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

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

________________________________________________________________________________________

Rubi [F]  time = 1.55, 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+p x^3}}{x^2 \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 + p*x^3])/(x^2*(a*q + b*x^2 + a*p*x^3)),x]

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 6.29, size = 2888, normalized size = 51.57 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[q + p*x^3])/(a*x) + (b*((-2*Sqrt[(q^(1/3)/p^(1/3) + x)/(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)*Sqrt[(((-1)^(2/3)*q^(1/3))/p^(1/3) + x)/(((-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)/(((-1)^(1/3) + (-1)^(
2/3))*q^(1/3))]], (-1)^(1/3)])/(a*Sqrt[(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x)/(-(((-1)^(1/3)*q^(1/3))/p^(1/3))
- ((-1)^(2/3)*q^(1/3))/p^(1/3))]*Sqrt[q + p*x^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)/(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)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x))/(((-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)/(((-1)^(1/3) + (-1)^(2/3))*q^(1/3))]], (
-1)^(1/3)])/(a*p*Sqrt[q + p*x^3]*(-(((-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)/(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)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x))/(((-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)/(((-1)^(1/3) + (-1)^(2/3))*q^(1/3))]], (-1)^(1/3)]*Roo
t[a*q + b*#1^2 + a*p*#1^3 & , 1]^3)/(a*Sqrt[q + p*x^3]*(-(((-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)/(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)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x))/(((-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)*Roo
t[a*q + b*#1^2 + a*p*#1^3 & , 2]), ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x)/(((-1)^(1/3) + (-1)^(2/3))*q^(
1/3))]], (-1)^(1/3)])/(a*p*Sqrt[q + p*x^3]*(-(((-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)/(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)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x))/(((-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)/(((-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^3]*(-(((-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])) + (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)/(q^(1/3)/p^(1/3) + ((-1)^(1/3)*q^(1/3))/p^(1/3))]*Sq
rt[((-(((-1)^(2/3)*q^(1/3))/p^(1/3)) - x)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x))/(((-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)/(((-1)^(1/3) + (-1
)^(2/3))*q^(1/3))]], (-1)^(1/3)])/(a*p*Sqrt[q + p*x^3]*(-(((-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)/(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)*(-(((-1)^(1/3)*q^(1/3))/p^(1/3)) + x))/(((-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)*Roo
t[a*q + b*#1^2 + a*p*#1^3 & , 3]), ArcSin[Sqrt[((-1)^(1/3)*q^(1/3) - p^(1/3)*x)/(((-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^3]*(-(((-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]))))/a

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.55, size = 56, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {q+p x^3}}{a x}+\frac {2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {q+p x^3}}\right )}{a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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*x^3+q)^(1/2)/x^2/(a*p*x^3+b*x^2+a*q),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.67, size = 877, normalized size = 15.66

method	result	size
elliptic	\(\frac {2 \sqrt {p \,x^{3}+q}}{a x}+\frac {2 i b \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{3 a^{2} p \sqrt {p \,x^{3}+q}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a p \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{2}+a q \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} b +3 a q \right ) \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i p \left (2 x +\frac {-i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {p \left (x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{-3 \left (-q \,p^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i p \left (2 x +\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{2 \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (-i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a p -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha p b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a p +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p +2 a \,p^{2} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} b \right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a p q -3 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (-q \,p^{2}\right )^{\frac {2}{3}} a p +i \sqrt {3}\, b p q -3 \underline {\hspace {1.25 ex}}\alpha a q \,p^{2}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 b p q}{2 p q b}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a p +2 b \right ) \sqrt {p \,x^{3}+q}}\right )}{a^{2} p^{2} q}\)	\(877\)
risch	\(\frac {2 \sqrt {p \,x^{3}+q}}{a x}-\frac {b \left (-\frac {2 i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{3 a p \sqrt {p \,x^{3}+q}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a p \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{2}+a q \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} b -3 a q \right ) \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i p \left (2 x +\frac {-i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {p \left (x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{-3 \left (-q \,p^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i p \left (2 x +\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{2 \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (-i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a p -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha p b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a p +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p +2 a \,p^{2} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} b \right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a p q -3 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (-q \,p^{2}\right )^{\frac {2}{3}} a p +i \sqrt {3}\, b p q -3 \underline {\hspace {1.25 ex}}\alpha a q \,p^{2}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 b p q}{2 p q b}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a p +2 b \right ) \sqrt {p \,x^{3}+q}}\right )}{a q \,p^{2} b}\right )}{a}\)	\(887\)
default	\(\frac {\frac {2 i b \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{3 a p \sqrt {p \,x^{3}+q}}-\frac {2 i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )+\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{p}\right )}{\sqrt {p \,x^{3}+q}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a p \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{2}+a q \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} b +3 a q \right ) \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i p \left (2 x +\frac {-i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {p \left (x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{-3 \left (-q \,p^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i p \left (2 x +\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{2 \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (-i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a p -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha p b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a p +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p +2 a \,p^{2} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} b \right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a p q -3 \underline {\hspace {1.25 ex}}\alpha ^{2} \left (-q \,p^{2}\right )^{\frac {2}{3}} a p +i \sqrt {3}\, b p q -3 \underline {\hspace {1.25 ex}}\alpha a q \,p^{2}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha b -3 b p q}{2 p q b}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \left (3 \underline {\hspace {1.25 ex}}\alpha a p +2 b \right ) \sqrt {p \,x^{3}+q}}\right )}{a \,p^{2} q}}{a}-\frac {2 \left (-\frac {\sqrt {p \,x^{3}+q}}{x}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )+\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{p}\right )}{\sqrt {p \,x^{3}+q}}\right )}{a}\)	\(1747\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((p*x^3-2*q)*(p*x^3+q)^(1/2)/x^2/(a*p*x^3+b*x^2+a*q),x,method=_RETURNVERBOSE)

