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

Optimal. Leaf size=92 \[ b \text {RootSum}\left [\text {$\#$1}^8 d-2 \text {$\#$1}^4 a d-\text {$\#$1}^4 b c+a^2 d+a b c+b^2\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^4 d-2 a d-b c}\& \right ] \]

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Rubi [B]  time = 0.90, antiderivative size = 589, normalized size of antiderivative = 6.40, number of steps used = 23, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2056, 911, 105, 63, 331, 298, 203, 206, 93, 205, 208} \begin {gather*} \frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a \left (c-\sqrt {c^2-4 d}\right )+2 b} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \left (c-\sqrt {c^2-4 d}\right )+2 b}}{\sqrt [4]{c-\sqrt {c^2-4 d}} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{c-\sqrt {c^2-4 d}} \sqrt {c^2-4 d} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a \left (\sqrt {c^2-4 d}+c\right )+2 b} \tan ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \left (\sqrt {c^2-4 d}+c\right )+2 b}}{\sqrt [4]{\sqrt {c^2-4 d}+c} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {c^2-4 d}+c} \sqrt {c^2-4 d} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a \left (c-\sqrt {c^2-4 d}\right )+2 b} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \left (c-\sqrt {c^2-4 d}\right )+2 b}}{\sqrt [4]{c-\sqrt {c^2-4 d}} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{c-\sqrt {c^2-4 d}} \sqrt {c^2-4 d} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a x^4-b x^3} \sqrt [4]{a \left (\sqrt {c^2-4 d}+c\right )+2 b} \tanh ^{-1}\left (\frac {\sqrt [4]{x} \sqrt [4]{a \left (\sqrt {c^2-4 d}+c\right )+2 b}}{\sqrt [4]{\sqrt {c^2-4 d}+c} \sqrt [4]{a x-b}}\right )}{x^{3/4} \sqrt [4]{\sqrt {c^2-4 d}+c} \sqrt {c^2-4 d} \sqrt [4]{a x-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*b + a*(c - Sqrt[c^2 - 4*d]))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[((2*b + a*(c - Sqrt[c^2 - 4*d]))^(1/4
)*x^(1/4))/((c - Sqrt[c^2 - 4*d])^(1/4)*(-b + a*x)^(1/4))])/((c - Sqrt[c^2 - 4*d])^(1/4)*Sqrt[c^2 - 4*d]*x^(3/
4)*(-b + a*x)^(1/4)) - (2*(2*b + a*(c + Sqrt[c^2 - 4*d]))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[((2*b + a*(c +
Sqrt[c^2 - 4*d]))^(1/4)*x^(1/4))/((c + Sqrt[c^2 - 4*d])^(1/4)*(-b + a*x)^(1/4))])/((c + Sqrt[c^2 - 4*d])^(1/4
)*Sqrt[c^2 - 4*d]*x^(3/4)*(-b + a*x)^(1/4)) - (2*(2*b + a*(c - Sqrt[c^2 - 4*d]))^(1/4)*(-(b*x^3) + a*x^4)^(1/4
)*ArcTanh[((2*b + a*(c - Sqrt[c^2 - 4*d]))^(1/4)*x^(1/4))/((c - Sqrt[c^2 - 4*d])^(1/4)*(-b + a*x)^(1/4))])/((c
- Sqrt[c^2 - 4*d])^(1/4)*Sqrt[c^2 - 4*d]*x^(3/4)*(-b + a*x)^(1/4)) + (2*(2*b + a*(c + Sqrt[c^2 - 4*d]))^(1/4)
*(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[((2*b + a*(c + Sqrt[c^2 - 4*d]))^(1/4)*x^(1/4))/((c + Sqrt[c^2 - 4*d])^(1/4)
*(-b + a*x)^(1/4))])/((c + Sqrt[c^2 - 4*d])^(1/4)*Sqrt[c^2 - 4*d]*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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 911

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

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]

