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

Optimal. Leaf size=63 \[ -\text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^4-a}\& \right ] \]

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Rubi [B]  time = 1.17, antiderivative size = 573, normalized size of antiderivative = 9.10, number of steps used = 23, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2056, 911, 105, 63, 331, 298, 203, 206, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt {a^2-4 b} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt {a^2-4 b} \sqrt [4]{a x+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 203

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

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 206

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

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 298

Int[(x_)^2/((a_) + (b_.)*(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), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 911

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

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

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Mathematica [B]  time = 11.53, size = 392, normalized size = 6.22 \begin {gather*} -\frac {\sqrt [4]{x^3 (a x+b)} \left (-\sqrt [4]{a-\sqrt {a^2-4 b}} \log \left (2^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}}-2 \sqrt [4]{a+\frac {b}{x}}\right )+\sqrt [4]{\sqrt {a^2-4 b}+a} \log \left (2^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a}-2 \sqrt [4]{a+\frac {b}{x}}\right )+\sqrt [4]{a-\sqrt {a^2-4 b}} \log \left (2^{3/4} \sqrt [4]{a-\sqrt {a^2-4 b}}+2 \sqrt [4]{a+\frac {b}{x}}\right )-\sqrt [4]{\sqrt {a^2-4 b}+a} \log \left (2^{3/4} \sqrt [4]{\sqrt {a^2-4 b}+a}+2 \sqrt [4]{a+\frac {b}{x}}\right )+2 \sqrt [4]{a-\sqrt {a^2-4 b}} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a+\frac {b}{x}}}{\sqrt [4]{a-\sqrt {a^2-4 b}}}\right )-2 \sqrt [4]{\sqrt {a^2-4 b}+a} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a+\frac {b}{x}}}{\sqrt [4]{\sqrt {a^2-4 b}+a}}\right )\right )}{\sqrt [4]{2} x \sqrt {a^2-4 b} \sqrt [4]{a+\frac {b}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.44, size = 63, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-RootSum[b - a*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1)/(-a + 2*#1^4) & ]

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fricas [B]  time = 0.58, size = 1918, normalized size = 30.44

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2)*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8
*a^2*b + 16*b^2)))*arctan(-1/8*sqrt(2)*(sqrt(2)*(a*x^4 + b*x^3)^(1/4)*(a^4 - 8*a^2*b + 16*b^2 - (a^7 - 12*a^5*
b + 48*a^3*b^2 - 64*a*b^3)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt
(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)) - ((a^4 - 8*a^2*b + 16*b^2)*x - (a^7 - 12*a^
5*b + 48*a^3*b^2 - 64*a*b^3)*x/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/
sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2))*sqrt((sqrt(2)*(a^2 - 4*b)*x^2*sqrt((a +
(a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)) + 2*sqrt(a*x^4
+ b*x^3))/x^2))*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a
^4 - 8*a^2*b + 16*b^2)))/(b*x)) - 2*sqrt(2)*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*
b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*arctan(1/8*sqrt(2)*(sqrt(2)*(a*x^4 + b*x^3)^(1/4)*(a^4 -
8*a^2*b + 16*b^2 + (a^7 - 12*a^5*b + 48*a^3*b^2 - 64*a*b^3)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sqrt((
a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)) - ((a^4 - 8
*a^2*b + 16*b^2)*x + (a^7 - 12*a^5*b + 48*a^3*b^2 - 64*a*b^3)*x/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))*sq
rt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2))*sqrt((s
qrt(2)*(a^2 - 4*b)*x^2*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8
*a^2*b + 16*b^2)) + 2*sqrt(a*x^4 + b*x^3))/x^2))*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12
*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))/(b*x)) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a + (a^4 - 8
*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*log((sqrt(2)*(a^4 - 8*
a^2*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/
(a^4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3) + 2*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt
(2)*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b
+ 16*b^2)))*log(-(sqrt(2)*(a^4 - 8*a^2*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a + (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6
- 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3) - 2*
(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b +
48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*log((sqrt(2)*(a^4 - 8*a^2*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a
- (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 1
2*a^4*b + 48*a^2*b^2 - 64*b^3) + 2*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2
*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^4 - 8*a^2*b + 16*b^2)))*log(-(sqrt(2)*(a^4 - 8*a^2
*b + 16*b^2)*x*sqrt(sqrt(2)*sqrt((a - (a^4 - 8*a^2*b + 16*b^2)/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3))/(a^
4 - 8*a^2*b + 16*b^2)))/sqrt(a^6 - 12*a^4*b + 48*a^2*b^2 - 64*b^3) - 2*(a*x^4 + b*x^3)^(1/4))/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x \left (a x +x^{2}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x\,\left (x^2+a\,x+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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