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

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

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Rubi [B]  time = 2.10, 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 = {2056, 6725, 47, 63, 331, 298, 203, 206, 908, 37, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{a x^4+b x^3} \sqrt [4]{a \sqrt {d}-b \sqrt {c}} \tan ^{-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}} \tan ^{-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 {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} \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 [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 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 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 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 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 (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 ) \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 (\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 ) \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 (4 \left (b \sqrt {c}-a \sqrt {d}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.33, size = 177, normalized size = 0.73 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x+b)} \left (\frac {x \left (b \sqrt {c}-a \sqrt {d}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a-\frac {b \sqrt {c}}{\sqrt {d}}\right ) x}{b+a x}\right )-x \left (a \sqrt {d}+b \sqrt {c}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a+\frac {b \sqrt {c}}{\sqrt {d}}\right ) x}{b+a x}\right )+6 \sqrt {d} (a x+b)}{\sqrt {d} (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[((d + c*x^2)*(b*x^3 + a*x^4)^(1/4))/(x^2*(-d + c*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*Sqrt[d]
*(b + a*x) + (b*Sqrt[c] - a*Sqrt[d])*x*Hypergeometric2F1[3/4, 1, 7/4, ((a - (b*Sqrt[c])/Sqrt[d])*x)/(b + a*x)]
- (b*Sqrt[c] + a*Sqrt[d])*x*Hypergeometric2F1[3/4, 1, 7/4, ((a + (b*Sqrt[c])/Sqrt[d])*x)/(b + a*x)])/(Sqrt[d]
*(b + a*x))))/(3*x)

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IntegrateAlgebraic [A]  time = 0.87, size = 240, 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 )+\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} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((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)*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)] + 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] +
b^2*c*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^2*d*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a*d*Log[x]*#1^4 + a*d*Log
[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1^3) + #1^7) & ]/d

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fricas [B]  time = 0.65, 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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \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]

sage2

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maple [F]  time = 0.01, 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*} \int \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (c x^{2} + d\right )}}{{\left (c x^{2} - d\right )} x^{2}}\,{d 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="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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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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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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