Rubi steps

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

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Mathematica [B]  time = 58.43, size = 491, normalized size = 5.34 \begin {gather*} \frac {\sqrt [4]{x^3 (a x-b)} \left (-\sqrt [4]{\frac {b \left (\sqrt {c^2-4 d}-c\right )}{d}-2 a} \log \left (\sqrt [4]{\frac {b \left (\sqrt {c^2-4 d}-c\right )}{2 d}-a}-\sqrt [4]{\frac {b}{x}-a}\right )+\sqrt [4]{-2 a-\frac {b \left (\sqrt {c^2-4 d}+c\right )}{d}} \log \left (\sqrt [4]{-a-\frac {b \left (\sqrt {c^2-4 d}+c\right )}{2 d}}-\sqrt [4]{\frac {b}{x}-a}\right )+\sqrt [4]{\frac {b \left (\sqrt {c^2-4 d}-c\right )}{d}-2 a} \log \left (\sqrt [4]{\frac {b \left (\sqrt {c^2-4 d}-c\right )}{2 d}-a}+\sqrt [4]{\frac {b}{x}-a}\right )-\sqrt [4]{-2 a-\frac {b \left (\sqrt {c^2-4 d}+c\right )}{d}} \log \left (\sqrt [4]{-a-\frac {b \left (\sqrt {c^2-4 d}+c\right )}{2 d}}+\sqrt [4]{\frac {b}{x}-a}\right )+2 \sqrt [4]{\frac {b \left (\sqrt {c^2-4 d}-c\right )}{d}-2 a} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2 b}{x}-2 a}}{\sqrt [4]{\frac {b \left (\sqrt {c^2-4 d}-c\right )}{d}-2 a}}\right )-2 \sqrt [4]{-2 a-\frac {b \left (\sqrt {c^2-4 d}+c\right )}{d}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2 b}{x}-2 a}}{\sqrt [4]{-2 a-\frac {b \left (\sqrt {c^2-4 d}+c\right )}{d}}}\right )\right )}{x \sqrt {c^2-4 d} \sqrt [4]{\frac {2 b}{x}-2 a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^3*(-b + a*x))^(1/4)*(2*(-2*a + (b*(-c + Sqrt[c^2 - 4*d]))/d)^(1/4)*ArcTan[(-2*a + (2*b)/x)^(1/4)/(-2*a + (
b*(-c + Sqrt[c^2 - 4*d]))/d)^(1/4)] - 2*(-2*a - (b*(c + Sqrt[c^2 - 4*d]))/d)^(1/4)*ArcTan[(-2*a + (2*b)/x)^(1/
4)/(-2*a - (b*(c + Sqrt[c^2 - 4*d]))/d)^(1/4)] - (-2*a + (b*(-c + Sqrt[c^2 - 4*d]))/d)^(1/4)*Log[(-a + (b*(-c
+ Sqrt[c^2 - 4*d]))/(2*d))^(1/4) - (-a + b/x)^(1/4)] + (-2*a - (b*(c + Sqrt[c^2 - 4*d]))/d)^(1/4)*Log[(-a - (b
*(c + Sqrt[c^2 - 4*d]))/(2*d))^(1/4) - (-a + b/x)^(1/4)] + (-2*a + (b*(-c + Sqrt[c^2 - 4*d]))/d)^(1/4)*Log[(-a
+ (b*(-c + Sqrt[c^2 - 4*d]))/(2*d))^(1/4) + (-a + b/x)^(1/4)] - (-2*a - (b*(c + Sqrt[c^2 - 4*d]))/d)^(1/4)*Lo
g[(-a - (b*(c + Sqrt[c^2 - 4*d]))/(2*d))^(1/4) + (-a + b/x)^(1/4)]))/(Sqrt[c^2 - 4*d]*(-2*a + (2*b)/x)^(1/4)*x
)

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IntegrateAlgebraic [A]  time = 0.00, size = 92, normalized size = 1.00 \begin {gather*} b \text {RootSum}\left [b^2+a b c+a^2 d-b c \text {$\#$1}^4-2 a d \text {$\#$1}^4+d \text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-b c-2 a d+2 d \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.70, size = 2783, normalized size = 30.25

