3.26.2 \(\int \frac {b+a x^6}{x^3 (-b+a x^3) \sqrt [4]{b x+a x^4}} \, dx\)

Optimal. Leaf size=208 \[ \frac {4 \left (a x^4+b x\right )^{3/4}}{9 b x^3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a} b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4+b x\right )^{3/4}}{a x^3+b}\right )}{3 \sqrt [4]{a} b} \]

________________________________________________________________________________________

Rubi [A]  time = 1.15, antiderivative size = 343, normalized size of antiderivative = 1.65, number of steps used = 17, number of rules used = 13, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2056, 6725, 264, 329, 275, 240, 212, 206, 203, 466, 465, 494, 453} \begin {gather*} \frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4+b x}}+\frac {2 \sqrt [4]{x} \sqrt [4]{a x^3+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} \sqrt [4]{a x^4+b x}}-\frac {2^{3/4} \sqrt [4]{x} (a+b) \sqrt [4]{a x^3+b} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x^3+b}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{a x^4+b x}}+\frac {4 (a+b) \left (a x^3+b\right )}{9 a b x^2 \sqrt [4]{a x^4+b x}}-\frac {4 \left (a x^3+b\right )}{9 a x^2 \sqrt [4]{a x^4+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(b + a*x^3))/(9*a*x^2*(b*x + a*x^4)^(1/4)) + (4*(a + b)*(b + a*x^3))/(9*a*b*x^2*(b*x + a*x^4)^(1/4)) + (2*
x^(1/4)*(b + a*x^3)^(1/4)*ArcTan[(a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(3*a^(1/4)*(b*x + a*x^4)^(1/4)) - (2^(3
/4)*(a + b)*x^(1/4)*(b + a*x^3)^(1/4)*ArcTan[(2^(1/4)*a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(3*a^(1/4)*b*(b*x +
a*x^4)^(1/4)) + (2*x^(1/4)*(b + a*x^3)^(1/4)*ArcTanh[(a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)])/(3*a^(1/4)*(b*x +
a*x^4)^(1/4)) - (2^(3/4)*(a + b)*x^(1/4)*(b + a*x^3)^(1/4)*ArcTanh[(2^(1/4)*a^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)
])/(3*a^(1/4)*b*(b*x + a*x^4)^(1/4))

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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

Rule 264

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

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 494

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 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 {b+a x^6}{x^3 \left (-b+a x^3\right ) \sqrt [4]{b x+a x^4}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {b+a x^6}{x^{13/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \left (\frac {b}{a x^{13/4} \sqrt [4]{b+a x^3}}+\frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}}+\frac {a b+b^2}{a x^{13/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}}\right ) \, dx}{\sqrt [4]{b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{b+a x^3}} \, dx}{\sqrt [4]{b x+a x^4}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}+\frac {\left (b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \int \frac {1}{x^{13/4} \left (-b+a x^3\right ) \sqrt [4]{b+a x^3}} \, dx}{a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \left (-b+a x^{12}\right ) \sqrt [4]{b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 b (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (-b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,x^{3/4}\right )}{3 a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1-a x^4}{x^4 \left (-b+2 a b x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 a \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )}{9 a b x^2 \sqrt [4]{b x+a x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}+\frac {\left (4 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-b+2 a b x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )}{9 a b x^2 \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}-\frac {\left (2 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b \sqrt [4]{b x+a x^4}}-\frac {\left (2 (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 b \sqrt [4]{b x+a x^4}}\\ &=-\frac {4 \left (b+a x^3\right )}{9 a x^2 \sqrt [4]{b x+a x^4}}+\frac {4 (a+b) \left (b+a x^3\right )}{9 a b x^2 \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{b x+a x^4}}+\frac {2 \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} \sqrt [4]{b x+a x^4}}-\frac {2^{3/4} (a+b) \sqrt [4]{x} \sqrt [4]{b+a x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )}{3 \sqrt [4]{a} b \sqrt [4]{b x+a x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 3.24, size = 188, normalized size = 0.90 \begin {gather*} \frac {4 \left (\frac {4 \Gamma \left (\frac {5}{4}\right ) (a+b) \left (a x^3+b\right ) \left (5 \left (-4 a^2 x^6+3 a b x^3+b^2\right ) \, _2F_1\left (1,1;\frac {5}{4};-\frac {2 a x^3}{b-a x^3}\right )-32 a x^3 \left (a x^3+b\right ) \, _2F_1\left (2,2;\frac {9}{4};-\frac {2 a x^3}{b-a x^3}\right )\right )}{a b \Gamma \left (\frac {1}{4}\right ) \left (b-a x^3\right )^2}+15 x^3 \sqrt [4]{\frac {a x^3}{b}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {a x^3}{b}\right )-\frac {5 \left (a x^3+b\right )}{a}\right )}{45 x^2 \sqrt [4]{x \left (a x^3+b\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(4*((-5*(b + a*x^3))/a + 15*x^3*(1 + (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, -((a*x^3)/b)] + (4*(a +
b)*(b + a*x^3)*Gamma[5/4]*(5*(b^2 + 3*a*b*x^3 - 4*a^2*x^6)*Hypergeometric2F1[1, 1, 5/4, (-2*a*x^3)/(b - a*x^3
)] - 32*a*x^3*(b + a*x^3)*Hypergeometric2F1[2, 2, 9/4, (-2*a*x^3)/(b - a*x^3)]))/(a*b*(b - a*x^3)^2*Gamma[1/4]
)))/(45*x^2*(x*(b + a*x^3))^(1/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.98, size = 208, normalized size = 1.00 \begin {gather*} \frac {4 \left (b x+a x^4\right )^{3/4}}{9 b x^3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a} b}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a}}-\frac {2^{3/4} (a+b) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )}{3 \sqrt [4]{a} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b*x + a*x^4)^(3/4))/(9*b*x^3) + (2*ArcTan[(a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) - (2^(3/
4)*(a + b)*ArcTan[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)*b) + (2*ArcTanh[(a^(1/4)*(b*x
+ a*x^4)^(3/4))/(b + a*x^3)])/(3*a^(1/4)) - (2^(3/4)*(a + b)*ArcTanh[(2^(1/4)*a^(1/4)*(b*x + a*x^4)^(3/4))/(b
+ a*x^3)])/(3*a^(1/4)*b)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^3/(a*x^3-b)/(a*x^4+b*x)^(1/4),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [B]  time = 0.21, size = 437, normalized size = 2.10 \begin {gather*} \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{3 \, a} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, a} - \frac {2^{\frac {1}{4}} {\left (a + b\right )} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} + \frac {2^{\frac {1}{4}} {\left (a + b\right )} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right )}{6 \, \left (-a\right )^{\frac {1}{4}} b} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, a b} - \frac {\sqrt {2} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} a + 2^{\frac {3}{4}} \left (-a\right )^{\frac {3}{4}} b\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{6 \, a b} + \frac {4 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}}}{9 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^3/(a*x^3-b)/(a*x^4+b*x)^(1/4),x, algorithm="giac")

