3.17.10 \(\int \frac {2 b+a x}{(b+a x+x^2) \sqrt [4]{b x^2+a x^3}} \, dx\)

Optimal. Leaf size=110 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3+b x^2}}{\sqrt {a x^3+b x^2}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3+b x^2}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3+b x^2}}\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 1.39, antiderivative size = 573, normalized size of antiderivative = 5.21, number of steps used = 13, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2056, 6728, 107, 106, 490, 1218} \begin {gather*} \frac {\sqrt {2} \sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \left (\sqrt {a^2-4 b}+a\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2-\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {-a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (-\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \left (a-\sqrt {a^2-4 b}\right ) \sqrt {-\frac {a x}{b}} \sqrt [4]{a x+b} \Pi \left (\frac {\sqrt {2} \sqrt {b}}{\sqrt {-a^2+\sqrt {a^2-4 b} a+2 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {a \sqrt {a^2-4 b}-a^2+2 b} \sqrt [4]{a x^3+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*b + a*x)/((b + a*x + x^2)*(b*x^2 + a*x^3)^(1/4)),x]

[Out]

(Sqrt[2]*(a + Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b)]*(b + a*x)^(1/4)*EllipticPi[-((Sqrt[2]*Sqrt[b])/Sqrt[-a
^2 - a*Sqrt[a^2 - 4*b] + 2*b]), ArcSin[(b + a*x)^(1/4)/b^(1/4)], -1])/(Sqrt[-a^2 - a*Sqrt[a^2 - 4*b] + 2*b]*(b
*x^2 + a*x^3)^(1/4)) - (Sqrt[2]*(a + Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b)]*(b + a*x)^(1/4)*EllipticPi[(Sqr
t[2]*Sqrt[b])/Sqrt[-a^2 - a*Sqrt[a^2 - 4*b] + 2*b], ArcSin[(b + a*x)^(1/4)/b^(1/4)], -1])/(Sqrt[-a^2 - a*Sqrt[
a^2 - 4*b] + 2*b]*(b*x^2 + a*x^3)^(1/4)) + (Sqrt[2]*(a - Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[-((a*x)/b)]*(b + a*x)^(
1/4)*EllipticPi[-((Sqrt[2]*Sqrt[b])/Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b]), ArcSin[(b + a*x)^(1/4)/b^(1/4)], -1
])/(Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b]*(b*x^2 + a*x^3)^(1/4)) - (Sqrt[2]*(a - Sqrt[a^2 - 4*b])*b^(1/4)*Sqrt[
-((a*x)/b)]*(b + a*x)^(1/4)*EllipticPi[(Sqrt[2]*Sqrt[b])/Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b], ArcSin[(b + a*x
)^(1/4)/b^(1/4)], -1])/(Sqrt[-a^2 + a*Sqrt[a^2 - 4*b] + 2*b]*(b*x^2 + a*x^3)^(1/4))

Rule 106

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

Rule 107

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(1/4)), x_Symbol] :> Dist[Sqrt[-((f*
(c + d*x))/(d*e - c*f))]/Sqrt[c + d*x], Int[1/((a + b*x)*Sqrt[-((c*f)/(d*e - c*f)) - (d*f*x)/(d*e - c*f)]*(e +
f*x)^(1/4)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[-(f/(d*e - c*f)), 0]

