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

Optimal. Leaf size=56 \[ \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {p x^6+q}}\right )}{a^{3/2}}+\frac {\sqrt {p x^6+q}}{a x^2} \]

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Rubi [F]  time = 2.21, 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^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Sqrt[q + p*x^6]/(a*x^2) - (3*p^(1/3)*Sqrt[q + p*x^6])/(a*((1 + Sqrt[3])*q^(1/3) + p^(1/3)*x^2)) + (3*3^(1/4)*S
qrt[2 - Sqrt[3]]*p^(1/3)*q^(1/3)*(q^(1/3) + p^(1/3)*x^2)*Sqrt[(q^(2/3) - p^(1/3)*q^(1/3)*x^2 + p^(2/3)*x^4)/((
1 + Sqrt[3])*q^(1/3) + p^(1/3)*x^2)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*q^(1/3) + p^(1/3)*x^2)/((1 + Sqrt[3])*q
^(1/3) + p^(1/3)*x^2)], -7 - 4*Sqrt[3]])/(2*a*Sqrt[(q^(1/3)*(q^(1/3) + p^(1/3)*x^2))/((1 + Sqrt[3])*q^(1/3) +
p^(1/3)*x^2)^2]*Sqrt[q + p*x^6]) - (Sqrt[2]*3^(3/4)*p^(1/3)*q^(1/3)*(q^(1/3) + p^(1/3)*x^2)*Sqrt[(q^(2/3) - p^
(1/3)*q^(1/3)*x^2 + p^(2/3)*x^4)/((1 + Sqrt[3])*q^(1/3) + p^(1/3)*x^2)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*q^(1
/3) + p^(1/3)*x^2)/((1 + Sqrt[3])*q^(1/3) + p^(1/3)*x^2)], -7 - 4*Sqrt[3]])/(a*Sqrt[(q^(1/3)*(q^(1/3) + p^(1/3
)*x^2))/((1 + Sqrt[3])*q^(1/3) + p^(1/3)*x^2)^2]*Sqrt[q + p*x^6]) + (b*Defer[Subst][Defer[Int][Sqrt[q + p*x^3]
/(a*q + b*x^2 + a*p*x^3), x], x, x^2])/a + (3*p*Defer[Subst][Defer[Int][(x*Sqrt[q + p*x^3])/(a*q + b*x^2 + a*p
*x^3), x], x, x^2])/2

Rubi steps

\begin {align*} \int \frac {\left (-2 q+p x^6\right ) \sqrt {q+p x^6}}{x^3 \left (a q+b x^4+a p x^6\right )} \, dx &=\int \left (-\frac {2 \sqrt {q+p x^6}}{a x^3}+\frac {x \left (2 b+3 a p x^2\right ) \sqrt {q+p x^6}}{a \left (a q+b x^4+a p x^6\right )}\right ) \, dx\\ &=\frac {\int \frac {x \left (2 b+3 a p x^2\right ) \sqrt {q+p x^6}}{a q+b x^4+a p x^6} \, dx}{a}-\frac {2 \int \frac {\sqrt {q+p x^6}}{x^3} \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(2 b+3 a p x) \sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )}{2 a}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {q+p x^3}}{x^2} \, dx,x,x^2\right )}{a}\\ &=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {\operatorname {Subst}\left (\int \left (\frac {2 b \sqrt {q+p x^3}}{a q+b x^2+a p x^3}+\frac {3 a p x \sqrt {q+p x^3}}{a q+b x^2+a p x^3}\right ) \, dx,x,x^2\right )}{2 a}-\frac {(3 p) \operatorname {Subst}\left (\int \frac {x}{\sqrt {q+p x^3}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {\sqrt {q+p x^6}}{a x^2}+\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )}{a}-\frac {\left (3 p^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x}{\sqrt {q+p x^3}} \, dx,x,x^2\right )}{2 a}+\frac {1}{2} (3 p) \operatorname {Subst}\left (\int \frac {x \sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )-\frac {\left (3 \sqrt {\frac {1}{2} \left (2-\sqrt {3}\right )} p^{2/3} \sqrt [3]{q}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {q+p x^3}} \, dx,x,x^2\right )}{a}\\ &=\frac {\sqrt {q+p x^6}}{a x^2}-\frac {3 \sqrt [3]{p} \sqrt {q+p x^6}}{a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right ) \sqrt {\frac {q^{2/3}-\sqrt [3]{p} \sqrt [3]{q} x^2+p^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}\right )|-7-4 \sqrt {3}\right )}{2 a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} \sqrt {q+p x^6}}-\frac {\sqrt {2} 3^{3/4} \sqrt [3]{p} \sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right ) \sqrt {\frac {q^{2/3}-\sqrt [3]{p} \sqrt [3]{q} x^2+p^{2/3} x^4}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}{\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2}\right )|-7-4 \sqrt {3}\right )}{a \sqrt {\frac {\sqrt [3]{q} \left (\sqrt [3]{q}+\sqrt [3]{p} x^2\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{q}+\sqrt [3]{p} x^2\right )^2}} \sqrt {q+p x^6}}+\frac {b \operatorname {Subst}\left (\int \frac {\sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )}{a}+\frac {1}{2} (3 p) \operatorname {Subst}\left (\int \frac {x \sqrt {q+p x^3}}{a q+b x^2+a p x^3} \, dx,x,x^2\right )\\ \end {align*}

