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

Optimal. Leaf size=209 \[ \frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^8-5 \text {$\#$1}^4 a+3 a^2-2 b\& ,\frac {-2 \text {$\#$1}^4 a \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 a \log (x)+3 a^2 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-2 b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-3 a^2 \log (x)+2 b \log (x)}{5 \text {$\#$1}^3 a-4 \text {$\#$1}^7}\& \right ]-\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )+\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right ) \]

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Rubi [B]  time = 4.30, antiderivative size = 750, normalized size of antiderivative = 3.59, number of steps used = 17, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2056, 905, 63, 331, 298, 203, 206, 6728, 93, 205, 208} \begin {gather*} \frac {\left (-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2+16 b}} \left (-a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}+\frac {\left (\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+16 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2+16 b}+a} \left (a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\left (-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{-a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a-\sqrt {a^2+16 b}} \left (-a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\left (\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}+a^2-2 b\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {a^2+16 b}+a^2-4 b}}{\sqrt [4]{\sqrt {a^2+16 b}+a} \sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {a^2+16 b}+a} \left (a \sqrt {a^2+16 b}+a^2-4 b\right )^{3/4} \sqrt [4]{a x+b}}-\frac {\sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {\sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4))) + ((a^2
- 2*b - (a*(a^2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^(1/4)*ArcTan[((a^2 - 4*b - a*Sqrt[a^2 + 16*b])^(1/4
)*x^(1/4))/((a - Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])/((a - Sqrt[a^2 + 16*b])^(1/4)*(a^2 - 4*b - a*Sqrt[
a^2 + 16*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a^2 - 2*b + (a*(a^2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^
(1/4)*ArcTan[((a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])
/((a + Sqrt[a^2 + 16*b])^(1/4)*(a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + (a^(1/4)*(b*x
^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) - ((a^2 - 2*b - (a*(a^
2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^(1/4)*ArcTanh[((a^2 - 4*b - a*Sqrt[a^2 + 16*b])^(1/4)*x^(1/4))/((a
- Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])/((a - Sqrt[a^2 + 16*b])^(1/4)*(a^2 - 4*b - a*Sqrt[a^2 + 16*b])^(
3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a^2 - 2*b + (a*(a^2 + 6*b))/Sqrt[a^2 + 16*b])*(b*x^3 + a*x^4)^(1/4)*ArcTanh[
((a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 16*b])^(1/4)*(b + a*x)^(1/4))])/((a + Sqrt[a
^2 + 16*b])^(1/4)*(a^2 - 4*b + a*Sqrt[a^2 + 16*b])^(3/4)*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 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 905

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Di
st[(e*g)/c, Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[(Simp[c*d*f - a*e*g + (c*e*f + c*d
*g - b*e*g)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 1))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g
}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] &
& GtQ[n, 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 {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{-2 b+a x+2 x^2} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {2 a b-\left (a^2-2 b\right ) x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-2 b+a x+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}}{\sqrt [4]{x} \left (a-\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}}+\frac {-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}}{\sqrt [4]{x} \left (a+\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (2 a \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a+\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a-\sqrt {a^2+16 b}+4 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (\sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2+16 b}-\left (-4 b+a \left (a+\sqrt {a^2+16 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2+16 b}-\left (-4 b+a \left (a-\sqrt {a^2+16 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+16 b}}-\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (-a^2+2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+16 b}}+\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b-a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+16 b}}-\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (-a^2+2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+16 b}}+\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-4 b+a \sqrt {a^2+16 b}} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^2-2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2+16 b}} \left (a^2-4 b-a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^2-2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2+16 b}} \left (a^2-4 b+a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^2-2 b-\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-4 b-a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a-\sqrt {a^2+16 b}} \left (a^2-4 b-a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^2-2 b+\frac {a \left (a^2+6 b\right )}{\sqrt {a^2+16 b}}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-4 b+a \sqrt {a^2+16 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+16 b}} \sqrt [4]{b+a x}}\right )}{\sqrt [4]{a+\sqrt {a^2+16 b}} \left (a^2-4 b+a \sqrt {a^2+16 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}

