3.30.98 \(\int \frac {x^4 (-2 q+p x^3) \sqrt {q+p x^3}}{b x^8+a (q+p x^3)^4} \, dx\)

Optimal. Leaf size=399 \[ -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {x \left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt {p x^3+q}}{-\sqrt [4]{a} p x^3-\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^3+q}}{\sqrt [4]{a} p x^3+\sqrt [4]{a} q-\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^3+q}}{\sqrt [4]{a} p x^3+\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^3+q}}{\sqrt [4]{a} p x^3+\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}} \]

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Rubi [F]  time = 2.33, 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 {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^4*(-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4),x]

[Out]

-2*q*Defer[Int][(x^4*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4), x] + p*Defer[Int][(x^7*Sqrt[q + p*x^3])/(b*x^
8 + a*(q + p*x^3)^4), x]

Rubi steps

\begin {align*} \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx &=\int \left (-\frac {2 q x^4 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}}+\frac {p x^7 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}}\right ) \, dx\\ &=p \int \frac {x^7 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}} \, dx-(2 q) \int \frac {x^4 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}} \, dx\\ &=p \int \frac {x^7 \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx-(2 q) \int \frac {x^4 \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx\\ \end {align*}

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Mathematica [C]  time = 7.82, size = 65821, normalized size = 164.96 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*(-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 3.50, size = 367, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^4*(-2*q + p*x^3)*Sqrt[q + p*x^3])/(b*x^8 + a*(q + p*x^3)^4),x]

[Out]

-1/4*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q - b^(1/4)*x^2
+ a^(1/4)*p*x^3)])/(a^(3/8)*b^(5/8)) + (Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q +
p*x^3])/(a^(1/4)*q - b^(1/4)*x^2 + a^(1/4)*p*x^3)])/(4*a^(3/8)*b^(5/8)) - (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2
- Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q + b^(1/4)*x^2 + a^(1/4)*p*x^3)])/(4*a^(3/8)*b^(5/8))
+ (Sqrt[2 - Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*b^(1/8)*x*Sqrt[q + p*x^3])/(a^(1/4)*q + b^(1/4)*x^2 +
a^(1/4)*p*x^3)])/(4*a^(3/8)*b^(5/8))

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fricas [B]  time = 15.79, size = 2875, normalized size = 7.21