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 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 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]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 7.21, size = 585, normalized size = 5.32 \begin {gather*} \frac {i \sqrt {2} a (a x+b)^{3/4} \sqrt {1-\frac {b}{a x+b}} \left (\left (a^2+\sqrt {a^4-4 a^2 b}-4 b\right ) \sqrt {\frac {-a^2+\sqrt {a^4-4 a^2 b}+2 b}{b^2}} \Pi \left (-\frac {\sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2-2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )+\left (-a^2-\sqrt {a^4-4 a^2 b}+4 b\right ) \sqrt {\frac {-a^2+\sqrt {a^4-4 a^2 b}+2 b}{b^2}} \Pi \left (\frac {\sqrt {2}}{\sqrt {b} \sqrt {-\frac {a^2-2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )+\sqrt {-\frac {a^2+\sqrt {a^4-4 a^2 b}-2 b}{b^2}} \left (-a^2+\sqrt {a^4-4 a^2 b}+4 b\right ) \left (\Pi \left (-\frac {\sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2+2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2}}{\sqrt {b} \sqrt {\frac {-a^2+2 b+\sqrt {a^4-4 a^2 b}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\sqrt {b}}}{\sqrt [4]{b+a x}}\right )\right |-1\right )\right )\right )}{\sqrt {-\sqrt {b}} b \sqrt {a^4-4 a^2 b} \sqrt {-\frac {a^2+\sqrt {a^4-4 a^2 b}-2 b}{b^2}} \sqrt {\frac {-a^2+\sqrt {a^4-4 a^2 b}+2 b}{b^2}} \sqrt [4]{x^2 (a x+b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*b + a*x)/((b + a*x + x^2)*(b*x^2 + a*x^3)^(1/4)),x]

[Out]

(I*Sqrt[2]*a*(b + a*x)^(3/4)*Sqrt[1 - b/(b + a*x)]*((a^2 - 4*b + Sqrt[a^4 - 4*a^2*b])*Sqrt[(-a^2 + 2*b + Sqrt[
a^4 - 4*a^2*b])/b^2]*EllipticPi[-(Sqrt[2]/(Sqrt[b]*Sqrt[-((a^2 - 2*b + Sqrt[a^4 - 4*a^2*b])/b^2)])), I*ArcSinh
[Sqrt[-Sqrt[b]]/(b + a*x)^(1/4)], -1] + (-a^2 + 4*b - Sqrt[a^4 - 4*a^2*b])*Sqrt[(-a^2 + 2*b + Sqrt[a^4 - 4*a^2
*b])/b^2]*EllipticPi[Sqrt[2]/(Sqrt[b]*Sqrt[-((a^2 - 2*b + Sqrt[a^4 - 4*a^2*b])/b^2)]), I*ArcSinh[Sqrt[-Sqrt[b]
]/(b + a*x)^(1/4)], -1] + Sqrt[-((a^2 - 2*b + Sqrt[a^4 - 4*a^2*b])/b^2)]*(-a^2 + 4*b + Sqrt[a^4 - 4*a^2*b])*(E
llipticPi[-(Sqrt[2]/(Sqrt[b]*Sqrt[(-a^2 + 2*b + Sqrt[a^4 - 4*a^2*b])/b^2])), I*ArcSinh[Sqrt[-Sqrt[b]]/(b + a*x
)^(1/4)], -1] - EllipticPi[Sqrt[2]/(Sqrt[b]*Sqrt[(-a^2 + 2*b + Sqrt[a^4 - 4*a^2*b])/b^2]), I*ArcSinh[Sqrt[-Sqr
t[b]]/(b + a*x)^(1/4)], -1])))/(Sqrt[-Sqrt[b]]*b*Sqrt[a^4 - 4*a^2*b]*Sqrt[-((a^2 - 2*b + Sqrt[a^4 - 4*a^2*b])/
b^2)]*Sqrt[(-a^2 + 2*b + Sqrt[a^4 - 4*a^2*b])/b^2]*(x^2*(b + a*x))^(1/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.51, size = 110, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b x^2+a x^3}}{\sqrt {2}}}{x \sqrt [4]{b x^2+a x^3}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{b x^2+a x^3}}{x^2+\sqrt {b x^2+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2*b + a*x)/((b + a*x + x^2)*(b*x^2 + a*x^3)^(1/4)),x]

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + 2 \, b}{{\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a x +2 b}{\left (a x +x^{2}+b \right ) \left (a \,x^{3}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + 2 \, b}{{\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x + x^{2} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*x + 2*b)/((x**2*(a*x + b))**(1/4)*(a*x + b + x**2)), x)

________________________________________________________________________________________