\(\int \frac {\sqrt {-x+x^4} (-b+a x^6)}{x^6} \, dx\) [729]

Optimal result

Integrand size = 24, antiderivative size = 56 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {\sqrt {-x+x^4} \left (2 b-2 b x^3+3 a x^6\right )}{9 x^5}-\frac {1}{3} a \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \]

[Out]

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

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 3.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2073, 2077, 2039, 2045, 2036, 335, 231} \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=-\frac {3^{3/4} a (1-x) x \sqrt {\frac {x^2+x+1}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x}{1-\left (1+\sqrt {3}\right ) x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {-\frac {(1-x) x}{\left (1-\left (1+\sqrt {3}\right ) x\right )^2}} \sqrt {x^4-x}}+\frac {a \left (x^4-x\right )^{3/2}}{3 x^3}+\frac {2 a \sqrt {x^4-x}}{5 x^3}-\frac {2 b \left (x^4-x\right )^{3/2}}{9 x^6} \]

[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 231

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]/(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)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^
2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x]

Rule 335

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 2036

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 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rule 2045

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 2073

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 2077

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*} \text {integral}& = \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 ) \text {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}} \operatorname {EllipticF}\left (\arccos \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*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {\sqrt {x \left (-1+x^3\right )} \left (\sqrt {-1+x^3} \left (3 a x^6-2 b \left (-1+x^3\right )\right )-3 a x^{9/2} \log \left (x^{3/2}+\sqrt {-1+x^3}\right )\right )}{9 x^5 \sqrt {-1+x^3}} \]

[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)*Log[x^(3/2) + Sqrt[-1 + x^3]]))/(
9*x^5*Sqrt[-1 + x^3])

Maple [A] (verified)

Time = 3.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98

method	result	size
trager	\(\frac {\sqrt {x^{4}-x}\, \left (3 a \,x^{6}-2 b \,x^{3}+2 b \right )}{9 x^{5}}-\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{6}\)	\(55\)
risch	\(\frac {\left (x^{3}-1\right ) \left (3 a \,x^{6}-2 b \,x^{3}+2 b \right )}{9 x^{4} \sqrt {x \left (x^{3}-1\right )}}-\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{6}\)	\(60\)
pseudoelliptic	\(\frac {\left (6 a \,x^{6}-4 b \,x^{3}+4 b \right ) \sqrt {x^{4}-x}+3 \left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right ) x^{5} a}{18 x^{5}}\)	\(80\)
meijerg	\(\frac {i a \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {-x^{3}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}+\frac {2 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, b \left (-x^{3}+1\right )^{\frac {3}{2}}}{9 \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, x^{\frac {9}{2}}}\)	\(89\)
default	\(a \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}\right )-b \left (-\frac {2 \sqrt {x^{4}-x}}{9 x^{5}}+\frac {2 \sqrt {x^{4}-x}}{9 x^{2}}\right )\)	\(91\)
elliptic	\(\frac {2 b \sqrt {x^{4}-x}}{9 x^{5}}-\frac {2 b \sqrt {x^{4}-x}}{9 x^{2}}+\frac {a x \sqrt {x^{4}-x}}{3}-\frac {a \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(333\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {3 \, a x^{5} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + 2 \, {\left (3 \, a x^{6} - 2 \, b x^{3} + 2 \, b\right )} \sqrt {x^{4} - x}}{18 \, x^{5}} \]

[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

Sympy [F]

\[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{6} - b\right )}{x^{6}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\int { \frac {{\left (a x^{6} - b\right )} \sqrt {x^{4} - x}}{x^{6}} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {1}{3} \, \sqrt {x^{4} - x} a x - \frac {2}{9} \, b {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} - \frac {1}{6} \, a \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{6} \, a \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 6.77 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {-x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {2\,a\,x\,\sqrt {x^4-x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{2};\ \frac {3}{2};\ x^3\right )}{3\,\sqrt {1-x^3}}-\frac {2\,b\,\sqrt {x^4-x}\,\left (x^3-1\right )}{9\,x^5} \]

[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)