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

Optimal. Leaf size=159 \[ \frac {\sqrt {2 a p q+b} \tanh ^{-1}\left (\frac {\sqrt {b} x^4 \sqrt {2 a p q+b}}{\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a p x^3+a q\right )+a p^2 x^6+2 a p q x^3+a q^2+b x^4}\right )}{a \sqrt {b}}+\frac {\log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )}{a}-\frac {2 \log (x)}{a} \]

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Rubi [F]  time = 6.06, 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}}{x \left (b x^4+a \left (q+p x^3\right )^2\right )} \, 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])/(x*(b*x^4 + a*(q + p*x^3)^2)),x]

[Out]

(-2*Defer[Int][Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]/x, x])/(a*q) + 5*p*Defer[Int][(x^2*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] + (2*b*Defer[Int][(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])/(a*q) + (2*p^2*Defer[Int][(x^5
*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])/q

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

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Mathematica [F]  time = 0.70, 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}}{x \left (b x^4+a \left (q+p x^3\right )^2\right )} \, 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])/(x*(b*x^4 + a*(q + p*x^3)^2)),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])/(x*(b*x^4 + a*(q + p*x^3)^2)), x]

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IntegrateAlgebraic [A]  time = 0.80, size = 159, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b+2 a p q} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+2 a p q} x^4}{a q^2+2 a p q x^3+b x^4+a p^2 x^6+\left (a q+a p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right )}{a \sqrt {b}}-\frac {2 \log (x)}{a}+\frac {\log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right )}{a} \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])/(x*(b*x^4 + a*(q + p*x^3)^2)),
x]

[Out]

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

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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)/x/(b*x^4+a*(p*x^3+q)^2),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 (p x^{3} - 2 \, q\right )}}{{\left (b x^{4} + {\left (p x^{3} + q\right )}^{2} a\right )} x}\,{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)/x/(b*x^4+a*(p*x^3+q)^2),x, algorithm="giac")

[Out]

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

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

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)/x/(b*x^4+a*(p*x^3+q)^2),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)/x/(b*x^4+a*(p*x^3+q)^2),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 (p x^{3} - 2 \, q\right )}}{{\left (b x^{4} + {\left (p x^{3} + q\right )}^{2} a\right )} x}\,{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)/x/(b*x^4+a*(p*x^3+q)^2),x, algorithm="maxima")

[Out]

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

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

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