3.17.55 \(\int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx\)

Optimal. Leaf size=112 \[ \frac {2 \sqrt {b} \sqrt {a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {a-b}}{\sqrt {b} (x-1) \left (x^2+x+1\right )}\right )}{3 a^2}+\frac {(2 b-a) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )}{3 a^2}+\frac {\sqrt {x^4-x} x}{3 a} \]

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Rubi [A]  time = 0.26, antiderivative size = 149, normalized size of antiderivative = 1.33, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2042, 466, 465, 478, 523, 217, 206, 377, 205} \begin {gather*} \frac {2 \sqrt {b} \sqrt {x^4-x} \sqrt {a-b} \tan ^{-1}\left (\frac {x^{3/2} \sqrt {a-b}}{\sqrt {b} \sqrt {x^3-1}}\right )}{3 a^2 \sqrt {x^3-1} \sqrt {x}}-\frac {\sqrt {x^4-x} (a-2 b) \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 a^2 \sqrt {x^3-1} \sqrt {x}}+\frac {\sqrt {x^4-x} x}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 377

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

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 478

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

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 2042

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]
:> Dist[(e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a
+ b*x^n)^FracPart[p]), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,
p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx &=\frac {\sqrt {-x+x^4} \int \frac {x^{7/2} \sqrt {-1+x^3}}{-b+a x^3} \, dx}{\sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt {-1+x^6}}{-b+a x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-1+x^2}}{-b+a x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {x \sqrt {-x+x^4}}{3 a}-\frac {\sqrt {-x+x^4} \operatorname {Subst}\left (\int \frac {b+(a-2 b) x^2}{\sqrt {-1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {x \sqrt {-x+x^4}}{3 a}-\frac {\left ((a-2 b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) b \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {x \sqrt {-x+x^4}}{3 a}-\frac {\left ((a-2 b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) b \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-b-(a-b) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {x \sqrt {-x+x^4}}{3 a}+\frac {2 \sqrt {a-b} \sqrt {b} \sqrt {-x+x^4} \tan ^{-1}\left (\frac {\sqrt {a-b} x^{3/2}}{\sqrt {b} \sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}-\frac {(a-2 b) \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.54, size = 136, normalized size = 1.21 \begin {gather*} \frac {x^2 \left (\frac {\sqrt {1-x^3} x^3 (a-2 b) F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};x^3,\frac {a x^3}{b}\right )}{b}+\frac {3 \sqrt {1-x^3} \sin ^{-1}\left (\frac {\sqrt {\frac {x^3 (b-a)}{b}}}{\sqrt {1-\frac {a x^3}{b}}}\right )}{\sqrt {\frac {x^3 (b-a)}{b}}}+3 \left (x^3-1\right )\right )}{9 a \sqrt {x \left (x^3-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^2*(3*(-1 + x^3) + ((a - 2*b)*x^3*Sqrt[1 - x^3]*AppellF1[3/2, 1/2, 1, 5/2, x^3, (a*x^3)/b])/b + (3*Sqrt[1 -
x^3]*ArcSin[Sqrt[((-a + b)*x^3)/b]/Sqrt[1 - (a*x^3)/b]])/Sqrt[((-a + b)*x^3)/b]))/(9*a*Sqrt[x*(-1 + x^3)])

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IntegrateAlgebraic [A]  time = 0.62, size = 112, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {-x+x^4}}{3 a}+\frac {2 \sqrt {a-b} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 a^2}+\frac {(-a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.78, size = 247, normalized size = 2.21 \begin {gather*} \left [\frac {2 \, \sqrt {x^{4} - x} a x - {\left (a - 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right ) + \sqrt {-a b + b^{2}} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} x^{6} + 2 \, {\left (3 \, a b - 4 \, b^{2}\right )} x^{3} + 4 \, {\left ({\left (a - 2 \, b\right )} x^{4} + b x\right )} \sqrt {x^{4} - x} \sqrt {-a b + b^{2}} + b^{2}}{a^{2} x^{6} - 2 \, a b x^{3} + b^{2}}\right )}{6 \, a^{2}}, \frac {2 \, \sqrt {x^{4} - x} a x - {\left (a - 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right ) + 2 \, \sqrt {a b - b^{2}} \arctan \left (-\frac {2 \, \sqrt {x^{4} - x} \sqrt {a b - b^{2}} x}{{\left (a - 2 \, b\right )} x^{3} + b}\right )}{6 \, a^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4-x)^(1/2)/(a*x^3-b),x, algorithm="fricas")

