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

Optimal. Leaf size=215 \[ a \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+a \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a \log (x)+b \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]+\frac {4 \sqrt [4]{a x^4+b x^3}}{x}-2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right ) \]

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Rubi [B]  time = 2.20, antiderivative size = 491, normalized size of antiderivative = 2.28, number of steps used = 23, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2056, 6725, 47, 63, 331, 298, 203, 206, 908, 37, 6688, 12, 93} \begin {gather*} \frac {4 \sqrt [4]{a x^4+b x^3}}{x}-\frac {2 \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 {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \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}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b*x^3 + a*x^4)^(1/4))/x - (2*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)) + (2*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/8)*(Sqrt[a] -
Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) + (2*a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*(b*
x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/
4)) + (2*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)) -
(2*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x^(1/4)
)/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) - (2*a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*Arc
Tanh[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 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 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 908

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

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6725

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

Rubi steps

\begin {align*} \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x} \left (b+a x^2\right )}{x^{5/4} \left (-b+a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {\sqrt [4]{b+a x}}{x^{5/4}}+\frac {2 b \sqrt [4]{b+a x}}{x^{5/4} \left (-b+a x^2\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x}}{x^{5/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\sqrt [4]{b+a x}}{x^{5/4} \left (-b+a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {-a b-a b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-b+a x^2\right )} \, dx}{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}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {a b (1+x)}{\sqrt [4]{x} (b+a x)^{3/4} \left (b-a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 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 {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (4 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 (2 a b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1+x}{\sqrt [4]{x} (b+a x)^{3/4} \left (b-a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (2 \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 \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 a b \sqrt [4]{b x^3+a x^4}\right ) \int \left (\frac {\sqrt {b}+\frac {b}{\sqrt {a}}}{2 b \sqrt [4]{x} \left (\sqrt {b}-\sqrt {a} x\right ) (b+a x)^{3/4}}+\frac {\sqrt {b}-\frac {b}{\sqrt {a}}}{2 b \sqrt [4]{x} \left (\sqrt {b}+\sqrt {a} x\right ) (b+a x)^{3/4}}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \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 {2 \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 \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt {b}+\sqrt {a} x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt {b}-\sqrt {a} x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \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 {2 \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 (4 a \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\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 (4 a \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \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 {2 \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 (2 a^{3/4} \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}-\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a^{3/4} \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}-\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^{3/4} \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}+\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a^{3/4} \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}+\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \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 {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \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 {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.37, size = 169, normalized size = 0.79 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \left (\frac {\left (\sqrt {a} \sqrt {b} x-a x\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-\sqrt {a} \sqrt {b} x}{b+a x}\right )-\left (\sqrt {a} \sqrt {b} x+a x\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt {a} \sqrt {b} x}{b+a x}\right )+6 (a x+b)}{a x+b}-\frac {3 \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\frac {a x}{b}\right )}{\sqrt [4]{\frac {a x}{b}+1}}\right )}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3*(b + a*x))^(1/4)*((-3*Hypergeometric2F1[-1/4, -1/4, 3/4, -((a*x)/b)])/(1 + (a*x)/b)^(1/4) + (6*(b + a*
x) + (-(a*x) + Sqrt[a]*Sqrt[b]*x)*Hypergeometric2F1[3/4, 1, 7/4, (a*x - Sqrt[a]*Sqrt[b]*x)/(b + a*x)] - (a*x +
Sqrt[a]*Sqrt[b]*x)*Hypergeometric2F1[3/4, 1, 7/4, (a*x + Sqrt[a]*Sqrt[b]*x)/(b + a*x)])/(b + a*x)))/(3*x)

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IntegrateAlgebraic [A]  time = 1.81, size = 214, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+a \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-b \log (x)-a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.74, size = 567, normalized size = 2.64 \begin {gather*} \frac {4 \, {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \arctan \left (\frac {{\left (a x - \sqrt {a b} x\right )} {\left (a + \sqrt {a b}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a + \sqrt {a b}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a + \sqrt {a b}\right )}^{\frac {3}{4}} {\left (a - \sqrt {a b}\right )}}{{\left (a^{2} - a b\right )} x}\right ) - 4 \, {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {{\left (a x + \sqrt {a b} x\right )} {\left (a - \sqrt {a b}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a - \sqrt {a b}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a + \sqrt {a b}\right )} {\left (a - \sqrt {a b}\right )}^{\frac {3}{4}}}{{\left (a^{2} - a b\right )} x}\right ) - 4 \, a^{\frac {1}{4}} x \arctan \left (\frac {a^{\frac {3}{4}} x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} a^{\frac {3}{4}}}{a x}\right ) - {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(a + sqrt(a*b))^(1/4)*x*arctan(((a*x - sqrt(a*b)*x)*(a + sqrt(a*b))^(3/4)*sqrt((sqrt(a + sqrt(a*b))*x^2 + s
qrt(a*x^4 + b*x^3))/x^2) - (a*x^4 + b*x^3)^(1/4)*(a + sqrt(a*b))^(3/4)*(a - sqrt(a*b)))/((a^2 - a*b)*x)) - 4*(
a - sqrt(a*b))^(1/4)*x*arctan(-((a*x + sqrt(a*b)*x)*(a - sqrt(a*b))^(3/4)*sqrt((sqrt(a - sqrt(a*b))*x^2 + sqrt
(a*x^4 + b*x^3))/x^2) - (a*x^4 + b*x^3)^(1/4)*(a + sqrt(a*b))*(a - sqrt(a*b))^(3/4))/((a^2 - a*b)*x)) - 4*a^(1
/4)*x*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))
- (a + sqrt(a*b))^(1/4)*x*log(2*((a + sqrt(a*b))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) + (a + sqrt(a*b))^(1/4)*x
*log(-2*((a + sqrt(a*b))^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) - (a - sqrt(a*b))^(1/4)*x*log(2*((a - sqrt(a*b))^
(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) + (a - sqrt(a*b))^(1/4)*x*log(-2*((a - sqrt(a*b))^(1/4)*x - (a*x^4 + b*x^3
)^(1/4))/x) + a^(1/4)*x*log((a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - a^(1/4)*x*log(-(a^(1/4)*x - (a*x^4 + b*x^
3)^(1/4))/x) + 4*(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^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{x^{2} \left (a \,x^{2}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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