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*(-1/(a^3*b^5))^(1/8)*arctan(-1/2*((sqrt(2)*a*b^2*x^2*(-1/(a^3*b^5))^(3/8) + sqrt(2)*(a^2*b^3*p*x^
3 + a^2*b^3*q)*(-1/(a^3*b^5))^(5/8))*sqrt(p*x^3 + q) + (2*p*x^4 + 2*q*x + (sqrt(2)*a*b^2*x^2*(-1/(a^3*b^5))^(3
/8) - sqrt(2)*(a^2*b^3*p*x^3 + a^2*b^3*q)*(-1/(a^3*b^5))^(5/8))*sqrt(p*x^3 + q))*sqrt((a*p^4*x^12 + 4*a*p^3*q*
x^9 + 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4 + 4*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*
p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) + 2*(sqrt(2)*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - sqrt(2)*(a^3
*b^5*p*x^8 + a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) - sqrt(2)*(a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2
*p*q^2*x^4 + a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) + sqrt(2)*(a*b*p^2*x^9 + 2*a*b*p*q*x^6 + a*b*q^2*x^3)*(-1/(a^
3*b^5))^(1/8))*sqrt(p*x^3 + q) - 4*(a*b^2*p*x^9 + a*b^2*q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12 + 4*a*p^3*q*x
^9 + 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4)))/(p*x^4 + q*x)) - 1/4*sqrt(2)*(-1/(a^3*b^5))^(1/8)*arct
an(-1/2*((sqrt(2)*a*b^2*x^2*(-1/(a^3*b^5))^(3/8) + sqrt(2)*(a^2*b^3*p*x^3 + a^2*b^3*q)*(-1/(a^3*b^5))^(5/8))*s
qrt(p*x^3 + q) - (2*p*x^4 + 2*q*x - (sqrt(2)*a*b^2*x^2*(-1/(a^3*b^5))^(3/8) - sqrt(2)*(a^2*b^3*p*x^3 + a^2*b^3
*q)*(-1/(a^3*b^5))^(5/8))*sqrt(p*x^3 + q))*sqrt((a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*
q^3*x^3 + a*q^4 + 4*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*
b^5))^(3/4) - 2*(sqrt(2)*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - sqrt(2)*(a^3*b^5*p*x^8 + a^3*b^5*q*x^5)*(-1/(a^3*b
^5))^(7/8) - sqrt(2)*(a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q^2*x^4 + a^2*b^2*q^3*x)*(-1/(a^3*b
^5))^(3/8) + sqrt(2)*(a*b*p^2*x^9 + 2*a*b*p*q*x^6 + a*b*q^2*x^3)*(-1/(a^3*b^5))^(1/8))*sqrt(p*x^3 + q) - 4*(a*
b^2*p*x^9 + a*b^2*q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q
^3*x^3 + a*q^4)))/(p*x^4 + q*x)) - 1/16*sqrt(2)*(-1/(a^3*b^5))^(1/8)*log((a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2
*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4 + 4*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^2*x^5 + a
^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) + 2*(sqrt(2)*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - sqrt(2)*(a^3*b^5*p*x^8 +
a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) - sqrt(2)*(a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q^2*x^4 +
a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) + sqrt(2)*(a*b*p^2*x^9 + 2*a*b*p*q*x^6 + a*b*q^2*x^3)*(-1/(a^3*b^5))^(1/8)