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)/x/(c*x+x^2+d),x, algorithm="fricas")

[Out]

-2*sqrt(2)*sqrt(sqrt(2)*sqrt((b*c + 2*a*d + (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c
^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)))*arctan(-1/8*sqrt(2)*(sqrt(2)*(b^3*c^4 - 8*b^3*c^2*d + 16*b^3
*d^2 - (b^2*c^7*d - 128*a*b*d^5 + 32*(3*a*b*c^2 - 2*b^2*c)*d^4 - 24*(a*b*c^4 - 2*b^2*c^3)*d^3 + 2*(a*b*c^6 - 6
*b^2*c^5)*d^2)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))*(a*x^4 - b*x^3)^(1/4)*sqrt((b*c + 2*a*d
+ (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 1
6*d^3)) + ((b*c^7*d - 128*a*d^5 + 32*(3*a*c^2 - 2*b*c)*d^4 - 24*(a*c^4 - 2*b*c^3)*d^3 + 2*(a*c^6 - 6*b*c^5)*d^
2)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5))*x - (b^2*c^4 - 8*b^2*c^2*d + 16*b^2*d^2)*x)*sqrt((b*
c + 2*a*d + (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^
2*d^2 + 16*d^3))*sqrt((sqrt(2)*(b^2*c^2 - 4*b^2*d)*x^2*sqrt((b*c + 2*a*d + (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b
^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)) + 2*sqrt(a*x^4 - b*x^3)*b^2)/x
^2))*sqrt(sqrt(2)*sqrt((b*c + 2*a*d + (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4
- 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)))/((a*b^3*c + a^2*b^2*d + b^4)*x)) + 2*sqrt(2)*sqrt(sqrt(2)*sqrt((b*
c + 2*a*d - (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^
2*d^2 + 16*d^3)))*arctan(1/8*sqrt(2)*(sqrt(2)*(b^3*c^4 - 8*b^3*c^2*d + 16*b^3*d^2 + (b^2*c^7*d - 128*a*b*d^5 +
32*(3*a*b*c^2 - 2*b^2*c)*d^4 - 24*(a*b*c^4 - 2*b^2*c^3)*d^3 + 2*(a*b*c^6 - 6*b^2*c^5)*d^2)*sqrt(b^2/(c^6*d^2
- 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))*(a*x^4 - b*x^3)^(1/4)*sqrt((b*c + 2*a*d - (c^4*d - 8*c^2*d^2 + 16*d^3)*s
qrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)) - ((b*c^7*d - 128*a*d^5 +
32*(3*a*c^2 - 2*b*c)*d^4 - 24*(a*c^4 - 2*b*c^3)*d^3 + 2*(a*c^6 - 6*b*c^5)*d^2)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3
+ 48*c^2*d^4 - 64*d^5))*x + (b^2*c^4 - 8*b^2*c^2*d + 16*b^2*d^2)*x)*sqrt((b*c + 2*a*d - (c^4*d - 8*c^2*d^2 +
16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3))*sqrt((sqrt(2)*(b
^2*c^2 - 4*b^2*d)*x^2*sqrt((b*c + 2*a*d - (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2
*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)) + 2*sqrt(a*x^4 - b*x^3)*b^2)/x^2))*sqrt(sqrt(2)*sqrt((b*c + 2*a
*d - (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 +
16*d^3)))/((a*b^3*c + a^2*b^2*d + b^4)*x)) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((b*c + 2*a*d + (c^4*d - 8*c^2*d^2
