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

Optimal. Leaf size=214 \[ \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (3 a p^3 x^9-3 a p^2 q x^7+9 a p^2 q x^6-3 a p q^2 x^4+9 a p q^2 x^3+3 a q^3+4 b p^2 x^8-8 b p q x^6+8 b p q x^5+4 b q^2 x^2+6 c p x^7+6 c q x^4\right )}{12 x^8}+\frac {1}{2} \left (-a p^2 q^2-2 c p q\right ) \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+\log (x) \left (a p^2 q^2+2 c p q\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.91, 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 (c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2\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]*(c*x^4 + b*x^2*(q + p*x^3) + a*(q + p*x^3)^2))
/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] - 2*b*q^2*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^7, 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*c*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*q*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^4, x] + c*p*Defer[
Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^2, x] + b*p^2*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 +
p^2*x^6]/x, 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 (c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2\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 {2 b q^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7}-\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 c q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}-\frac {b p q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^4}+\frac {c p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+\frac {b p^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x}\right ) \, dx\\ &=(c p) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx+\left (b p^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, 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 c q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx-(b p q) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^4} \, dx-\left (2 b q^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7} \, 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 = 1.27, 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 (c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2\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]*(c*x^4 + b*x^2*(q + p*x^3) + a*(q + p*x^
3)^2))/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]*(c*x^4 + b*x^2*(q + p*x^3) + a*(q + p*x^
3)^2))/x^9, x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.71, size = 214, 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 (3 a q^3+4 b q^2 x^2+9 a p q^2 x^3+6 c q x^4-3 a p q^2 x^4+8 b p q x^5-8 b p q x^6+9 a p^2 q x^6+6 c p x^7-3 a p^2 q x^7+4 b p^2 x^8+3 a p^3 x^9\right )}{12 x^8}+\left (2 c p q+a p^2 q^2\right ) \log (x)+\frac {1}{2} \left (-2 c 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]*(c*x^4 + b*x^2*(q + p*x^3) + a*
(q + p*x^3)^2))/x^9,x]

[Out]

(Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(3*a*q^3 + 4*b*q^2*x^2 + 9*a*p*q^2*x^3 + 6*c*q*x^4 - 3*a*p*q^2*x^
4 + 8*b*p*q*x^5 - 8*b*p*q*x^6 + 9*a*p^2*q*x^6 + 6*c*p*x^7 - 3*a*p^2*q*x^7 + 4*b*p^2*x^8 + 3*a*p^3*x^9))/(12*x^
8) + (2*c*p*q + a*p^2*q^2)*Log[x] + ((-2*c*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)*(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2)/x^9,x, alg
orithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (c x^{4} + {\left (p x^{3} + q\right )} b x^{2} + {\left (p x^{3} + q\right )}^{2} a\right )} {\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)*(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2)/x^9,x, alg
orithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (c \,x^{4}+b \,x^{2} \left (p \,x^{3}+q \right )+a \left (p \,x^{3}+q \right )^{2}\right )}{x^{9}}\, dx\]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (c x^{4} + {\left (p x^{3} + q\right )} b x^{2} + {\left (p x^{3} + q\right )}^{2} a\right )} {\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)*(c*x^4+b*x^2*(p*x^3+q)+a*(p*x^3+q)^2)/x^9,x, alg
orithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(-((2*q - p*x^3)*(a*(q + p*x^3)^2 + c*x^4 + b*x^2*(q + p*x^3))*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2
))/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 p x^{5} + b q x^{2} + c 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)*(c*x**4+b*x**2*(p*x**3+q)+a*(p*x**3+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*p*x**5 + b*q*x**2 + c*x**4)/x**9, x)

________________________________________________________________________________________