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

Optimal. Leaf size=236 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8 d-2 \text {$\#$1}^4 a d+a^2 d+b^2 c\& ,\frac {-\text {$\#$1}^4 a d \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 a d \log (x)+a^2 d \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )+b^2 c \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-a^2 d \log (x)-b^2 c \log (x)}{\text {$\#$1}^3 a-\text {$\#$1}^7}\& \right ]}{2 c d}-\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{c}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^3}}\right )}{c} \]

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Rubi [B]  time = 1.60, antiderivative size = 561, normalized size of antiderivative = 2.38, number of steps used = 17, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2056, 906, 63, 331, 298, 203, 206, 6725, 93, 205, 208} \begin {gather*} \frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}+\frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{a x-b}}-\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 )}{c 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 )}{c x^{3/4} \sqrt [4]{a x-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(c*x^(3/4)*(-b + a*x)^(1/4))
+ ((-(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))])/(c*d^(1/8)*x^(3/4)*(-b + a*x)^(1/4)) + ((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))])/(c*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)])/(c*x^
(3/4)*(-b + a*x)^(1/4)) - ((-(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))])/(c*d^(1/8)*x^(3/4)*(-b + a*x)^(1/4)) - ((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))])/(c*d^(1/8)*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 906

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(e*g)/c, In
t[(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)*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] && 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 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 {\sqrt [4]{-b x^3+a x^4}}{d+c x^2} \, dx &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{-b+a x}}{d+c x^2} \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {-a d-b c x}{\sqrt [4]{x} (-b+a x)^{3/4} \left (d+c x^2\right )} \, dx}{c 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}{c x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\sqrt [4]{-b x^3+a x^4} \int \left (\frac {-\frac {b c d}{\sqrt {-c}}-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\frac {b c d}{\sqrt {-c}}-a d^{3/2}}{2 d \sqrt [4]{x} (-b+a x)^{3/4} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx}{c 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 )}{c 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}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{c 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}{2 c 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}{2 c 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 )}{c 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 )}{c 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 {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 )}{c 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 {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 )}{c x^{3/4} \sqrt [4]{-b+a 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 )}{c 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 )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\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 )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\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 )}{c 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 ) \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 )}{c \sqrt {b \sqrt {-c}+a \sqrt {d}} 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 ) \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 )}{c \sqrt {b \sqrt {-c}+a \sqrt {d}} x^{3/4} \sqrt [4]{-b+a 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 )}{c x^{3/4} \sqrt [4]{-b+a x}}+\frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\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 )}{c \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 )}{c x^{3/4} \sqrt [4]{-b+a x}}-\frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\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 )}{c \sqrt [8]{d} x^{3/4} \sqrt [4]{-b+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 182, normalized size = 0.77 \begin {gather*} -\frac {2 \sqrt [4]{x^3 (a x-b)} \left (\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 )-\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 )+2 a \sqrt {d} \left (1-\frac {a x}{b}\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {a x}{b}\right )\right )}{3 c \sqrt {d} (b-a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(x^3*(-b + a*x))^(1/4)*(2*a*Sqrt[d]*(1 - (a*x)/b)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (a*x)/b] + (b*Sqr
t[-c] - a*Sqrt[d])*Hypergeometric2F1[3/4, 1, 7/4, -(((a - (b*Sqrt[-c])/Sqrt[d])*x)/(b - a*x))] - (b*Sqrt[-c] +
a*Sqrt[d])*Hypergeometric2F1[3/4, 1, 7/4, -(((a + (b*Sqrt[-c])/Sqrt[d])*x)/(b - a*x))]))/(3*c*Sqrt[d]*(b - a*
x))

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IntegrateAlgebraic [A]  time = 0.64, size = 235, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\frac {2 \sqrt [4]{a} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{c}+\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 ]}{2 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.78, size = 940, normalized size = 3.98 \begin {gather*} -2 \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (c^{7} d x \sqrt {-\frac {b^{2}}{c^{7} d}} - a c^{3} d x\right )} \sqrt {\frac {c^{2} x^{2} \sqrt {\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {3}{4}} - {\left (c^{7} d \sqrt {-\frac {b^{2}}{c^{7} d}} - a c^{3} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c + a^{2} d\right )} x}\right ) + 2 \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (c^{7} d x \sqrt {-\frac {b^{2}}{c^{7} d}} + a c^{3} d x\right )} \sqrt {\frac {c^{2} x^{2} \sqrt {-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {3}{4}} - {\left (c^{7} d \sqrt {-\frac {b^{2}}{c^{7} d}} + a c^{3} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {3}{4}}}{{\left (b^{2} c + a^{2} d\right )} x}\right ) - 4 \, \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {c^{3} x \sqrt {\frac {c^{2} x^{2} \sqrt {\frac {a}{c^{4}}} + \sqrt {a x^{4} - b x^{3}}}{x^{2}}} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} c^{3} \left (\frac {a}{c^{4}}\right )^{\frac {3}{4}}}{a x}\right ) - \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} + a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (-\frac {c^{4} \sqrt {-\frac {b^{2}}{c^{7} d}} - a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} + {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {c x \left (\frac {a}{c^{4}}\right )^{\frac {1}{4}} - {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4)*arctan(((c^7*d*x*sqrt(-b^2/(c^7*d)) - a*c^3*d*x)*sqrt((c^2*x^2*sqr
t((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4) + sqrt(a*x^4 - b*x^3))/x^2)*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(3/4) - (c^
7*d*sqrt(-b^2/(c^7*d)) - a*c^3*d)*(a*x^4 - b*x^3)^(1/4)*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(3/4))/((b^2*c + a^
2*d)*x)) + 2*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)^(1/4)*arctan(((c^7*d*x*sqrt(-b^2/(c^7*d)) + a*c^3*d*x)*sqrt((
c^2*x^2*sqrt(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4) + sqrt(a*x^4 - b*x^3))/x^2)*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4
)^(3/4) - (c^7*d*sqrt(-b^2/(c^7*d)) + a*c^3*d)*(a*x^4 - b*x^3)^(1/4)*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)^(3/4)
)/((b^2*c + a^2*d)*x)) - 4*(a/c^4)^(1/4)*arctan((c^3*x*sqrt((c^2*x^2*sqrt(a/c^4) + sqrt(a*x^4 - b*x^3))/x^2)*(
a/c^4)^(3/4) - (a*x^4 - b*x^3)^(1/4)*c^3*(a/c^4)^(3/4))/(a*x)) - 1/2*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4)*
log((c*x*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) + 1/2*((c^4*sqrt(-b^2/(c^7*d)) +
a)/c^4)^(1/4)*log(-(c*x*((c^4*sqrt(-b^2/(c^7*d)) + a)/c^4)^(1/4) - (a*x^4 - b*x^3)^(1/4))/x) - 1/2*(-(c^4*sqr
t(-b^2/(c^7*d)) - a)/c^4)^(1/4)*log((c*x*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x)
+ 1/2*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)^(1/4)*log(-(c*x*(-(c^4*sqrt(-b^2/(c^7*d)) - a)/c^4)^(1/4) - (a*x^4
- b*x^3)^(1/4))/x) + (a/c^4)^(1/4)*log((c*x*(a/c^4)^(1/4) + (a*x^4 - b*x^3)^(1/4))/x) - (a/c^4)^(1/4)*log(-(c*
x*(a/c^4)^(1/4) - (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((a*x^4-b*x^3)^(1/4)/(c*x^2+d),x, algorithm="giac")

[Out]

sage2

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}-b \,x^{3}\right )^{\frac {1}{4}}}{c \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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