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Mathematica [C]  time = 6.79, size = 2948, normalized size = 52.64 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[q + p*x^6]/(a*x^2) + (b*((-2*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p^(1/3) + ((-1)^(1/3)*q^(1/3))/p^(1/3)
)]*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x^2)*Sqrt[(((-1)^(2/3)*q^(1/3))/p^(1/3) + x^2)/(((-1)^(1/3)*q^(1/3))/p^(
1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))]*EllipticF[ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x^2)/(((-1)^(1/3) +
(-1)^(2/3))*q^(1/3))]], (-1)^(1/3)])/(a*Sqrt[(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x^2)/(-(((-1)^(1/3)*q^(1/3))/p
^(1/3)) - ((-1)^(2/3)*q^(1/3))/p^(1/3))]*Sqrt[q + p*x^6]) + (4*(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(
1/3))/p^(1/3))*q*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p^(1/3) + ((-1)^(1/3)*q^(1/3))/p^(1/3))]*Sqrt[((-(((-1)
^(2/3)*q^(1/3))/p^(1/3)) - x^2)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x^2))/(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)
^(2/3)*q^(1/3))/p^(1/3))^2]*EllipticPi[((-1)^(1/3)*q^(1/3) + (-1)^(2/3)*q^(1/3))/((-1)^(1/3)*q^(1/3) - p^(1/3)
*Root[a*q + b*#1^2 + a*p*#1^3 & , 1]), ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x^2)/(((-1)^(1/3) + (-1)^(2/3
))*q^(1/3))]], (-1)^(1/3)])/(a*p*Sqrt[q + p*x^6]*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + Root[a*q + b*#1^2 + a*p*#1
^3 & , 1])*(Root[a*q + b*#1^2 + a*p*#1^3 & , 1] - Root[a*q + b*#1^2 + a*p*#1^3 & , 2])*(Root[a*q + b*#1^2 + a*
p*#1^3 & , 1] - Root[a*q + b*#1^2 + a*p*#1^3 & , 3])) - (2*(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3)
)/p^(1/3))*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p^(1/3) + ((-1)^(1/3)*q^(1/3))/p^(1/3))]*Sqrt[((-(((-1)^(2/3)
*q^(1/3))/p^(1/3)) - x^2)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x^2))/(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)
*q^(1/3))/p^(1/3))^2]*EllipticPi[((-1)^(1/3)*q^(1/3) + (-1)^(2/3)*q^(1/3))/((-1)^(1/3)*q^(1/3) - p^(1/3)*Root[
a*q + b*#1^2 + a*p*#1^3 & , 1]), ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x^2)/(((-1)^(1/3) + (-1)^(2/3))*q^(
1/3))]], (-1)^(1/3)]*Root[a*q + b*#1^2 + a*p*#1^3 & , 1]^3)/(a*Sqrt[q + p*x^6]*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)
) + Root[a*q + b*#1^2 + a*p*#1^3 & , 1])*(Root[a*q + b*#1^2 + a*p*#1^3 & , 1] - Root[a*q + b*#1^2 + a*p*#1^3 &
, 2])*(Root[a*q + b*#1^2 + a*p*#1^3 & , 1] - Root[a*q + b*#1^2 + a*p*#1^3 & , 3])) + (4*(((-1)^(1/3)*q^(1/3))
/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))*q*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p^(1/3) + ((-1)^(1/3)*q^(1/3)
)/p^(1/3))]*Sqrt[((-(((-1)^(2/3)*q^(1/3))/p^(1/3)) - x^2)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x^2))/(((-1)^(1/3
)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))^2]*EllipticPi[((-1)^(1/3)*q^(1/3) + (-1)^(2/3)*q^(1/3))/((-
1)^(1/3)*q^(1/3) - p^(1/3)*Root[a*q + b*#1^2 + a*p*#1^3 & , 2]), ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x^2
)/(((-1)^(1/3) + (-1)^(2/3))*q^(1/3))]], (-1)^(1/3)])/(a*p*Sqrt[q + p*x^6]*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) +
Root[a*q + b*#1^2 + a*p*#1^3 & , 2])*(-Root[a*q + b*#1^2 + a*p*#1^3 & , 1] + Root[a*q + b*#1^2 + a*p*#1^3 & ,