+ 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)))*log((sqrt(2)*
(c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5))*x*sqrt(sqrt(2)*sqrt((b*c +
2*a*d + (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d
^2 + 16*d^3))) + 2*(a*x^4 - b*x^3)^(1/4)*b)/x) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((b*c + 2*a*d + (c^4*d - 8*c^2*d
^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)))*log(-(sqrt
(2)*(c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5))*x*sqrt(sqrt(2)*sqrt((b
*c + 2*a*d + (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c
^2*d^2 + 16*d^3))) - 2*(a*x^4 - b*x^3)^(1/4)*b)/x) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((b*c + 2*a*d - (c^4*d - 8*c
^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)))*log((s
qrt(2)*(c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5))*x*sqrt(sqrt(2)*sqrt
((b*c + 2*a*d - (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d -
8*c^2*d^2 + 16*d^3))) + 2*(a*x^4 - b*x^3)^(1/4)*b)/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((b*c + 2*a*d - (c^4*d -
8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*d - 8*c^2*d^2 + 16*d^3)))*log
(-(sqrt(2)*(c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5))*x*sqrt(sqrt(2)*
sqrt((b*c + 2*a*d - (c^4*d - 8*c^2*d^2 + 16*d^3)*sqrt(b^2/(c^6*d^2 - 12*c^4*d^3 + 48*c^2*d^4 - 64*d^5)))/(c^4*
d - 8*c^2*d^2 + 16*d^3))) - 2*(a*x^4 - b*x^3)^(1/4)*b)/x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{201326592,[1,14,10,12,0,4]%%%}+%%%{-218103808,[1,14,10,11,0,6]%
%%}+%%%{88080384,[1,14,10,10,0,8]%%%}+%%%{-15728640,[1,14,10,9,0,10]%%%}+%%%{1048576,[1,14,10,8,0,12]%%%}+%%%{
1610612736,[1,13,10,13,1,3]%%%}+%%%{-1811939328,[1,13,10,12,1,5]%%%}+%%%{754974720,[1,13,10,11,1,7]%%%}+%%%{-1
38412032,[1,13,10,10,1,9]%%%}+%%%{9437184,[1,13,10,9,1,11]%%%}+%%%{4831838208,[1,12,10,14,2,2]%%%}+%%%{-583847
1168,[1,12,10,13,2,4]%%%}+%%%{2566914048,[1,12,10,12,2,6]%%%}+%%%{-490733568,[1,12,10,11,2,8]%%%}+%%%{34603008
,[1,12,10,10,2,10]%%%}+%%%{-18446744062972133376,[1,11,10,15,3,1]%%%}+%%%{-9126805504,[1,11,10,14,3,3]%%%}+%%%
{4429185024,[1,11,10,13,3,5]%%%}+%%%{-905969664,[1,11,10,12,3,7]%%%}+%%%{67108864,[1,11,10,11,3,9]%%%}+%%%{327
68,[1,11,6,12,0,3]%%%}+%%%{-8192,[1,11,6,11,0,5]%%%}+%%%{3221225472,[1,10,10,16,4,0]%%%}+%%%{-7247757312,[1,10
,10,15,4,2]%%%}+%%%{4227858432,[1,10,10,14,4,4]%%%}+%%%{-956301312,[1,10,10,13,4,6]%%%}+%%%{75497472,[1,10,10,
12,4,8]%%%}+%%%{196608,[1,10,6,13,1,2]%%%}+%%%{-49152,[1,10,6,12,1,4]%%%}+%%%{-3221225472,[1,9,10,16,5,1]%%%}+
