3.8.29 \(\int \frac {\sqrt {-x+x^4} (-b+a x^6)}{x^6} \, dx\)

Optimal. Leaf size=56 \[ \frac {\sqrt {x^4-x} \left (3 a x^6-2 b x^3+2 b\right )}{9 x^5}-\frac {1}{3} a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) \]

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Rubi [C]  time = 0.32, antiderivative size = 178, normalized size of antiderivative = 3.18, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2048, 2052, 2014, 2020, 2011, 329, 225} \begin {gather*} \frac {a \left (x^4-x\right )^{3/2}}{3 x^3}+\frac {2 a \sqrt {x^4-x}}{5 x^3}-\frac {3^{3/4} a (1-x) x \sqrt {\frac {x^2+x+1}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x^4-x}}-\frac {2 b \left (x^4-x\right )^{3/2}}{9 x^6} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a*Sqrt[-x + x^4])/(5*x^3) - (2*b*(-x + x^4)^(3/2))/(9*x^6) + (a*(-x + x^4)^(3/2))/(3*x^3) - (3^(3/4)*a*(1 -
x)*x*Sqrt[(1 + x + x^2)/(1 - (1 + Sqrt[3])*x)^2]*EllipticF[ArcCos[(1 - (1 - Sqrt[3])*x)/(1 - (1 + Sqrt[3])*x)
], (2 + Sqrt[3])/4])/(5*Sqrt[-(((1 - x)*x)/(1 - (1 + Sqrt[3])*x)^2)]*Sqrt[-x + x^4])

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(x*(s
+ r*x^2)*Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2
)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4])/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[(r*x^2*(s + r*x^2))/(s + (1
+ Sqrt[3])*r*x^2)^2]), x]] /; FreeQ[{a, b}, x]

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 2011

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2014

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

Rule 2020

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b*
x^n)^p)/(c*(m + j*p + 1)), x] - Dist[(b*p*(n - j))/(c^n*(m + j*p + 1)), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2048

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]},
With[{Pqq = Coeff[Pq, x, q]}, Int[(c*x)^m*ExpandToSum[Pq - Pqq*x^q - (a*Pqq*(m + q - n + 1)*x^(q - n))/(b*(m +
q + n*p + 1)), x]*(a*x^j + b*x^n)^p, x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a*x^j + b*x^n)^(p + 1))/(b*c^(q -
n + 1)*(m + q + n*p + 1)), x]] /; GtQ[q, n - 1] && NeQ[m + q + n*p + 1, 0] && (IntegerQ[2*p] || IntegerQ[p + (
q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] &&  !IntegerQ[p] && IGtQ[j, 0] && IGtQ[n, 0] && L
tQ[j, n]

Rule 2052

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx &=\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}+\int \frac {\left (-b-a x^2\right ) \sqrt {-x+x^4}}{x^6} \, dx\\ &=\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}+\int \left (-\frac {b \sqrt {-x+x^4}}{x^6}-\frac {a \sqrt {-x+x^4}}{x^4}\right ) \, dx\\ &=\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}-a \int \frac {\sqrt {-x+x^4}}{x^4} \, dx-b \int \frac {\sqrt {-x+x^4}}{x^6} \, dx\\ &=\frac {2 a \sqrt {-x+x^4}}{5 x^3}-\frac {2 b \left (-x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}-\frac {1}{5} (3 a) \int \frac {1}{\sqrt {-x+x^4}} \, dx\\ &=\frac {2 a \sqrt {-x+x^4}}{5 x^3}-\frac {2 b \left (-x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}-\frac {\left (3 a \sqrt {x} \sqrt {-1+x^3}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^3}} \, dx}{5 \sqrt {-x+x^4}}\\ &=\frac {2 a \sqrt {-x+x^4}}{5 x^3}-\frac {2 b \left (-x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}-\frac {\left (6 a \sqrt {x} \sqrt {-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^6}} \, dx,x,\sqrt {x}\right )}{5 \sqrt {-x+x^4}}\\ &=\frac {2 a \sqrt {-x+x^4}}{5 x^3}-\frac {2 b \left (-x+x^4\right )^{3/2}}{9 x^6}+\frac {a \left (-x+x^4\right )^{3/2}}{3 x^3}-\frac {3^{3/4} a (1-x) x \sqrt {\frac {1+x+x^2}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {-x+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 71, normalized size = 1.27 \begin {gather*} \frac {\sqrt {x \left (x^3-1\right )} \left (\sqrt {1-x^3} \left (3 a x^6-2 b \left (x^3-1\right )\right )+3 a x^{9/2} \sin ^{-1}\left (x^{3/2}\right )\right )}{9 x^5 \sqrt {1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(-1 + x^3)]*(Sqrt[1 - x^3]*(3*a*x^6 - 2*b*(-1 + x^3)) + 3*a*x^(9/2)*ArcSin[x^(3/2)]))/(9*x^5*Sqrt[1 -
x^3])