)*sqrt(p*x^3 + q) - 4*(a*b^2*p*x^9 + a*b^2*q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*
q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4)) + 1/16*sqrt(2)*(-1/(a^3*b^5))^(1/8)*log((a*p^4*x^12 + 4*a*p^3*q*x^9
+ 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4 + 4*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^
2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) - 2*(sqrt(2)*a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - sqrt(2)*(a^3*b^5
*p*x^8 + a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) - sqrt(2)*(a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q
^2*x^4 + a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) + sqrt(2)*(a*b*p^2*x^9 + 2*a*b*p*q*x^6 + a*b*q^2*x^3)*(-1/(a^3*b^
5))^(1/8))*sqrt(p*x^3 + q) - 4*(a*b^2*p*x^9 + a*b^2*q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12 + 4*a*p^3*q*x^9 +
6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4)) - 1/2*(-1/(a^3*b^5))^(1/8)*arctan(a*b^2*x*(-1/(a^3*b^5))^(3
/8)/sqrt(p*x^3 + q)) - 1/8*(-1/(a^3*b^5))^(1/8)*log((a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 - b*x^8 + 4*
a*p*q^3*x^3 + a*q^4 + 2*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^8 + 3*a^3*b^4*p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(
a^3*b^5))^(3/4) + 2*(a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - (a^3*b^5*p*x^8 + a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) +
(a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q^2*x^4 + a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) - (a*b*p^
2*x^9 + 2*a*b*p*q*x^6 + a*b*q^2*x^3)*(-1/(a^3*b^5))^(1/8))*sqrt(p*x^3 + q) - 2*(a^2*b^3*p^2*x^10 + 2*a^2*b^3*p
*q*x^7 + a^2*b^3*q^2*x^4)*sqrt(-1/(a^3*b^5)) + 2*(a*b^2*p*x^9 + a*b^2*q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12
+ 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a*q^4)) + 1/8*(-1/(a^3*b^5))^(1/8)*log((a*p^4*x^1
2 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 - b*x^8 + 4*a*p*q^3*x^3 + a*q^4 + 2*(a^3*b^4*p^3*x^11 + 3*a^3*b^4*p^2*q*x^
8 + 3*a^3*b^4*p*q^2*x^5 + a^3*b^4*q^3*x^2)*(-1/(a^3*b^5))^(3/4) - 2*(a^2*b^4*x^7*(-1/(a^3*b^5))^(5/8) - (a^3*b
^5*p*x^8 + a^3*b^5*q*x^5)*(-1/(a^3*b^5))^(7/8) + (a^2*b^2*p^3*x^10 + 3*a^2*b^2*p^2*q*x^7 + 3*a^2*b^2*p*q^2*x^4
+ a^2*b^2*q^3*x)*(-1/(a^3*b^5))^(3/8) - (a*b*p^2*x^9 + 2*a*b*p*q*x^6 + a*b*q^2*x^3)*(-1/(a^3*b^5))^(1/8))*sqr
t(p*x^3 + q) - 2*(a^2*b^3*p^2*x^10 + 2*a^2*b^3*p*q*x^7 + a^2*b^3*q^2*x^4)*sqrt(-1/(a^3*b^5)) + 2*(a*b^2*p*x^9
+ a*b^2*q*x^6)*(-1/(a^3*b^5))^(1/4))/(a*p^4*x^12 + 4*a*p^3*q*x^9 + 6*a*p^2*q^2*x^6 + b*x^8 + 4*a*p*q^3*x^3 + a
*q^4))

