3.27.73 \(\int \frac {(d+c x^2) \sqrt [4]{b x^3+a x^4}}{x^2 (-d+c x^2)} \, dx\) [2673]

Optimal. Leaf size=241 \[ \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\frac {\text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\& ,\frac {b^2 c \log (x)-a^2 d \log (x)-b^2 c \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^2 d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-a d \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\& \right ]}{d} \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(519\) vs. \(2(241)=482\).
time = 1.15, antiderivative size = 519, normalized size of antiderivative = 2.15, number of steps used = 21, number of rules used = 13, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2081, 6857, 49, 65, 338, 304, 209, 212, 922, 37, 95, 211, 214} \begin {gather*} \frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \text {ArcTan}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \text {ArcTan}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \text {ArcTan}\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 [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}-b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \sqrt {d}+b \sqrt {c}}}{\sqrt [8]{d} \sqrt [4]{a x+b}}\right )}{\sqrt [8]{d} x^{3/4} \sqrt [4]{a x+b}}+\frac {4 \sqrt [4]{a x^4+b x^3}}{x}+\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c*x^2)*(b*x^3 + a*x^4)^(1/4))/(x^2*(-d + c*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*(-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[((-(b*Sqrt[c]) + a*Sq
rt[d])^(1/4)*x^(1/4))/(d^(1/8)*(b + a*x)^(1/4))])/(d^(1/8)*x^(3/4)*(b + a*x)^(1/4)) + (2*(b*Sqrt[c] + a*Sqrt[d
])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(b + a*x)^(1/4))])/(d^(
1/8)*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*(-(b*Sqrt[c]) + a*Sqrt[d])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[((-(b*Sqrt[c]) +
a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(b + a*x)^(1/4))])/(d^(1/8)*x^(3/4)*(b + a*x)^(1/4)) - (2*(b*Sqrt[c] + a*Sq
rt[d])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[((b*Sqrt[c] + a*Sqrt[d])^(1/4)*x^(1/4))/(d^(1/8)*(b + a*x)^(1/4))])
/(d^(1/8)*x^(3/4)*(b + a*x)^(1/4))

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 49

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 65

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 95

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 209

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

Rule 211

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

Rule 212

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

Rule 214

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

Rule 304

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 338

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 922

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 2081

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 6857

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

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Mathematica [A]
time = 0.00, size = 269, normalized size = 1.12 \begin {gather*} \frac {x^2 (b+a x)^{3/4} \left (2 d \left (2 \sqrt [4]{b+a x}-\sqrt [4]{a} \sqrt [4]{x} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\sqrt [4]{a} \sqrt [4]{x} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )-\frac {1}{4} \sqrt [4]{x} \text {RootSum}\left [b^2 c-a^2 d+2 a d \text {$\#$1}^4-d \text {$\#$1}^8\&,\frac {b^2 c \log (x)-a^2 d \log (x)-4 b^2 c \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^2 d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+a d \log (x) \text {$\#$1}^4-4 a d \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{d \left (x^3 (b+a x)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.48, size = 687, normalized size = 2.85 \begin {gather*} -\frac {4 \, {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \arctan \left (\frac {{\left (a d x - \sqrt {\frac {b^{2} c}{d}} d x\right )} {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a + \sqrt {\frac {b^{2} c}{d}}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a d - \sqrt {\frac {b^{2} c}{d}} d\right )} {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\right )} x}\right ) - 4 \, {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {{\left (a d x + \sqrt {\frac {b^{2} c}{d}} d x\right )} {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}} \sqrt {\frac {\sqrt {a - \sqrt {\frac {b^{2} c}{d}}} x^{2} + \sqrt {a x^{4} + b x^{3}}}{x^{2}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a d + \sqrt {\frac {b^{2} c}{d}} d\right )} {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {3}{4}}}{{\left (b^{2} c - a^{2} d\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 {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {\frac {b^{2} c}{d}}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {\frac {b^{2} c}{d}}\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((c*x^2+d)*(a*x^4+b*x^3)^(1/4)/x^2/(c*x^2-d),x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 104.52, size = 491, normalized size = 2.04 \begin {gather*} \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 ) + \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}} \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}{2} \, \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 ) - 2 \, \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) - 2 \, \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d}{{\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}}}\right ) - \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) - \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d + \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} + \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + \left (\frac {a d - \sqrt {c d} b}{d}\right )^{\frac {1}{4}} \log \left ({\left | -{\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} d + {\left (a d^{4} - \sqrt {c d} b d^{3}\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4)) + 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)*log(sqrt(2
)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) - 1/2*sqrt(2)*(-a)^(1/4)*log(-sqrt(2)*(-a)^(1/4)*(a +
b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) - 2*((a*d + sqrt(c*d)*b)/d)^(1/4)*arctan((a + b/x)^(1/4)*d/(a*d^4 + sq
rt(c*d)*b*d^3)^(1/4)) - 2*((a*d - sqrt(c*d)*b)/d)^(1/4)*arctan((a + b/x)^(1/4)*d/(a*d^4 - sqrt(c*d)*b*d^3)^(1/
4)) - ((a*d + sqrt(c*d)*b)/d)^(1/4)*log(abs((a + b/x)^(1/4)*d + (a*d^4 + sqrt(c*d)*b*d^3)^(1/4))) - ((a*d - sq
rt(c*d)*b)/d)^(1/4)*log(abs((a + b/x)^(1/4)*d + (a*d^4 - sqrt(c*d)*b*d^3)^(1/4))) + ((a*d + sqrt(c*d)*b)/d)^(1
/4)*log(abs(-(a + b/x)^(1/4)*d + (a*d^4 + sqrt(c*d)*b*d^3)^(1/4))) + ((a*d - sqrt(c*d)*b)/d)^(1/4)*log(abs(-(a
+ b/x)^(1/4)*d + (a*d^4 - sqrt(c*d)*b*d^3)^(1/4))) + 4*(a + b/x)^(1/4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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