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

Optimal. Leaf size=180 \[ \frac {2 \sqrt {2 a^2 p q-b^2} \tan ^{-1}\left (\frac {x^2 \sqrt {2 a^2 p q-b^2}}{a \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+a p x^3+a q+b x^2}\right )}{a^2}-\frac {b \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )}{a^2}+\frac {2 b \log (x)}{a^2}+\frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{a x^2} \]

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Rubi [F]  time = 2.92, 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^3 \left (a q+b x^2+a p x^3\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^3*(a*q + b*x^2 + a*p*x^3)),x]

[Out]

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

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Mathematica [F]  time = 3.27, 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}}{x^3 \left (a q+b x^2+a p x^3\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^3*(a*q + b*x^2 + a*p*x^3)),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^3*(a*q + b*x^2 + a*p*x^3)), x]

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IntegrateAlgebraic [A]  time = 0.99, size = 180, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{a x^2}+\frac {2 \sqrt {-b^2+2 a^2 p q} \tan ^{-1}\left (\frac {\sqrt {-b^2+2 a^2 p q} x^2}{a q+b x^2+a p x^3+a \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right )}{a^2}+\frac {2 b \log (x)}{a^2}-\frac {b \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^2} \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^3*(a*q + b*x^2 + a*p*x^3)),
x]

[Out]

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

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

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

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

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sympy [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**3/(a*p*x**3+b*x**2+a*q),x)

[Out]

Timed out

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