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IntegrateAlgebraic [A]  time = 0.48, size = 56, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-x+x^4} \left (2 b-2 b x^3+3 a x^6\right )}{9 x^5}-\frac {1}{3} a \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-x + x^4]*(2*b - 2*b*x^3 + 3*a*x^6))/(9*x^5) - (a*ArcTanh[x^2/Sqrt[-x + x^4]])/3

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fricas [A]  time = 0.69, size = 59, normalized size = 1.05 \begin {gather*} \frac {3 \, a x^{5} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + 2 \, {\left (3 \, a x^{6} - 2 \, b x^{3} + 2 \, b\right )} \sqrt {x^{4} - x}}{18 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="fricas")

[Out]

1/18*(3*a*x^5*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1) + 2*(3*a*x^6 - 2*b*x^3 + 2*b)*sqrt(x^4 - x))/x^5

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giac [A]  time = 0.38, size = 57, normalized size = 1.02 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} - x} a x - \frac {2}{9} \, b {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} - \frac {1}{6} \, a \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{6} \, a \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.32, size = 55, normalized size = 0.98

method	result	size
trager	\(\frac {\sqrt {x^{4}-x}\, \left (3 a \,x^{6}-2 b \,x^{3}+2 b \right )}{9 x^{5}}-\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{6}\)	\(55\)
meijerg	\(\frac {i a \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {-x^{3}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}}+\frac {2 \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, b \left (-x^{3}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\)	\(89\)
risch	\(\frac {\left (x^{3}-1\right ) \left (3 a \,x^{6}-2 b \,x^{3}+2 b \right )}{9 x^{4} \sqrt {x \left (x^{3}-1\right )}}-\frac {a \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(325\)
elliptic	\(\frac {2 b \sqrt {x^{4}-x}}{9 x^{5}}-\frac {2 b \sqrt {x^{4}-x}}{9 x^{2}}+\frac {a x \sqrt {x^{4}-x}}{3}-\frac {a \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(333\)
default	\(a \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-b \left (-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\right )\)	\(336\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x)^(1/2)*(a*x^6-b)/x^6,x,method=_RETURNVERBOSE)

[Out]

1/9*(x^4-x)^(1/2)*(3*a*x^6-2*b*x^3+2*b)/x^5-1/6*a*ln(-2*x^3-2*x*(x^4-x)^(1/2)+1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="maxima")

[Out]

integrate((a*x^6 - b)*sqrt(x^4 - x)/x^6, x)

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mupad [B]  time = 1.05, size = 51, normalized size = 0.91 \begin {gather*} \frac {2\,a\,x\,\sqrt {x^4-x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{2};\ \frac {3}{2};\ x^3\right )}{3\,\sqrt {1-x^3}}-\frac {2\,b\,\sqrt {x^4-x}\,\left (x^3-1\right )}{9\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 - x)^(1/2)*(b - a*x^6))/x^6,x)

[Out]

(2*a*x*(x^4 - x)^(1/2)*hypergeom([-1/2, 1/2], 3/2, x^3))/(3*(1 - x^3)^(1/2)) - (2*b*(x^4 - x)^(1/2)*(x^3 - 1))
/(9*x^5)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{6} - b\right )}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x)**(1/2)*(a*x**6-b)/x**6,x)

[Out]

Integral(sqrt(x*(x - 1)*(x**2 + x + 1))*(a*x**6 - b)/x**6, x)

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