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

Optimal. Leaf size=167 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {p x^3+q}}{\sqrt {a} p x^3+\sqrt {a} q-\sqrt {b} x^2}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} p x^3}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}}{x \sqrt {p x^3+q}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \]

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Rubi [A]  time = 0.46, antiderivative size = 242, normalized size of antiderivative = 1.45, number of steps used = 10, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {6712, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a} \sqrt {p x^3+q}}+1\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^3+q}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {p x^3+q}}+\sqrt {a}+\frac {\sqrt {b} x^2}{p x^3+q}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 6712

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ
[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

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

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Mathematica [C]  time = 6.51, size = 13567, normalized size = 81.24 \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])/(b*x^4 + a*(q + p*x^3)^2),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.07, size = 167, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x \sqrt {q+p x^3}}{\sqrt {a} q-\sqrt {b} x^2+\sqrt {a} p x^3}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{a} q}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b} x^2}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} p x^3}{\sqrt {2} \sqrt [4]{b}}}{x \sqrt {q+p x^3}}\right )}{\sqrt {2} a^{3/4} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.43, size = 354, normalized size = 2.12 \begin {gather*} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} b x \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}}{\sqrt {p x^{3} + q}}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} + 2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{4} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} - 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} \log \left (\frac {a p^{2} x^{6} + 2 \, a p q x^{3} - b x^{4} + a q^{2} - 2 \, {\left (a b x^{3} \left (-\frac {1}{a^{3} b}\right )^{\frac {1}{4}} + {\left (a^{3} b p x^{4} + a^{3} b q x\right )} \left (-\frac {1}{a^{3} b}\right )^{\frac {3}{4}}\right )} \sqrt {p x^{3} + q} - 2 \, {\left (a^{2} b p x^{5} + a^{2} b q x^{2}\right )} \sqrt {-\frac {1}{a^{3} b}}}{a p^{2} x^{6} + 2 \, a p q x^{3} + b x^{4} + a q^{2}}\right ) \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)/(b*x^4+a*(p*x^3+q)^2),x, algorithm="fricas")

[Out]

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

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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 )}}{b x^{4} + {\left (p x^{3} + q\right )}^{2} a}\,{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)/(b*x^4+a*(p*x^3+q)^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.38, size = 1164, normalized size = 6.97

