\(\int \frac {(-q+2 p x^3) (a q+b x+a p x^3) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3 (c q+d x+c p x^3)} \, dx\) [2678]

Optimal result

Integrand size = 71, antiderivative size = 242 \[ \int \frac {\left (-q+2 p x^3\right ) \left (a q+b x+a p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3 \left (c q+d x+c p x^3\right )} \, dx=\frac {\left (a c q+2 b c x-2 a d x+a c p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{2 c^2 x^2}-\frac {2 (-b c+a d) \sqrt {-d^2+2 c^2 p q} \arctan \left (\frac {\sqrt {-d^2+2 c^2 p q} x}{c q+d x+c p x^3+c \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}\right )}{c^3}+\frac {\left (b c d-a d^2+a c^2 p q\right ) \log (x)}{c^3}+\frac {\left (-b c d+a d^2-a c^2 p q\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right )}{c^3} \]

[Out]

1/2*(a*c*p*x^3+a*c*q-2*a*d*x+2*b*c*x)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)/c^2/x^2-2*(a*d-b*c)*(2*c^2*p*q-d
^2)^(1/2)*arctan((2*c^2*p*q-d^2)^(1/2)*x/(c*q+d*x+c*p*x^3+c*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)))/c^3+(a*c
^2*p*q-a*d^2+b*c*d)*ln(x)/c^3+(-a*c^2*p*q+a*d^2-b*c*d)*ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2))/c^3

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.61 \[ \int \frac {\left (-q+2 p x^3\right ) \left (a q+b x+a p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}}{x^3 \left (c q+d x+c p x^3\right )} \, dx=\frac {a c^2 q \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}+2 b c^2 x \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}-2 a c d x \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}+a c^2 p x^3 \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}-4 (-b c+a d) \sqrt {-d^2+2 c^2 p q} x^2 \arctan \left (\frac {\sqrt {-d^2+2 c^2 p q} x}{d x+c \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}\right )+2 b c d x^2 \log (x)-2 a d^2 x^2 \log (x)+2 a c^2 p q x^2 \log (x)-2 b c d x^2 \log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )+2 a d^2 x^2 \log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )-2 a c^2 p q x^2 \log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )}{2 c^3 x^2} \]

[In]

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

[Out]

(a*c^2*q*Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6] + 2*b*c^2*x*Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6] - 2*a*c
*d*x*Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6] + a*c^2*p*x^3*Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6] - 4*(-(b*
c) + a*d)*Sqrt[-d^2 + 2*c^2*p*q]*x^2*ArcTan[(Sqrt[-d^2 + 2*c^2*p*q]*x)/(d*x + c*(q + p*x^3 + Sqrt[q^2 + 2*p*q*
(-1 + x)*x^2 + p^2*x^6]))] + 2*b*c*d*x^2*Log[x] - 2*a*d^2*x^2*Log[x] + 2*a*c^2*p*q*x^2*Log[x] - 2*b*c*d*x^2*Lo
g[q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]] + 2*a*d^2*x^2*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x
)*x^2 + p^2*x^6]] - 2*a*c^2*p*q*x^2*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]])/(2*c^3*x^2)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(-\frac {2 \left (-\frac {c^{2} \sqrt {\frac {-2 c^{2} p q +d^{2}}{c^{2}}}\, \left (\left (a p \,x^{3}+a q +2 b x \right ) c -2 a d x \right ) \sqrt {p^{2} x^{6}+2 x^{2} p q \left (-1+x \right )+q^{2}}}{4}+\left (\frac {c \left (a \,c^{2} p q -a \,d^{2}+b c d \right ) \ln \left (\frac {q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 x^{2} p q \left (-1+x \right )+q^{2}}}{x}\right ) \sqrt {\frac {-2 c^{2} p q +d^{2}}{c^{2}}}}{2}+\left (\ln \left (\frac {\sqrt {p^{2} x^{6}+2 x^{2} p q \left (-1+x \right )+q^{2}}\, \sqrt {\frac {-2 c^{2} p q +d^{2}}{c^{2}}}\, c -2 c p q x -\left (p \,x^{3}+q \right ) d}{\left (p \,x^{3}+q \right ) c +d x}\right )+\ln \left (2\right )\right ) \left (c^{2} p q -\frac {d^{2}}{2}\right ) \left (a d -b c \right )\right ) x^{2}\right )}{\sqrt {\frac {-2 c^{2} p q +d^{2}}{c^{2}}}\, c^{4} x^{2}}\)	\(272\)

[In]

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

[Out]

-2*(-1/4*c^2*((-2*c^2*p*q+d^2)/c^2)^(1/2)*((a*p*x^3+a*q+2*b*x)*c-2*a*d*x)*(p^2*x^6+2*x^2*p*q*(-1+x)+q^2)^(1/2)
+(1/2*c*(a*c^2*p*q-a*d^2+b*c*d)*ln((q+p*x^3+(p^2*x^6+2*x^2*p*q*(-1+x)+q^2)^(1/2))/x)*((-2*c^2*p*q+d^2)/c^2)^(1
/2)+(ln(((p^2*x^6+2*x^2*p*q*(-1+x)+q^2)^(1/2)*((-2*c^2*p*q+d^2)/c^2)^(1/2)*c-2*c*p*q*x-(p*x^3+q)*d)/((p*x^3+q)
*c+d*x))+ln(2))*(c^2*p*q-1/2*d^2)*(a*d-b*c))*x^2)/((-2*c^2*p*q+d^2)/c^2)^(1/2)/c^4/x^2

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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