[Out]

[1/6*(2*sqrt(x^4 - x)*a*x - (a - 2*b)*log(-2*x^3 - 2*sqrt(x^4 - x)*x + 1) + sqrt(-a*b + b^2)*log(-((a^2 - 8*a*
b + 8*b^2)*x^6 + 2*(3*a*b - 4*b^2)*x^3 + 4*((a - 2*b)*x^4 + b*x)*sqrt(x^4 - x)*sqrt(-a*b + b^2) + b^2)/(a^2*x^
6 - 2*a*b*x^3 + b^2)))/a^2, 1/6*(2*sqrt(x^4 - x)*a*x - (a - 2*b)*log(-2*x^3 - 2*sqrt(x^4 - x)*x + 1) + 2*sqrt(
a*b - b^2)*arctan(-2*sqrt(x^4 - x)*sqrt(a*b - b^2)*x/((a - 2*b)*x^3 + b)))/a^2]

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giac [A]  time = 0.56, size = 100, normalized size = 0.89 \begin {gather*} \frac {\sqrt {x^{4} - x} x}{3 \, a} - \frac {{\left (a - 2 \, b\right )} \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a^{2}} + \frac {{\left (a - 2 \, b\right )} \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a^{2}} - \frac {2 \, \sqrt {a b - b^{2}} \arctan \left (\frac {b \sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {a b - b^{2}}}\right )}{3 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.35, size = 665, normalized size = 5.94

method	result	size
elliptic	\(\frac {x \sqrt {x^{4}-x}}{3 a}+\frac {2 \left (-\frac {a -b}{a^{2}}+\frac {1}{2 a}\right ) \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 )}}-\frac {2 b \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 a^{2}}\)	\(665\)
risch	\(\frac {x^{2} \left (x^{3}-1\right )}{3 a \sqrt {x \left (x^{3}-1\right )}}-\frac {\frac {2 \left (a -2 b \right ) \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 )}{a \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 )}}+\frac {4 b \left (a -b \right ) \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (a -b \right ) \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 a}}{2 a}\)	\(682\)
default	\(\frac {\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 )}}}{a}+\frac {b \left (\frac {2 \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 )}{a \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 )}}-\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 a}\right )}{a}\)	\(946\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^4-x)^(1/2)/(a*x^3-b),x,method=_RETURNVERBOSE)

[Out]

1/3*x*(x^4-x)^(1/2)/a+2*(-(a-b)/a^2+1/2/a)*(1/2-1/2*I*3^(1/2))*((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-
1+x))^(1/2)*(-1+x)^2*((x+1/2+1/2*I*3^(1/2))/(-1/2-1/2*I*3^(1/2))/(-1+x))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(-1/2+1/
2*I*3^(1/2))/(-1+x))^(1/2)/(-3/2+1/2*I*3^(1/2))/(x*(-1+x)*(x+1/2+1/2*I*3^(1/2))*(x+1/2-1/2*I*3^(1/2)))^(1/2)*(
EllipticF(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/
(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-EllipticPi(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x)
)^(1/2),(-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))
/(3/2-1/2*I*3^(1/2)))^(1/2)))-2/3/a^2*b*4^(1/2)*sum(1/_alpha*(-1+x)^2*(_alpha^2+_alpha+1)*(1-I*3^(1/2))*(x/(-1
+x)*(-3+I*3^(1/2))/(-1+I*3^(1/2)))^(1/2)*(1/(-1+x)*(I*3^(1/2)+2*x+1)/(-1-I*3^(1/2)))^(1/2)*(1/(-1+x)*(1+2*x-I*
3^(1/2))/(-1+I*3^(1/2)))^(1/2)/(-3+I*3^(1/2))/(x*(-1+x)*(I*3^(1/2)+2*x+1)*(1+2*x-I*3^(1/2)))^(1/2)*(EllipticF(
((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I
*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-_alpha^2*a/b*EllipticPi(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1
+x))^(1/2),1/6*(I*_alpha^2*3^(1/2)*a-3*_alpha^2*a-I*3^(1/2)*b+3*b)/b,((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/
(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))),_alpha=RootOf(_Z^3*a-b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} - x} x^{3}}{a x^{3} - b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x^4 - x)*x^3/(a*x^3 - b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^3\,\sqrt {x^4-x}}{b-a\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{a x^{3} - b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(x*(x - 1)*(x**2 + x + 1))/(a*x**3 - b), x)

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