method	result	size
default	\(-\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^{2} \textit {\_Z}^{6}+2 a q p \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{4}+a \,q^{2}\right )}{\sum }\frac {\left (3 \underline {\hspace {1.25 ex}}\alpha ^{3} a p q +\underline {\hspace {1.25 ex}}\alpha ^{4} b +3 a \,q^{2}\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 (2 q \,p^{2} \left (a \,p^{2} \underline {\hspace {1.25 ex}}\alpha ^{4}+a q p \underline {\hspace {1.25 ex}}\alpha +b \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a q \,p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{3}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q b +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} p b -\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{3} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,q^{2} p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} b p +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha q p b +i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a q -\left (-q \,p^{2}\right )^{\frac {1}{3}} a \,p^{2} q^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p q -\left (-q \,p^{2}\right )^{\frac {2}{3}} b q \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 \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} 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}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p -3 p^{4} \underline {\hspace {1.25 ex}}\alpha ^{5} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, a p q -i \sqrt {3}\, b p q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{2} a q -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{3} p^{2} b +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} a q p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,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 ^{2} \left (3 \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}+3 a q p +2 \underline {\hspace {1.25 ex}}\alpha b \right ) \sqrt {p \,x^{3}+q}}\right )}{2 a \,p^{2} q^{2} b}\)	\(1164\)
elliptic	\(-\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^{2} \textit {\_Z}^{6}+2 a q p \,\textit {\_Z}^{3}+b \,\textit {\_Z}^{4}+a \,q^{2}\right )}{\sum }\frac {\left (3 \underline {\hspace {1.25 ex}}\alpha ^{3} a p q +\underline {\hspace {1.25 ex}}\alpha ^{4} b +3 a \,q^{2}\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 (2 q \,p^{2} \left (a \,p^{2} \underline {\hspace {1.25 ex}}\alpha ^{4}+a q p \underline {\hspace {1.25 ex}}\alpha +b \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a q \,p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{3}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q b +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} p b -\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{3} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,q^{2} p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} b p +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha q p b +i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a q -\left (-q \,p^{2}\right )^{\frac {1}{3}} a \,p^{2} q^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p q -\left (-q \,p^{2}\right )^{\frac {2}{3}} b q \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 \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} 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}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p -3 p^{4} \underline {\hspace {1.25 ex}}\alpha ^{5} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, a p q -i \sqrt {3}\, b p q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{2} a q -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{3} p^{2} b +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} a q p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,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 ^{2} \left (3 \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}+3 a q p +2 \underline {\hspace {1.25 ex}}\alpha b \right ) \sqrt {p \,x^{3}+q}}\right )}{2 a \,p^{2} q^{2} b}\)	\(1164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*I/a*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))-1/2*I/a/p^2/q^2/b*2^(1/2)*s
um((3*_alpha^3*a*p*q+_alpha^4*b+3*a*q^2)/_alpha^2/(3*_alpha^3*a*p^2+3*a*p*q+2*_alpha*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)*(2*q*p^2*(_alpha^4*a*p^2+_alpha*a*p*q+_alpha^2*b)+I*(-q*p^2)^(2/3)*3^(1/2)*
_alpha^2*a*q*p^2+(-q*p^2)^(2/3)*_alpha^5*a*p^3+I*(-q*p^2)^(2/3)*p^3*3^(1/2)*_alpha^5*a-I*(-q*p^2)^(2/3)*3^(1/2
)*q*b+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^3*p*b-(-q*p^2)^(1/3)*_alpha^3*a*p^3*q+(-q*p^2)^(2/3)*_alpha^2*a*p^2*q+I*
(-q*p^2)^(1/3)*3^(1/2)*a*q^2*p^2+(-q*p^2)^(2/3)*_alpha^3*b*p+I*(-q*p^2)^(1/3)*3^(1/2)*_alpha*q*p*b+I*(-q*p^2)^
(1/3)*p^3*3^(1/2)*_alpha^3*a*q-(-q*p^2)^(1/3)*a*p^2*q^2-(-q*p^2)^(1/3)*_alpha*b*p*q-(-q*p^2)^(2/3)*b*q)*Ellipt
icPi(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/q*(I*3^(1/2)*_alpha^2*a*p^3*q+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha*b-2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^2*b*p-3*
p^4*_alpha^5*a+I*3^(1/2)*_alpha^5*a*p^4+3*(-q*p^2)^(2/3)*_alpha^3*a*p^2-2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha*a*p^
2*q+I*(-q*p^2)^(2/3)*3^(1/2)*a*p*q-I*3^(1/2)*b*p*q+I*(-q*p^2)^(2/3)*3^(1/2)*_alpha^3*a*p^2-3*p^3*_alpha^2*a*q-
2*I*(-q*p^2)^(1/3)*3^(1/2)*_alpha^4*a*p^3-3*_alpha^3*p^2*b+3*(-q*p^2)^(2/3)*a*q*p+I*3^(1/2)*_alpha^3*b*p^2+3*(
-q*p^2)^(2/3)*_alpha*b+3*b*p*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^6*a*p^2+2*_Z^3*a*p*q+_Z^4*b+a*q^2))

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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 )}}{b x^{4} + {\left (p x^{3} + q\right )}^{2} a}\,{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)/(b*x^4+a*(p*x^3+q)^2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {p\,x^3+q}\,\left (2\,q-p\,x^3\right )}{a\,{\left (p\,x^3+q\right )}^2+b\,x^4} \,d x \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))/(a*(q + p*x^3)^2 + b*x^4),x)

[Out]

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

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

[Out]

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

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