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

Optimal. Leaf size=117 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]

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Rubi [A]  time = 0.38, antiderivative size = 163, normalized size of antiderivative = 1.39, number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2056, 1254, 466, 490, 1211, 220, 1699, 205, 208} \begin {gather*} \frac {\sqrt {a x^3+b x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x} \sqrt {a x^2+b}}-\frac {\sqrt {a x^3+b x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2+b}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} \sqrt {x} \sqrt {a x^2+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 205

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

Rule 208

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

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1211

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1254

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(f*x)^m*(d +
e*x^2)^(q + p)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, c, d, e, f, q, m, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Integ
erQ[p]

Rule 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

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

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Mathematica [C]  time = 0.91, size = 111, normalized size = 0.95 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {a} x}{\sqrt {b}}} \sqrt {\frac {a x^2}{b}+1} \left (\Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {a} x}{\sqrt {b}}}\right )\right |-1\right )-\Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {a} x}{\sqrt {b}}}\right )\right |-1\right )\right )}{a \sqrt {x \left (a x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(I*Sqrt[a]*x)/Sqrt[b]]*Sqrt[1 + (a*x^2)/b]*(-EllipticPi[-I, I*ArcSinh[Sqrt[(I*Sqrt[a]*x)/Sqrt[b]]], -1]
+ EllipticPi[I, I*ArcSinh[Sqrt[(I*Sqrt[a]*x)/Sqrt[b]]], -1]))/(a*Sqrt[x*(b + a*x^2)])

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IntegrateAlgebraic [A]  time = 0.35, size = 117, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}{b+a x^2}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.56, size = 344, normalized size = 2.94 \begin {gather*} \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {a x^{3} + b x} a b \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}}{a x^{2} + b}\right ) - \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} + 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} + 6 \, a b x^{2} + b^{2} - 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} + a b^{2}\right )} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} + b x} + 4 \, {\left (a^{3} b^{2} x^{3} + a^{2} b^{3} x\right )} \sqrt {\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(1/4)^(1/4)*(1/(a^3*b^3))^(1/4)*arctan(2*(1/4)^(1/4)*sqrt(a*x^3 + b*x)*a*b*(1/(a^3*b^3))^(1/4)/(a*x^2 + b)
) - 1/8*(1/4)^(1/4)*(1/(a^3*b^3))^(1/4)*log((a^2*x^4 + 6*a*b*x^2 + b^2 + 4*(4*(1/4)^(3/4)*a^3*b^3*x*(1/(a^3*b^
3))^(3/4) + (1/4)^(1/4)*(a^2*b*x^2 + a*b^2)*(1/(a^3*b^3))^(1/4))*sqrt(a*x^3 + b*x) + 4*(a^3*b^2*x^3 + a^2*b^3*
x)*sqrt(1/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2)) + 1/8*(1/4)^(1/4)*(1/(a^3*b^3))^(1/4)*log((a^2*x^4 + 6*a*b*
x^2 + b^2 - 4*(4*(1/4)^(3/4)*a^3*b^3*x*(1/(a^3*b^3))^(3/4) + (1/4)^(1/4)*(a^2*b*x^2 + a*b^2)*(1/(a^3*b^3))^(1/
4))*sqrt(a*x^3 + b*x) + 4*(a^3*b^2*x^3 + a^2*b^3*x)*sqrt(1/(a^3*b^3)))/(a^2*x^4 - 2*a*b*x^2 + b^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x}}{a^{2} x^{4} - b^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.12, size = 296, normalized size = 2.53

method	result	size
elliptic	\(\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}\)	\(296\)
default	\(-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{2 b a \sqrt {a \,x^{3}+b x}}+\frac {\frac {2 b \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}-\frac {b \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}+\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {b \sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}}{2 b}\)	\(631\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^2*(-a*b)^(1/2)*(x*a/(-a*b)^(1/2)+1)^(1/2)*(-2*x*a/(-a*b)^(1/2)+2)^(1/2)*(-x*a/(-a*b)^(1/2))^(1/2)/(a*x^3
+b*x)^(1/2)/(-1/a*(-a*b)^(1/2)-1/a*(a*b)^(1/2))*EllipticPi(((x+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),-1/a*(-
a*b)^(1/2)/(-1/a*(-a*b)^(1/2)-1/a*(a*b)^(1/2)),1/2*2^(1/2))+1/2/a^2*(-a*b)^(1/2)*(x*a/(-a*b)^(1/2)+1)^(1/2)*(-
2*x*a/(-a*b)^(1/2)+2)^(1/2)*(-x*a/(-a*b)^(1/2))^(1/2)/(a*x^3+b*x)^(1/2)/(-1/a*(-a*b)^(1/2)+1/a*(a*b)^(1/2))*El
lipticPi(((x+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),-1/a*(-a*b)^(1/2)/(-1/a*(-a*b)^(1/2)+1/a*(a*b)^(1/2)),1/2
*2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(a*x**2 + b))/((a*x**2 - b)*(a*x**2 + b)), x)

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