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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(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.85, size = 1596, normalized size = 4.00

method	result	size
default	\(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,p^{4} \textit {\_Z}^{12}+4 a \,p^{3} q \,\textit {\_Z}^{9}+6 a \,p^{2} q^{2} \textit {\_Z}^{6}+b \,\textit {\_Z}^{8}+4 a p \,q^{3} \textit {\_Z}^{3}+a \,q^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{6} p^{2}-\underline {\hspace {1.25 ex}}\alpha ^{3} p q -2 q^{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2} \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 (\left (-q \,p^{2}\right )^{\frac {2}{3}} b \,q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} q p b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,q^{2} p b +2 q \,p^{2} \left (a \,p^{5} \underline {\hspace {1.25 ex}}\alpha ^{9}+3 a \,p^{4} q \,\underline {\hspace {1.25 ex}}\alpha ^{6}+3 a \,p^{3} q^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+b p \,\underline {\hspace {1.25 ex}}\alpha ^{5}+a \,p^{2} q^{3}-b q \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,q^{3}-\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} b p q +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p \,q^{2}+3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{6} q +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{5} q +3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{5} q^{2}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{4} q^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q^{2} b -3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,q^{3}+i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} q \,p^{2} b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{7}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{6}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{2}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{4} q^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a \,p^{3} q^{3}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} p^{2} b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{7} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \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 {3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{6} q^{2}+3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{5} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{2}-3 q^{3} b p -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,p^{3} q^{4}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{7} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{3} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{4} q^{4}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{6}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b p q -3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{5} q -6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{5} q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4} q^{2}-6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{4} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} q^{3}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} b \,p^{2} q +2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p \,q^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{6} q -3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{6} q b -3 p^{4} \underline {\hspace {1.25 ex}}\alpha a \,q^{4}+3 \underline {\hspace {1.25 ex}}\alpha ^{3} q^{2} p^{2} b -3 p^{7} \underline {\hspace {1.25 ex}}\alpha ^{10} a q -9 p^{6} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,q^{2}-9 p^{5} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{3}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,q^{2} b -3 \underline {\hspace {1.25 ex}}\alpha ^{11} p^{6} \left (-q \,p^{2}\right )^{\frac {2}{3}} a -3 \underline {\hspace {1.25 ex}}\alpha ^{7} \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{2} b -9 \underline {\hspace {1.25 ex}}\alpha ^{5} p^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} p^{3} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} q p b -9 \underline {\hspace {1.25 ex}}\alpha ^{8} p^{5} \left (-q \,p^{2}\right )^{\frac {2}{3}} a q +i \sqrt {3}\, b p \,q^{3}-i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,p^{2} q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b \,q^{2}}{2 p \,q^{3} 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 \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{4}-9 \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{3} q -9 \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2} q^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{5} b -3 a p \,q^{3}\right ) \sqrt {p \,x^{3}+q}}\right )}{4 p^{2} q^{3} b}\)	\(1596\)
elliptic	\(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,p^{4} \textit {\_Z}^{12}+4 a \,p^{3} q \,\textit {\_Z}^{9}+6 a \,p^{2} q^{2} \textit {\_Z}^{6}+b \,\textit {\_Z}^{8}+4 a p \,q^{3} \textit {\_Z}^{3}+a \,q^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{6} p^{2}-\underline {\hspace {1.25 ex}}\alpha ^{3} p q -2 q^{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2} \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 (\left (-q \,p^{2}\right )^{\frac {2}{3}} b \,q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} q p b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,q^{2} p b +2 q \,p^{2} \left (a \,p^{5} \underline {\hspace {1.25 ex}}\alpha ^{9}+3 a \,p^{4} q \,\underline {\hspace {1.25 ex}}\alpha ^{6}+3 a \,p^{3} q^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+b p \,\underline {\hspace {1.25 ex}}\alpha ^{5}+a \,p^{2} q^{3}-b q \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,q^{3}-\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} b p q +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p \,q^{2}+3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{6} q +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{5} q +3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{5} q^{2}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{4} q^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q^{2} b -3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,q^{3}+i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} q \,p^{2} b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{7}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{6}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{2}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{4} q^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a \,p^{3} q^{3}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} p^{2} b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{7} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \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 {3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{6} q^{2}+3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{5} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{2}-3 q^{3} b p -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,p^{3} q^{4}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{7} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{3} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{4} q^{4}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{6}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b p q -3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{5} q -6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{5} q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4} q^{2}-6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{4} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} q^{3}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} b \,p^{2} q +2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p \,q^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{6} q -3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{6} q b -3 p^{4} \underline {\hspace {1.25 ex}}\alpha a \,q^{4}+3 \underline {\hspace {1.25 ex}}\alpha ^{3} q^{2} p^{2} b -3 p^{7} \underline {\hspace {1.25 ex}}\alpha ^{10} a q -9 p^{6} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,q^{2}-9 p^{5} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{3}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,q^{2} b -3 \underline {\hspace {1.25 ex}}\alpha ^{11} p^{6} \left (-q \,p^{2}\right )^{\frac {2}{3}} a -3 \underline {\hspace {1.25 ex}}\alpha ^{7} \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{2} b -9 \underline {\hspace {1.25 ex}}\alpha ^{5} p^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} p^{3} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} q p b -9 \underline {\hspace {1.25 ex}}\alpha ^{8} p^{5} \left (-q \,p^{2}\right )^{\frac {2}{3}} a q +i \sqrt {3}\, b p \,q^{3}-i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,p^{2} q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b \,q^{2}}{2 p \,q^{3} 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 \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{4}-9 \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{3} q -9 \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2} q^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{5} b -3 a p \,q^{3}\right ) \sqrt {p \,x^{3}+q}}\right )}{4 p^{2} q^{3} b}\)	\(1596\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x,method=_RETURNVERBOSE)