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Mathematica [F]  time = 14.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{b x^3+a x^4}}{-2 b+a x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.49, size = 209, normalized size = 1.00 \begin {gather*} -\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {1}{2} \text {RootSum}\left [3 a^2-2 b-5 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {3 a^2 \log (x)-2 b \log (x)-3 a^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-2 a \log (x) \text {$\#$1}^4+2 a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 a \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^(1/4)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)]) + a^(1/4)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + Ro
otSum[3*a^2 - 2*b - 5*a*#1^4 + 2*#1^8 & , (3*a^2*Log[x] - 2*b*Log[x] - 3*a^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]
+ 2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 2*a*Log[x]*#1^4 + 2*a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-5*a
*#1^3 + 4*#1^7) & ]/2

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fricas [B]  time = 1.52, size = 4158, normalized size = 19.89

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

[Out]

-2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^
2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*arctan(1/512
*(sqrt(2)*sqrt(1/2)*((a^10 + 58*a^8*b + 1232*a^6*b^2 + 11008*a^4*b^3 + 28672*a^2*b^4 - 65536*b^5)*x*sqrt((a^8
+ 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)) + (a^11 + 40*a^9*b +
516*a^7*b^2 + 2160*a^5*b^3 + 512*a^3*b^4 - 4096*a*b^5)*x)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b +
256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/
(a^4 + 32*a^2*b + 256*b^2))*sqrt((sqrt(1/2)*((a^11 + 54*a^9*b + 1048*a^7*b^2 + 8320*a^5*b^3 + 18432*a^3*b^4 -
32768*a*b^5)*x^2*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096
*b^3)) + (a^12 + 36*a^10*b + 436*a^8*b^2 + 1920*a^6*b^3 + 320*a^4*b^4 - 10752*a^2*b^5 + 8192*b^6)*x^2)*sqrt((a
^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/
(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)) + 32*(a^8*b^2 + 12*a^6*b^3 + 20*a^4*b^
4 - 96*a^2*b^5 + 64*b^6)*sqrt(a*x^4 + b*x^3))/x^2) + 8*sqrt(1/2)*(a^15*b + 46*a^13*b^2 + 748*a^11*b^3 + 4936*a
^9*b^4 + 9344*a^7*b^5 - 18304*a^5*b^6 - 28672*a^3*b^7 + 32768*a*b^8 + (a^14*b + 64*a^12*b^2 + 1572*a^10*b^3 +
17936*a^8*b^4 + 84864*a^6*b^5 + 18432*a^4*b^6 - 622592*a^2*b^7 + 524288*b^8)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2
- 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))*(a*x^4 + b*x^3)^(1/4)*sqrt((a^5 + 14*a^3*b
+ 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4
*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a
^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^
2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))/((3*a^10*b^4 + 34*a^8*b^5 + 36*a^6*b^6 - 328*a^4*b^7 + 384*a^2*b^
8 - 128*b^9)*x)) + 2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12
*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b
^2)))*arctan(1/512*(sqrt(2)*sqrt(1/2)*((a^10 + 58*a^8*b + 1232*a^6*b^2 + 11008*a^4*b^3 + 28672*a^2*b^4 - 65536
*b^5)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)) -
(a^11 + 40*a^9*b + 516*a^7*b^2 + 2160*a^5*b^3 + 512*a^3*b^4 - 4096*a*b^5)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3
*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a
^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b
+ 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)
))/(a^4 + 32*a^2*b + 256*b^2))*sqrt(-(sqrt(1/2)*((a^11 + 54*a^9*b + 1048*a^7*b^2 + 8320*a^5*b^3 + 18432*a^3*b^
4 - 32768*a*b^5)*x^2*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 +
4096*b^3)) - (a^12 + 36*a^10*b + 436*a^8*b^2 + 1920*a^6*b^3 + 320*a^4*b^4 - 10752*a^2*b^5 + 8192*b^6)*x^2)*sqr
t((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b
^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)) - 32*(a^8*b^2 + 12*a^6*b^3 + 20*a^