%%%{2415919104,[1,9,10,15,5,3]%%%}+%%%{-603979776,[1,9,10,14,5,5]%%%}+%%%{50331648,[1,9,10,13,5,7]%%%}+%%%{393
216,[1,9,6,14,2,1]%%%}+%%%{-98304,[1,9,6,13,2,3]%%%}+%%%{-1073741824,[1,8,10,17,6,0]%%%}+%%%{805306368,[1,8,10
,16,6,2]%%%}+%%%{-201326592,[1,8,10,15,6,4]%%%}+%%%{16777216,[1,8,10,14,6,6]%%%}+%%%{262144,[1,8,6,15,3,0]%%%}
+%%%{-65536,[1,8,6,14,3,2]%%%}+%%%{1073741824,[0,15,11,12,0,4]%%%}+%%%{-1073741824,[0,15,11,11,0,6]%%%}+%%%{40
2653184,[0,15,11,10,0,8]%%%}+%%%{-67108864,[0,15,11,9,0,10]%%%}+%%%{4194304,[0,15,11,8,0,12]%%%}+%%%{858993459
2,[0,14,11,13,1,3]%%%}+%%%{-7516192768,[0,14,11,12,1,5]%%%}+%%%{2147483648,[0,14,11,11,1,7]%%%}+%%%{-134217728
,[0,14,11,10,1,9]%%%}+%%%{-33554432,[0,14,11,9,1,11]%%%}+%%%{4194304,[0,14,11,8,1,13]%%%}+%%%{25769803776,[0,1
3,11,14,2,2]%%%}+%%%{-16106127360,[0,13,11,13,2,4]%%%}+%%%{2013265920,[0,13,11,11,2,8]%%%}+%%%{-503316480,[0,1
3,11,10,2,10]%%%}+%%%{37748736,[0,13,11,9,2,12]%%%}+%%%{34359738368,[0,12,11,15,3,1]%%%}+%%%{-21474836480,[0,1
2,11,13,3,5]%%%}+%%%{10737418240,[0,12,11,12,3,7]%%%}+%%%{-2013265920,[0,12,11,11,3,9]%%%}+%%%{134217728,[0,12
,11,10,3,11]%%%}+%%%{262144,[0,12,7,12,0,3]%%%}+%%%{-131072,[0,12,7,11,0,5]%%%}+%%%{16384,[0,12,7,10,0,7]%%%}+
%%%{17179869184,[0,11,11,16,4,0]%%%}+%%%{42949672960,[0,11,11,15,4,2]%%%}+%%%{-53687091200,[0,11,11,14,4,4]%%%
}+%%%{21474836480,[0,11,11,13,4,6]%%%}+%%%{-3690987520,[0,11,11,12,4,8]%%%}+%%%{234881024,[0,11,11,11,4,10]%%%
}+%%%{1572864,[0,11,7,13,1,2]%%%}+%%%{-786432,[0,11,7,12,1,4]%%%}+%%%{98304,[0,11,7,11,1,6]%%%}+%%%{5153960755
2,[0,10,11,16,5,1]%%%}+%%%{-51539607552,[0,10,11,15,5,3]%%%}+%%%{19327352832,[0,10,11,14,5,5]%%%}+%%%{-3221225
472,[0,10,11,13,5,7]%%%}+%%%{201326592,[0,10,11,12,5,9]%%%}+%%%{3145728,[0,10,7,14,2,1]%%%}+%%%{-1572864,[0,10
,7,13,2,3]%%%}+%%%{196608,[0,10,7,12,2,5]%%%}+%%%{17179869184,[0,9,11,17,6,0]%%%}+%%%{-17179869184,[0,9,11,16,
6,2]%%%}+%%%{6442450944,[0,9,11,15,6,4]%%%}+%%%{-1073741824,[0,9,11,14,6,6]%%%}+%%%{67108864,[0,9,11,13,6,8]%%
%}+%%%{2097152,[0,9,7,15,3,0]%%%}+%%%{-1048576,[0,9,7,14,3,2]%%%}+%%%{131072,[0,9,7,13,3,4]%%%} / %%%{-4096,[0
,5,5,5,0,3]%%%}+%%%{1024,[0,5,5,4,0,5]%%%}+%%%{-24576,[0,4,5,6,1,2]%%%}+%%%{6144,[0,4,5,5,1,4]%%%}+%%%{-49152,
[0,3,5,7,2,1]%%%}+%%%{12288,[0,3,5,6,2,3]%%%}+%%%{-32768,[0,2,5,8,3,0]%%%}+%%%{8192,[0,2,5,7,3,2]%%%} Error: B
ad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (x^2+c\,x+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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