[Out]

1/4*I/p^2/q^3/b*2^(1/2)*sum((_alpha^6*p^2-_alpha^3*p*q-2*q^2)*_alpha^2/(-3*_alpha^9*a*p^4-9*_alpha^6*a*p^3*q-9
*_alpha^3*a*p^2*q^2-2*_alpha^5*b-3*a*p*q^3)*(-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)*((-q*p^2)
^(2/3)*b*q^2+I*(-q*p^2)^(2/3)*p^3*3^(1/2)*_alpha*a*q^3+I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^4*q*p^2*b+3*I*(-q*p^2)^
(2/3)*p^4*3^(1/2)*_alpha^4*a*q^2+2*q*p^2*(_alpha^9*a*p^5+3*_alpha^6*a*p^4*q+3*_alpha^3*a*p^3*q^2+_alpha^5*b*p+
a*p^2*q^3-_alpha^2*b*q)-I*(-q*p^2)^(1/3)*p^4*3^(1/2)*_alpha^2*a*q^3-3*I*(-q*p^2)^(1/3)*p^6*3^(1/2)*_alpha^8*a*
q+3*I*(-q*p^2)^(2/3)*p^5*3^(1/2)*_alpha^7*a*q-3*I*(-q*p^2)^(1/3)*p^5*3^(1/2)*_alpha^5*a*q^2-I*(-q*p^2)^(2/3)*3
^(1/2)*_alpha^3*q*p*b-I*(-q*p^2)^(1/3)*3^(1/2)*_alpha*q^2*p*b-(-q*p^2)^(2/3)*_alpha^3*b*p*q+(-q*p^2)^(1/3)*_al
pha*b*p*q^2+3*(-q*p^2)^(1/3)*_alpha^8*a*p^6*q+3*(-q*p^2)^(2/3)*_alpha^7*a*p^5*q+3*(-q*p^2)^(1/3)*_alpha^5*a*p^
5*q^2+3*(-q*p^2)^(2/3)*_alpha^4*a*p^4*q^2+(-q*p^2)^(1/3)*_alpha^11*a*p^7+(-q*p^2)^(2/3)*_alpha^10*a*p^6+(-q*p^
2)^(1/3)*_alpha^7*b*p^3+(-q*p^2)^(2/3)*_alpha^6*b*p^2+(-q*p^2)^(1/3)*_alpha^2*a*p^4*q^3+(-q*p^2)^(2/3)*_alpha*
a*p^3*q^3-(-q*p^2)^(1/3)*_alpha^4*b*p^2*q+I*(-q*p^2)^(2/3)*3^(1/2)*q^2*b+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^6*p^2
*b-I*(-q*p^2)^(1/3)*p^7*3^(1/2)*_alpha^11*a-I*(-q*p^2)^(1/3)*p^3*3^(1/2)*_alpha^7*b+I*(-q*p^2)^(2/3)*p^6*3^(1/
2)*_alpha^10*a)*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*(-3*q^3*b*p+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^4*b*p*q-3*I*(-q*p^2)^(2/3)*3^(1/2)*_alp
ha^8*a*p^5*q-6*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^6*a*p^5*q^2-3*I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^5*a*p^4*q^2-6*I*(
-q*p^2)^(1/3)*3^(1/2)*_alpha^3*a*p^4*q^3-I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^2*a*p^3*q^3-2*I*(-q*p^2)^(1/3)*3^(1/2
)*_alpha^5*b*p^2*q+2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^2*b*p*q^2-2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^9*a*p^6*q+I*3
^(1/2)*b*p*q^3-2*I*(-q*p^2)^(1/3)*3^(1/2)*a*p^3*q^4-3*p^3*_alpha^6*q*b-3*p^4*_alpha*a*q^4+3*_alpha^3*q^2*p^2*b
-3*p^7*_alpha^10*a*q-9*p^6*_alpha^7*a*q^2-9*p^5*_alpha^4*a*q^3-3*(-q*p^2)^(2/3)*_alpha*q^2*b-3*_alpha^11*p^6*(
-q*p^2)^(2/3)*a-3*_alpha^7*(-q*p^2)^(2/3)*p^2*b-9*_alpha^5*p^4*(-q*p^2)^(2/3)*a*q^2-3*_alpha^2*p^3*(-q*p^2)^(2
/3)*a*q^3+3*_alpha^4*(-q*p^2)^(2/3)*q*p*b-9*_alpha^8*p^5*(-q*p^2)^(2/3)*a*q+I*3^(1/2)*_alpha^10*a*p^7*q+I*3^(1
/2)*_alpha^6*b*p^3*q+I*3^(1/2)*_alpha*a*p^4*q^4-I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^11*a*p^6+3*I*3^(1/2)*_alpha^7*
a*p^6*q^2+3*I*3^(1/2)*_alpha^4*a*p^5*q^3-I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^7*b*p^2-I*3^(1/2)*_alpha^3*b*p^2*q^2-
I*(-q*p^2)^(2/3)*3^(1/2)*_alpha*b*q^2)/q^3/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^12*a*p^4+4*_Z^9*a*p^3*q+6*_Z^6*a*p^2*q^2+_Z^8*b+4*_Z^3*a*p*q^3+a*q^
4))

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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 )} x^{4}}{b x^{8} + {\left (p x^{3} + q\right )}^{4} a}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(p*x^3-2*q)*(p*x^3+q)^(1/2)/(b*x^8+a*(p*x^3+q)^4),x, algorithm="maxima")

[Out]

integrate(sqrt(p*x^3 + q)*(p*x^3 - 2*q)*x^4/(b*x^8 + (p*x^3 + q)^4*a), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*(q + p*x^3)^(1/2)*(2*q - p*x^3))/(a*(q + p*x^3)^4 + b*x^8),x)

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{4} x^{12} + 4 a p^{3} q x^{9} + 6 a p^{2} q^{2} x^{6} + 4 a p q^{3} x^{3} + a q^{4} + b x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(p*x**3-2*q)*(p*x**3+q)**(1/2)/(b*x**8+a*(p*x**3+q)**4),x)

[Out]

Integral(x**4*(p*x**3 - 2*q)*sqrt(p*x**3 + q)/(a*p**4*x**12 + 4*a*p**3*q*x**9 + 6*a*p**2*q**2*x**6 + 4*a*p*q**
3*x**3 + a*q**4 + b*x**8), x)

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