3.28.31 \(\int \frac {(-2 q+p x^3) (a q+b x^2+a p x^3) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 (c q+d x^2+c p x^3)} \, dx\)

Optimal. Leaf size=251 \[ \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a c p x^3+a c q-2 a d x^2+2 b c x^2\right )}{2 c^2 x^4}+\frac {\log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right ) \left (-a c^2 p q+a d^2-b c d\right )}{c^3}-\frac {2 (a d-b c) \sqrt {2 c^2 p q-d^2} \tan ^{-1}\left (\frac {x^2 \sqrt {2 c^2 p q-d^2}}{c \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+c p x^3+c q+d x^2}\right )}{c^3}+\frac {2 \log (x) \left (a c^2 p q-a d^2+b c d\right )}{c^3} \]

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Rubi [F]  time = 13.94, 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 ) \left (a q+b x^2+a p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 \left (c q+d x^2+c p x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 4.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 q+p x^3\right ) \left (a q+b x^2+a p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5 \left (c q+d x^2+c p x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.22, size = 251, normalized size = 1.00 \begin {gather*} \frac {\left (a c q+2 b c x^2-2 a d x^2+a c p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{2 c^2 x^4}-\frac {2 (-b c+a d) \sqrt {-d^2+2 c^2 p q} \tan ^{-1}\left (\frac {\sqrt {-d^2+2 c^2 p q} x^2}{c q+d x^2+c p x^3+c \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}\right )}{c^3}+\frac {2 \left (b c d-a d^2+a c^2 p q\right ) \log (x)}{c^3}+\frac {\left (-b c d+a d^2-a c^2 p q\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 )}{c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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)*(a*p*x^3+b*x^2+a*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)/x^5/(c*p*x^3+d*x^2+c*q),x, a
lgorithm="fricas")

[Out]

Timed out

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giac [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)*(a*p*x^3+b*x^2+a*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)/x^5/(c*p*x^3+d*x^2+c*q),x, a
lgorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{3}-2 q \right ) \left (a p \,x^{3}+b \,x^{2}+a q \right ) \sqrt {p^{2} x^{6}-2 p q \,x^{4}+2 p q \,x^{3}+q^{2}}}{x^{5} \left (c p \,x^{3}+d \,x^{2}+c q \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((p*x^3-2*q)*(a*p*x^3+b*x^2+a*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)/x^5/(c*p*x^3+d*x^2+c*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 (a p x^{3} + b x^{2} + a q\right )} {\left (p x^{3} - 2 \, q\right )}}{{\left (c p x^{3} + d x^{2} + c q\right )} x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^3-2*q)*(a*p*x^3+b*x^2+a*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)/x^5/(c*p*x^3+d*x^2+c*q),x, a
lgorithm="maxima")

[Out]

integrate(sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(a*p*x^3 + b*x^2 + a*q)*(p*x^3 - 2*q)/((c*p*x^3 + d*x^2
+ c*q)*x^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (2\,q-p\,x^3\right )\,\left (a\,p\,x^3+b\,x^2+a\,q\right )\,\sqrt {p^2\,x^6-2\,p\,q\,x^4+2\,p\,q\,x^3+q^2}}{x^5\,\left (c\,p\,x^3+d\,x^2+c\,q\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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