3.24.15 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (a q^2+2 a p q x^3+b x^4+a p^2 x^6)}{x^9} \, dx\)

Optimal. Leaf size=178 \[ \frac {1}{2} \left (-a p^2 q^2-2 b p q\right ) \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+\log (x) \left (a p^2 q^2+2 b p q\right )+\frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p^3 x^9-a p^2 q x^7+3 a p^2 q x^6-a p q^2 x^4+3 a p q^2 x^3+a q^3+2 b p x^7+2 b q x^4\right )}{4 x^8} \]

________________________________________________________________________________________

Rubi [F]  time = 1.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^2 + 2*a*p*q*x^3 + b*x^4 + a*p^2*x^6))/x^9
,x]

[Out]

a*p^3*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6], x] - 2*a*q^3*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2
*p*q*x^4 + p^2*x^6]/x^9, x] - 3*a*p*q^2*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^6, 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] + b*p*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*
x^4 + p^2*x^6]/x^2, x]

Rubi steps

\begin {align*} \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 (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx &=\int \left (a p^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}-\frac {2 a q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^9}-\frac {3 a p q^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^6}-\frac {2 b q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {b p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}\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^3\right ) \int \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 (3 a p q^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^6} \, dx-\left (2 a q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^9} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.78, size = 0, normalized size = 0.00 \begin {gather*} \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 (a q^2+2 a p q x^3+b x^4+a p^2 x^6\right )}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^2 + 2*a*p*q*x^3 + b*x^4 + a*p^2*x^6
))/x^9,x]

[Out]

Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^2 + 2*a*p*q*x^3 + b*x^4 + a*p^2*x^6
))/x^9, x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.40, size = 178, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (a q^3+3 a p q^2 x^3+2 b q x^4-a p q^2 x^4+3 a p^2 q x^6+2 b p x^7-a p^2 q x^7+a p^3 x^9\right )}{4 x^8}+\left (2 b p q+a p^2 q^2\right ) \log (x)+\frac {1}{2} \left (-2 b p q-a p^2 q^2\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^2 + 2*a*p*q*x^3 + b*x^4 +
a*p^2*x^6))/x^9,x]

[Out]

(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(a*q^3 + 3*a*p*q^2*x^3 + 2*b*q*x^4 - a*p*q^2*x^4 + 3*a*p^2*q*x^6
+ 2*b*p*x^7 - a*p^2*q*x^7 + a*p^3*x^9))/(4*x^8) + (2*b*p*q + a*p^2*q^2)*Log[x] + ((-2*b*p*q - a*p^2*q^2)*Log[q
+ p*x^3 + Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]])/2

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*(a*p^2*x^6+2*a*p*q*x^3+b*x^4+a*q^2)/x^9,x, algor
ithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}\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^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*(a*p^2*x^6+2*a*p*q*x^3+b*x^4+a*q^2)/x^9,x, algor
ithm="giac")

[Out]

integrate((a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/
x^9, x)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (a \,p^{2} x^{6}+2 a p q \,x^{3}+b \,x^{4}+a \,q^{2}\right )}{x^{9}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)*(a*p^2*x^6+2*a*p*q*x^3+b*x^4+a*q^2)/x^9,x)

[Out]

int((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(a*p^2*x^6+2*a*p*q*x^3+b*x^4+a*q^2)/x^9,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}\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^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*(a*p^2*x^6+2*a*p*q*x^3+b*x^4+a*q^2)/x^9,x, algor
ithm="maxima")

[Out]

integrate((a*p^2*x^6 + 2*a*p*q*x^3 + b*x^4 + a*q^2)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/
x^9, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (2\,q-p\,x^3\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}\,\left (a\,p^2\,x^6+2\,a\,p\,q\,x^3+a\,q^2+b\,x^4\right )}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2)*(a*q^2 + b*x^4 + a*p^2*x^6 + 2*a*p*q*x^3))/x
^9,x)

[Out]

int(-((2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2)*(a*q^2 + b*x^4 + a*p^2*x^6 + 2*a*p*q*x^3))/x
^9, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b x^{4}\right )}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*(a*p**2*x**6+2*a*p*q*x**3+b*x**4+a*q**2)/
x**9,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**2*x**6 + 2*a*p*q*x**3 + a*q**2
+ b*x**4)/x**9, x)

________________________________________________________________________________________