2])*(Root[a*q + b*#1^2 + a*p*#1^3 & , 2] - Root[a*q + b*#1^2 + a*p*#1^3 & , 3])) - (2*(((-1)^(1/3)*q^(1/3))/p^
(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p^(1/3) + ((-1)^(1/3)*q^(1/3))/p^(
1/3))]*Sqrt[((-(((-1)^(2/3)*q^(1/3))/p^(1/3)) - x^2)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x^2))/(((-1)^(1/3)*q^(
1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))^2]*EllipticPi[((-1)^(1/3)*q^(1/3) + (-1)^(2/3)*q^(1/3))/((-1)^(1
/3)*q^(1/3) - p^(1/3)*Root[a*q + b*#1^2 + a*p*#1^3 & , 2]), ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x^2)/(((
-1)^(1/3) + (-1)^(2/3))*q^(1/3))]], (-1)^(1/3)]*Root[a*q + b*#1^2 + a*p*#1^3 & , 2]^3)/(a*Sqrt[q + p*x^6]*(-((
(-1)^(1/3)*q^(1/3))/p^(1/3)) + Root[a*q + b*#1^2 + a*p*#1^3 & , 2])*(-Root[a*q + b*#1^2 + a*p*#1^3 & , 1] + Ro
ot[a*q + b*#1^2 + a*p*#1^3 & , 2])*(Root[a*q + b*#1^2 + a*p*#1^3 & , 2] - Root[a*q + b*#1^2 + a*p*#1^3 & , 3])
) + (4*(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))*q*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p
^(1/3) + ((-1)^(1/3)*q^(1/3))/p^(1/3))]*Sqrt[((-(((-1)^(2/3)*q^(1/3))/p^(1/3)) - x^2)*(-(((-1)^(1/3)*q^(1/3))/
p^(1/3)) + x^2))/(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))^2]*EllipticPi[((-1)^(1/3)*q^(1/
3) + (-1)^(2/3)*q^(1/3))/((-1)^(1/3)*q^(1/3) - p^(1/3)*Root[a*q + b*#1^2 + a*p*#1^3 & , 3]), ArcSin[Sqrt[((-1)
^(1/3)*q^(1/3) - p^(1/3)*x^2)/(((-1)^(1/3) + (-1)^(2/3))*q^(1/3))]], (-1)^(1/3)])/(a*p*Sqrt[q + p*x^6]*(-(((-1
)^(1/3)*q^(1/3))/p^(1/3)) + Root[a*q + b*#1^2 + a*p*#1^3 & , 3])*(-Root[a*q + b*#1^2 + a*p*#1^3 & , 1] + Root[
a*q + b*#1^2 + a*p*#1^3 & , 3])*(-Root[a*q + b*#1^2 + a*p*#1^3 & , 2] + Root[a*q + b*#1^2 + a*p*#1^3 & , 3]))
- (2*(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))*Sqrt[(q^(1/3)/p^(1/3) + x^2)/(q^(1/3)/p^(1/
3) + ((-1)^(1/3)*q^(1/3))/p^(1/3))]*Sqrt[((-(((-1)^(2/3)*q^(1/3))/p^(1/3)) - x^2)*(-(((-1)^(1/3)*q^(1/3))/p^(1
/3)) + x^2))/(((-1)^(1/3)*q^(1/3))/p^(1/3) + ((-1)^(2/3)*q^(1/3))/p^(1/3))^2]*EllipticPi[((-1)^(1/3)*q^(1/3) +
(-1)^(2/3)*q^(1/3))/((-1)^(1/3)*q^(1/3) - p^(1/3)*Root[a*q + b*#1^2 + a*p*#1^3 & , 3]), ArcSin[Sqrt[((-1)^(1/
3)*q^(1/3) - p^(1/3)*x^2)/(((-1)^(1/3) + (-1)^(2/3))*q^(1/3))]], (-1)^(1/3)]*Root[a*q + b*#1^2 + a*p*#1^3 & ,
3]^3)/(a*Sqrt[q + p*x^6]*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + Root[a*q + b*#1^2 + a*p*#1^3 & , 3])*(-Root[a*q +
b*#1^2 + a*p*#1^3 & , 1] + Root[a*q + b*#1^2 + a*p*#1^3 & , 3])*(-Root[a*q + b*#1^2 + a*p*#1^3 & , 2] + Root[a
*q + b*#1^2 + a*p*#1^3 & , 3]))))/(2*a)

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IntegrateAlgebraic [A]  time = 11.55, size = 56, normalized size = 1.00 \begin {gather*} \frac {\sqrt {q+p x^6}}{a x^2}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a} \sqrt {q+p x^6}}\right )}{a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[q + p*x^6]/(a*x^2) + (Sqrt[b]*ArcTan[(Sqrt[b]*x^2)/(Sqrt[a]*Sqrt[q + p*x^6])])/a^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((p*x^6-2*q)*(p*x^6+q)^(1/2)/x^3/(a*p*x^6+b*x^4+a*q),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (p \,x^{6}-2 q \right ) \sqrt {p \,x^{6}+q}}{x^{3} \left (a p \,x^{6}+b \,x^{4}+a q \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {\sqrt {p\,x^6+q}\,\left (2\,q-p\,x^6\right )}{x^3\,\left (a\,p\,x^6+b\,x^4+a\,q\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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