4*b^4 - 96*a^2*b^5 + 64*b^6)*sqrt(a*x^4 + b*x^3))/x^2) - 8*sqrt(1/2)*(a^15*b + 46*a^13*b^2 + 748*a^11*b^3 + 49
36*a^9*b^4 + 9344*a^7*b^5 - 18304*a^5*b^6 - 28672*a^3*b^7 + 32768*a*b^8 - (a^14*b + 64*a^12*b^2 + 1572*a^10*b^
3 + 17936*a^8*b^4 + 84864*a^6*b^5 + 18432*a^4*b^6 - 622592*a^2*b^7 + 524288*b^8)*sqrt((a^8 + 12*a^6*b + 20*a^4
*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))*(a*x^4 + b*x^3)^(1/4)*sqrt(sqrt(1/2)*s
qrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64
*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))*sqrt((a^5 + 14*a^3*b + 8*a*b^2
+ (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^
2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2)))/((3*a^10*b^4 + 34*a^8*b^5 + 36*a^6*b^6 - 328*a^4*b^7 + 384*a^
2*b^8 - 128*b^9)*x)) + 1/2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^
8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b +
256*b^2)))*log(-(((a^5 + 32*a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^
6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)) - (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5
+ 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^
6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) + 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 +
b*x^3)^(1/4))/x) - 1/2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 +
12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 25
6*b^2)))*log((((a^5 + 32*a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 +
48*a^4*b + 768*a^2*b^2 + 4096*b^3)) - (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14
*a^3*b + 8*a*b^2 + (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 +
48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) - 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x
^3)^(1/4))/x) - 1/2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*
a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^
2)))*log(-(((a^5 + 32*a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*
a^4*b + 768*a^2*b^2 + 4096*b^3)) + (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^
3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*
a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) + 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)
^(1/4))/x) + 1/2*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b + 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6
*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))
)*log((((a^5 + 32*a^3*b + 256*a*b^2)*x*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*
b + 768*a^2*b^2 + 4096*b^3)) + (a^6 + 22*a^4*b + 88*a^2*b^2 - 128*b^3)*x)*sqrt(sqrt(1/2)*sqrt((a^5 + 14*a^3*b
+ 8*a*b^2 - (a^4 + 32*a^2*b + 256*b^2)*sqrt((a^8 + 12*a^6*b + 20*a^4*b^2 - 96*a^2*b^3 + 64*b^4)/(a^6 + 48*a^4*
b + 768*a^2*b^2 + 4096*b^3)))/(a^4 + 32*a^2*b + 256*b^2))) - 8*(a^4*b + 6*a^2*b^2 - 8*b^3)*(a*x^4 + b*x^3)^(1/
4))/x) - 2*a^(1/4)*arctan((a^(3/4)*x*sqrt((sqrt(a)*x^2 + sqrt(a*x^4 + b*x^3))/x^2) - (a*x^4 + b*x^3)^(1/4)*a^(
3/4))/(a*x)) + 1/2*a^(1/4)*log((a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - 1/2*a^(1/4)*log(-(a^(1/4)*x - (a*x^4 +
b*x^3)^(1/4))/x)

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giac [B]  time = 2.96, size = 174, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{2} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4)) + 1/2*sqrt(2)*(
-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4)) + 1/4*sqrt(2)*(-a)^(1/4)*lo
g(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) - 1/4*sqrt(2)*(-a)^(1/4)*log(-sqrt(2)*(-a)^(1
/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x +2 x^{2}-2 b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^4+b*x^3)^(1/4)/(a*x+2*x^2-2*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}}}{a x + 2 \, x^{2} - 2 \, b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^4 + b*x^3)^(1/4)/(a*x - 2*b + 2*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 )}}{a x - 2 b + 2 x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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