[Out]

2*(p*x^3+q)^(1/2)/a/x+2/3*I/a^2*b*3^(1/2)/p*(-q*p^2)^(1/3)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2)/p*(-q*p^2)
^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2)*((x-1/p*(-q*p^2)^(1/3))/(-3/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2
)^(1/3)))^(1/2)*(-I*(x+1/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2)/(p
*x^3+q)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2)/p*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p
^2)^(1/3))^(1/2),(I*3^(1/2)/p*(-q*p^2)^(1/3)/(-3/2/p*(-q*p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1/2))-I/
a^2/p^2/q*2^(1/2)*sum((_alpha^2*b+3*a*q)/_alpha/(3*_alpha*a*p+2*b)*(-q*p^2)^(1/3)*(1/2*I*p*(2*x+1/p*(-I*3^(1/2
)*(-q*p^2)^(1/3)+(-q*p^2)^(1/3)))/(-q*p^2)^(1/3))^(1/2)*(p*(x-1/p*(-q*p^2)^(1/3))/(-3*(-q*p^2)^(1/3)+I*3^(1/2)
*(-q*p^2)^(1/3)))^(1/2)*(-1/2*I*p*(2*x+1/p*(I*3^(1/2)*(-q*p^2)^(1/3)+(-q*p^2)^(1/3)))/(-q*p^2)^(1/3))^(1/2)/(p
*x^3+q)^(1/2)*(-I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^2*a*p^2+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha*a*p-I*(-q*p^2)^(1/3)*3
^(1/2)*_alpha*p*b+(-q*p^2)^(1/3)*_alpha^2*a*p^2+I*(-q*p^2)^(2/3)*3^(1/2)*b+(-q*p^2)^(2/3)*_alpha*a*p+(-q*p^2)^
(1/3)*_alpha*b*p+2*a*p^2*q+(-q*p^2)^(2/3)*b)*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/p*(-q*p^2)^(1/3)-1/2*I*3^(1/2)/p
*(-q*p^2)^(1/3))*3^(1/2)*p/(-q*p^2)^(1/3))^(1/2),-1/2/p*(-I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^2*a*p+I*3^(1/2)*_alp
ha*a*p^2*q-I*(-q*p^2)^(2/3)*3^(1/2)*_alpha*b-2*I*(-q*p^2)^(1/3)*3^(1/2)*a*p*q-3*_alpha^2*(-q*p^2)^(2/3)*a*p+I*
3^(1/2)*b*p*q-3*_alpha*a*q*p^2-3*(-q*p^2)^(2/3)*_alpha*b-3*b*p*q)/q/b,(I*3^(1/2)/p*(-q*p^2)^(1/3)/(-3/2/p*(-q*
p^2)^(1/3)+1/2*I*3^(1/2)/p*(-q*p^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*a*p+_Z^2*b+a*q))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 5.99, size = 102, normalized size = 1.82 \begin {gather*} \frac {2\,\sqrt {p\,x^3+q}}{a\,x}+\frac {\sqrt {b}\,\ln \left (\frac {a^5\,b\,p^4\,x^2-a^6\,p^4\,\left (p\,x^3+q\right )+a^{11/2}\,\sqrt {b}\,p^4\,x\,\sqrt {p\,x^3+q}\,2{}\mathrm {i}}{4\,b^2\,q\,x^2+4\,a\,b\,q\,\left (p\,x^3+q\right )}\right )\,1{}\mathrm {i}}{a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(q + p*x^3)^(1/2))/(a*x) + (b^(1/2)*log((a^5*b*p^4*x^2 - a^6*p^4*(q + p*x^3) + a^(11/2)*b^(1/2)*p^4*x*(q +
p*x^3)^(1/2)*2i)/(4*b^2*q*x^2 + 4*a*b*q*(q + p*x^3)))*1i)/a^(3/2)

________________________________________________________________________________________

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*x**3+q)**(1/2)/x**2/(a*p*x**3+b*x**2+a*q),x)

[Out]

Timed out

________________________________________________________________________________________