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

Optimal. Leaf size=192 \[ \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (6 a p^4 x^{12}-4 a p^3 q x^{10}+24 a p^3 q x^9-16 a p^2 q^2 x^8-8 a p^2 q^2 x^7+36 a p^2 q^2 x^6-4 a p q^3 x^4+24 a p q^3 x^3+6 a q^4+15 b p x^9+15 b q x^6\right )}{30 x^{10}}-b p q \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+2 b p q \log (x) \]

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Rubi [F]  time = 1.33, 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 (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, 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]*(b*x^6 + a*(q + p*x^3)^3))/x^11,x]

[Out]

-2*a*q^4*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^11, x] - 5*a*p*q^3*Defer[Int][Sqrt[q^2 + 2*p
*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^8, x] - q*(2*b + 3*a*p^2*q)*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*
x^6]/x^5, x] + p*(b + a*p^2*q)*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^2, x] + a*p^4*Defer[In
t][x*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6], 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 (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, dx &=\int \left (-\frac {2 a q^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}}-\frac {5 a p q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8}-\frac {q \left (2 b+3 a p^2 q\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {p \left (b+a p^2 q\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+a p^4 x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \, dx\\ &=\left (a p^4\right ) \int x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-\left (5 a p q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8} \, dx-\left (2 a q^4\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}} \, dx+\left (p \left (b+a p^2 q\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx-\left (q \left (2 b+3 a p^2 q\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.65, 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 (b x^6+a \left (q+p x^3\right )^3\right )}{x^{11}} \, 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]*(b*x^6 + a*(q + p*x^3)^3))/x^11,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]*(b*x^6 + a*(q + p*x^3)^3))/x^11, x]

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IntegrateAlgebraic [A]  time = 0.58, size = 192, 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 (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 ) \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]*(b*x^6 + a*(q + p*x^3)^3))/x^11
,x]

[Out]

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

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

[Out]

Timed out

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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 (b x^{6} + {\left (p x^{3} + q\right )}^{3} a\right )} {\left (p x^{3} - 2 \, q\right )}}{x^{11}}\,{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)*(b*x^6+a*(p*x^3+q)^3)/x^11,x, algorithm="giac")

[Out]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (b \,x^{6}+a \left (p \,x^{3}+q \right )^{3}\right )}{x^{11}}\, dx\]

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

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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 (b x^{6} + {\left (p x^{3} + q\right )}^{3} a\right )} {\left (p x^{3} - 2 \, q\right )}}{x^{11}}\,{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)*(b*x^6+a*(p*x^3+q)^3)/x^11,x, algorithm="maxima"
)

[Out]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

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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^{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 \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)*(b*x**6+a*(p*x**3+q)**3)/x**11,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**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)

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