3.28.33 \(\int \frac {(-2 q+p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} (b x^8+a (q+p x^3)^4)}{x^{13}} \, dx\) [2733]

Optimal. Leaf size=252 \[ \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (2 a q^5+10 a p q^4 x^3-a p q^4 x^4+20 a p^2 q^3 x^6-3 a p^2 q^3 x^7+6 b q x^8-3 a p^2 q^3 x^8+20 a p^3 q^2 x^9-3 a p^3 q^2 x^{10}+6 b p x^{11}-3 a p^3 q^2 x^{11}+10 a p^4 q x^{12}-a p^4 q x^{13}+2 a p^5 x^{15}\right )}{12 x^{12}}+\left (2 b p q+a p^3 q^3\right ) \log (x)+\frac {1}{2} \left (-2 b p q-a p^3 q^3\right ) \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \]

[Out]

1/12*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(2*a*p^5*x^15-a*p^4*q*x^13+10*a*p^4*q*x^12-3*a*p^3*q^2*x^11-3*a*p
^3*q^2*x^10+20*a*p^3*q^2*x^9-3*a*p^2*q^3*x^8-3*a*p^2*q^3*x^7+6*b*p*x^11+20*a*p^2*q^3*x^6-a*p*q^4*x^4+6*b*q*x^8
+10*a*p*q^4*x^3+2*a*q^5)/x^12+(a*p^3*q^3+2*b*p*q)*ln(x)+1/2*(-a*p^3*q^3-2*b*p*q)*ln(q+p*x^3+(p^2*x^6-2*p*q*x^4
+2*p*q*x^3+q^2)^(1/2))

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Rubi [F]
time = 1.25, 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^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, 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^8 + a*(q + p*x^3)^4))/x^13,x]

[Out]

-2*a*q^5*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^13, x] - 7*a*p*q^4*Defer[Int][Sqrt[q^2 + 2*p
*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^10, x] - 8*a*p^2*q^3*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^
7, 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] - 2*a*p^3*q^2*Defer[Int][Sqrt[q^2
+ 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x^4, 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] + 2*a*p^4*q*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x, x] + a*p^5*Defer[Int][x^2*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^8+a \left (q+p x^3\right )^4\right )}{x^{13}} \, dx &=\int \left (-\frac {2 a q^5 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{13}}-\frac {7 a p q^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{10}}-\frac {8 a p^2 q^3 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7}-\frac {2 b q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}-\frac {2 a p^3 q^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^4}+\frac {b p \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+\frac {2 a p^4 q \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x}+a p^5 x^2 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\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^5\right ) \int x^2 \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 (2 a p^4 q\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x} \, dx-\left (2 a p^3 q^2\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^4} \, dx-\left (8 a p^2 q^3\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^7} \, dx-\left (7 a p q^4\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{10}} \, dx-\left (2 a q^5\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{13}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.44, size = 160, normalized size = 0.63 \begin {gather*} \frac {1}{12} \left (\frac {\left (q+p x^3\right ) \sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (6 b x^8+a \left (2 q^4-p q^3 (-8+x) x^3-p^3 q (-8+x) x^9+2 p^4 x^{12}+p^2 q^2 x^6 \left (12-2 x-3 x^2\right )\right )\right )}{x^{12}}-6 p q \left (2 b+a p^2 q^2\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}}{q+p x^3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[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^8 + a*(q + p*x^3)^4))/x^13,x]

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}\, \left (b \,x^{8}+a \left (p \,x^{3}+q \right )^{4}\right )}{x^{13}}\, dx\]

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^8+a*(p*x^3+q)^4)/x^13,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^8+a*(p*x^3+q)^4)/x^13,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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^8+a*(p*x^3+q)^4)/x^13,x, algorithm="maxima"
)

[Out]

integrate((b*x^8 + (p*x^3 + q)^4*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^13, x)

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Fricas [A]
time = 97.08, size = 234, normalized size = 0.93 \begin {gather*} -\frac {6 \, {\left (a p^{3} q^{3} + 2 \, b p q\right )} x^{12} \log \left (-\frac {p x^{3} + q + \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}}}{x^{2}}\right ) - {\left (2 \, a p^{5} x^{15} - a p^{4} q x^{13} + 10 \, a p^{4} q x^{12} - 3 \, a p^{3} q^{2} x^{10} + 20 \, a p^{3} q^{2} x^{9} - 3 \, a p^{2} q^{3} x^{7} + 20 \, a p^{2} q^{3} x^{6} - 3 \, {\left (a p^{3} q^{2} - 2 \, b p\right )} x^{11} - a p q^{4} x^{4} + 10 \, a p q^{4} x^{3} - 3 \, {\left (a p^{2} q^{3} - 2 \, b q\right )} x^{8} + 2 \, a q^{5}\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}}}{12 \, x^{12}} \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^8+a*(p*x^3+q)^4)/x^13,x, algorithm="fricas"
)

[Out]

-1/12*(6*(a*p^3*q^3 + 2*b*p*q)*x^12*log(-(p*x^3 + q + sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2))/x^2) - (2*a
*p^5*x^15 - a*p^4*q*x^13 + 10*a*p^4*q*x^12 - 3*a*p^3*q^2*x^10 + 20*a*p^3*q^2*x^9 - 3*a*p^2*q^3*x^7 + 20*a*p^2*
q^3*x^6 - 3*(a*p^3*q^2 - 2*b*p)*x^11 - a*p*q^4*x^4 + 10*a*p*q^4*x^3 - 3*(a*p^2*q^3 - 2*b*q)*x^8 + 2*a*q^5)*sqr
t(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2))/x^12

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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^{4} x^{12} + 4 a p^{3} q x^{9} + 6 a p^{2} q^{2} x^{6} + 4 a p q^{3} x^{3} + a q^{4} + b x^{8}\right )}{x^{13}}\, 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**8+a*(p*x**3+q)**4)/x**13,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**4*x**12 + 4*a*p**3*q*x**9 + 6*a
*p**2*q**2*x**6 + 4*a*p*q**3*x**3 + a*q**4 + b*x**8)/x**13, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^8+a*(p*x^3+q)^4)/x^13,x, algorithm="giac")

[Out]

integrate((b*x^8 + (p*x^3 + q)^4*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^13, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (a\,{\left (p\,x^3+q\right )}^4+b\,x^8\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^{13}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((a*(q + p*x^3)^4 + b*x^8)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2))/x^13,x)

[Out]

int(-((a*(q + p*x^3)^4 + b*x^8)*(2*q - p*x^3)*(p^2*x^6 + q^2 + 2*p*q*x^3 - 2*p*q*x^4)^(1/2))/x^13, x)

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