x)+2 p^3 q x^8 (-1+6 x)+2 p^2 q^2 x^4 \left (-4-2 x+9 x^2\right )\right )\right )}{30 x^5}+b p q \log (x)-b p q \log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right ) \]

[In]

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

[Out]

(Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]*(15*b*x^3*(q + p*x^3) + 2*a*(3*q^4 + 3*p^4*x^12 + 2*p*q^3*x^2*(-1 +
6*x) + 2*p^3*q*x^8*(-1 + 6*x) + 2*p^2*q^2*x^4*(-4 - 2*x + 9*x^2))))/(30*x^5) + b*p*q*Log[x] - b*p*q*Log[q + p*
x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]]

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\left (6 a \,p^{4} x^{12}+24 a \,p^{3} q \,x^{9}-4 a \,p^{3} q \,x^{8}+\left (36 q^{2} a \,p^{2}+15 b p \right ) x^{6}-8 a \,p^{2} q^{2} x^{5}-16 a \,p^{2} q^{2} x^{4}+\left (24 a \,q^{3} p +15 q b \right ) x^{3}-4 a p \,q^{3} x^{2}+6 a \,q^{4}\right ) \sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}-30 b p q \ln \left (\frac {q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}}{x}\right ) x^{5}}{30 x^{5}}\)	\(178\)

[In]

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

[Out]

1/30*((6*a*p^4*x^12+24*a*p^3*q*x^9-4*a*p^3*q*x^8+(36*a*p^2*q^2+15*b*p)*x^6-8*a*p^2*q^2*x^5-16*a*p^2*q^2*x^4+(2
4*a*p*q^3+15*b*q)*x^3-4*a*p*q^3*x^2+6*a*q^4)*(p^2*x^6+2*p*q*x^2*(-1+x)+q^2)^(1/2)-30*b*p*q*ln((q+p*x^3+(p^2*x^
6+2*p*q*x^2*(-1+x)+q^2)^(1/2))/x)*x^5)/x^5

Fricas [A] (verification not implemented)

none

Time = 9.31 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^3+a \left (q+p x^3\right )^3\right )}{x^6} \, dx=-\frac {30 \, b p q x^{5} \log \left (\frac {p x^{3} + q + \sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}}}{x}\right ) - {\left (6 \, a p^{4} x^{12} + 24 \, a p^{3} q x^{9} - 4 \, a p^{3} q x^{8} - 8 \, a p^{2} q^{2} x^{5} - 16 \, a p^{2} q^{2} x^{4} - 4 \, a p q^{3} x^{2} + 3 \, {\left (12 \, a p^{2} q^{2} + 5 \, b p\right )} x^{6} + 6 \, a q^{4} + 3 \, {\left (8 \, a p q^{3} + 5 \, b q\right )} x^{3}\right )} \sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}}}{30 \, x^{5}} \]

[In]

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

[Out]

-1/30*(30*b*p*q*x^5*log((p*x^3 + q + sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2))/x) - (6*a*p^4*x^12 + 24*a*p^
3*q*x^9 - 4*a*p^3*q*x^8 - 8*a*p^2*q^2*x^5 - 16*a*p^2*q^2*x^4 - 4*a*p*q^3*x^2 + 3*(12*a*p^2*q^2 + 5*b*p)*x^6 +
6*a*q^4 + 3*(8*a*p*q^3 + 5*b*q)*x^3)*sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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