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

Optimal. Leaf size=157 \[ \frac {2^{3/4} \left (a^{3/4} b+a^{7/4}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )}{3 b^2}+\frac {2^{3/4} \left (a^{3/4} b+a^{7/4}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \left (a x^4-b x\right )^{3/4}}{a x^3-b}\right )}{3 b^2}+\frac {4 \left (a x^4-b x\right )^{3/4} \left (b-a x^3\right )}{21 b^2 x^6} \]

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Rubi [A]  time = 1.19, antiderivative size = 269, normalized size of antiderivative = 1.71, number of steps used = 14, number of rules used = 11, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2056, 6725, 271, 264, 466, 465, 494, 461, 212, 206, 203} \begin {gather*} \frac {2^{3/4} a^{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 b^2 \sqrt [4]{a x^4-b x}}+\frac {2^{3/4} a^{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 b^2 \sqrt [4]{a x^4-b x}}-\frac {4 (a+b) \left (b-a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{a x^4-b x}}+\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{a x^4-b x}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \sqrt [4]{a x^4-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b - a*x^3))/(21*a*x^5*(-(b*x) + a*x^4)^(1/4)) - (4*(b - a*x^3))/(21*b*x^2*(-(b*x) + a*x^4)^(1/4)) - (4*(a
+ b)*(b - a*x^3)^2)/(21*a*b^2*x^5*(-(b*x) + a*x^4)^(1/4)) + (2^(3/4)*a^(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*b^2*(-(b*x) + a*x^4)^(1/4)) + (2^(3/4)*a^(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*b^2*(-(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 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 271

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

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

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^6 \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^{25/4} \sqrt [4]{-b+a x^3} \left (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 \left (-\frac {b}{a x^{25/4} \sqrt [4]{-b+a x^3}}+\frac {1}{x^{13/4} \sqrt [4]{-b+a x^3}}+\frac {a b+b^2}{a x^{25/4} \sqrt [4]{-b+a x^3} \left (b+a x^3\right )}\right ) \, dx}{\sqrt [4]{-b x+a x^4}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{x^{13/4} \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^{25/4} \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^{25/4} \sqrt [4]{-b+a x^3} \left (b+a x^3\right )} \, dx}{a \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{9 b x^2 \sqrt [4]{-b x+a x^4}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \int \frac {1}{x^{13/4} \sqrt [4]{-b+a x^3}} \, dx}{7 \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^{22} \sqrt [4]{-b+a x^{12}} \left (b+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{a \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \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^8 \sqrt [4]{-b+a x^4} \left (b+a x^4\right )} \, dx,x,x^{3/4}\right )}{3 a \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \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 {\left (1-a x^4\right )^2}{x^8 \left (b-2 a b x^4\right )} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 a b \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \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 \left (\frac {1}{b x^8}-\frac {a^2}{b \left (-1+2 a x^4\right )}\right ) \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 a b \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \sqrt [4]{-b x+a x^4}}-\frac {4 (a+b) \left (b-a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{-b x+a x^4}}-\frac {\left (4 a (a+b) \sqrt [4]{x} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+2 a x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-b+a x^3}}\right )}{3 b^2 \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \sqrt [4]{-b x+a x^4}}-\frac {4 (a+b) \left (b-a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{-b x+a x^4}}+\frac {\left (2 a (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^2 \sqrt [4]{-b x+a x^4}}+\frac {\left (2 a (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^2 \sqrt [4]{-b x+a x^4}}\\ &=\frac {4 \left (b-a x^3\right )}{21 a x^5 \sqrt [4]{-b x+a x^4}}-\frac {4 \left (b-a x^3\right )}{21 b x^2 \sqrt [4]{-b x+a x^4}}-\frac {4 (a+b) \left (b-a x^3\right )^2}{21 a b^2 x^5 \sqrt [4]{-b x+a x^4}}+\frac {2^{3/4} a^{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 b^2 \sqrt [4]{-b x+a x^4}}+\frac {2^{3/4} a^{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 b^2 \sqrt [4]{-b x+a x^4}}\\ \end {align*}

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Mathematica [C]  time = 5.18, size = 114, normalized size = 0.73 \begin {gather*} \frac {28 a x^6 (a+b) \sqrt [4]{1-\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 a x^3}{a x^3+b}\right )-4 \left (b-a x^3\right )^2 \sqrt [4]{\frac {a x^3}{b}+1}}{21 b^2 x^5 \sqrt [4]{\frac {a x^3}{b}+1} \sqrt [4]{a x^4-b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-4*(b - a*x^3)^2*(1 + (a*x^3)/b)^(1/4) + 28*a*(a + b)*x^6*(1 - (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5
/4, (2*a*x^3)/(b + a*x^3)])/(21*b^2*x^5*(1 + (a*x^3)/b)^(1/4)*(-(b*x) + a*x^4)^(1/4))

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IntegrateAlgebraic [A]  time = 0.94, size = 157, normalized size = 1.00 \begin {gather*} \frac {4 \left (b-a x^3\right ) \left (-b x+a x^4\right )^{3/4}}{21 b^2 x^6}+\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \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 b^2}+\frac {2^{3/4} \left (a^{7/4}+a^{3/4} b\right ) \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 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(b - a*x^3)*(-(b*x) + a*x^4)^(3/4))/(21*b^2*x^6) + (2^(3/4)*(a^(7/4) + a^(3/4)*b)*ArcTan[(2^(1/4)*a^(1/4)*(
-(b*x) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b^2) + (2^(3/4)*(a^(7/4) + a^(3/4)*b)*ArcTanh[(2^(1/4)*a^(1/4)*(-(b*x
) + a*x^4)^(3/4))/(-b + a*x^3)])/(3*b^2)

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

[Out]

Timed out

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giac [B]  time = 0.32, size = 279, normalized size = 1.78 \begin {gather*} \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 \, b^{2}} + \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 \, b^{2}} - \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 )} \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 )}{12 \, b^{2}} + \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 )} \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 )}{12 \, b^{2}} - \frac {4 \, {\left (a - \frac {b}{x^{3}}\right )}^{\frac {7}{4}}}{21 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))/b^2 + 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))/b^2 - 1/12*sqrt(2)*(2^(3/4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/
4)*b)*log(2^(3/4)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a - b/x^3))/b^2 + 1/12*sqrt(2)*(2^(3/
4)*(-a)^(3/4)*a + 2^(3/4)*(-a)^(3/4)*b)*log(-2^(3/4)*(-a)^(1/4)*(a - b/x^3)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a
- b/x^3))/b^2 - 4/21*(a - b/x^3)^(7/4)/b^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^6+b)/x^6/(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^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^6+b}{x^6\,{\left (a\,x^4-b\,x\right )}^{1/4}\,\left (a\,x^3+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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