[Out]

1/3*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/a + 1/3*sqrt(
2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/a - 1/6*sqrt(2)*(-a)^
(3/4)*log(sqrt(2)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(-a) + sqrt(a + b/x^3))/a + 1/6*sqrt(2)*(-a)^(3/4)*log(-s
qrt(2)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(-a) + sqrt(a + b/x^3))/a - 1/6*2^(1/4)*(a + b)*log(2^(3/4)*(-a)^(1/
4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/((-a)^(1/4)*b) + 1/6*2^(1/4)*(a + b)*log(-2^(3/4)*(
-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^3))/((-a)^(1/4)*b) - 1/6*sqrt(2)*(2^(3/4)*(-a)^(
3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x^3)^(1/4))/(-a)^(1/4))/(a*b)
- 1/6*sqrt(2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a +
b/x^3)^(1/4))/(-a)^(1/4))/(a*b) + 4/9*(a + b/x^3)^(3/4)/b

________________________________________________________________________________________

maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{6}+b}{x^{3} \left (a \,x^{3}-b \right ) \left (a \,x^{4}+b x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^6+b)/x^3/(a*x^3-b)/(a*x^4+b*x)^(1/4),x)

[Out]

int((a*x^6+b)/x^3/(a*x^3-b)/(a*x^4+b*x)^(1/4),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{{\left (a x^{4} + b x\right )}^{\frac {1}{4}} {\left (a x^{3} - b\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^3/(a*x^3-b)/(a*x^4+b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)/((a*x^4 + b*x)^(1/4)*(a*x^3 - b)*x^3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {a\,x^6+b}{x^3\,{\left (a\,x^4+b\,x\right )}^{1/4}\,\left (b-a\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b + a*x^6)/(x^3*(b*x + a*x^4)^(1/4)*(b - a*x^3)),x)

[Out]

-int((b + a*x^6)/(x^3*(b*x + a*x^4)^(1/4)*(b - a*x^3)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{6} + b}{x^{3} \sqrt [4]{x \left (a x^{3} + b\right )} \left (a x^{3} - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**6+b)/x**3/(a*x**3-b)/(a*x**4+b*x)**(1/4),x)

[Out]

Integral((a*x**6 + b)/(x**3*(x*(a*x**3 + b))**(1/4)*(a*x**3 - b)), x)

________________________________________________________________________________________