\(\int \frac {x^4}{(a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\) [109]

Optimal result

Integrand size = 26, antiderivative size = 368 \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[Out]

7/243*x^2/a^3/b/((b*x^3+a)^2)^(1/2)-1/12*x^2/b/(b*x^3+a)^3/((b*x^3+a)^2)^(1/2)+1/54*x^2/a/b/(b*x^3+a)^2/((b*x^
3+a)^2)^(1/2)+7/324*x^2/a^2/b/(b*x^3+a)/((b*x^3+a)^2)^(1/2)-7/729*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)/b^(
5/3)/((b*x^3+a)^2)^(1/2)+7/1458*(b*x^3+a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/b^(5/3)/((b*x^3+a
)^2)^(1/2)-7/729*(b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(10/3)/b^(5/3)*3^(1/2)/((b*x^3+
a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1369, 294, 296, 298, 31, 648, 631, 210, 642} \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}} \]

[In]

Int[x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(7*x^2)/(243*a^3*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x^2/(12*b*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
+ x^2/(54*a*b*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (7*x^2)/(324*a^2*b*(a + b*x^3)*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6]) - (7*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(10/3)*b^
(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(10/3)*b^(5/3)*Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6]) + (7*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(10/3)*b^(
5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 294

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

Rule 296

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

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac {x^4}{\left (a b+b^2 x^3\right )^5} \, dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^4} \, dx}{6 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 b \left (a b+b^2 x^3\right )\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^3} \, dx}{54 a \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = -\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \int \frac {x}{\left (a b+b^2 x^3\right )^2} \, dx}{81 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \int \frac {x}{a b+b^2 x^3} \, dx}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{729 a^{10/3} b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \int \frac {\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{729 a^{10/3} b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \int \frac {-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{1458 a^{10/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \int \frac {1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{486 a^3 b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {\left (7 \left (a b+b^2 x^3\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{243 a^{10/3} b^{8/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ & = \frac {7 x^2}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x^2}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{54 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {7 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{10/3} b^{5/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.62 \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{10/3} b^{2/3} x^2+54 a^{7/3} b^{2/3} x^2 \left (a+b x^3\right )+63 a^{4/3} b^{2/3} x^2 \left (a+b x^3\right )^2+84 \sqrt [3]{a} b^{2/3} x^2 \left (a+b x^3\right )^3+28 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )-28 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+14 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{10/3} b^{5/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

[In]

Integrate[x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

((a + b*x^3)*(-243*a^(10/3)*b^(2/3)*x^2 + 54*a^(7/3)*b^(2/3)*x^2*(a + b*x^3) + 63*a^(4/3)*b^(2/3)*x^2*(a + b*x
^3)^2 + 84*a^(1/3)*b^(2/3)*x^2*(a + b*x^3)^3 + 28*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))] - 28*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 14*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2]))/(2916*a^(10/3)*b^(5/3)*((a + b*x^3)^2)^(5/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 3.70 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.30

method	result	size
risch	\(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {7 b^{2} x^{11}}{243 a^{3}}+\frac {35 b \,x^{8}}{324 a^{2}}+\frac {4 x^{5}}{27 a}-\frac {7 x^{2}}{486 b}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {7 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{729 \left (b \,x^{3}+a \right ) a^{3} b^{2}}\)	\(109\)
default	\(-\frac {\left (28 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{4} x^{12}+28 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}-14 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{4} x^{12}-84 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4} x^{11}+112 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a \,b^{3} x^{9}+112 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}-56 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a \,b^{3} x^{9}-315 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{3} x^{8}+168 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} b^{2} x^{6}+168 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}-84 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} b^{2} x^{6}-432 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b^{2} x^{5}+112 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{3} b \,x^{3}+112 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}-56 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{3} b \,x^{3}+42 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b \,x^{2}+28 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{4}+28 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}-14 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2} a^{3} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\)	\(521\)

[In]

int(x^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

((b*x^3+a)^2)^(1/2)/(b*x^3+a)^5*(7/243*b^2/a^3*x^11+35/324*b/a^2*x^8+4/27/a*x^5-7/486/b*x^2)+7/729*((b*x^3+a)^
2)^(1/2)/(b*x^3+a)/a^3/b^2*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.99 \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\left [\frac {84 \, a b^{5} x^{11} + 315 \, a^{2} b^{4} x^{8} + 432 \, a^{3} b^{3} x^{5} - 42 \, a^{4} b^{2} x^{2} + 42 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + 14 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 28 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{2916 \, {\left (a^{4} b^{7} x^{12} + 4 \, a^{5} b^{6} x^{9} + 6 \, a^{6} b^{5} x^{6} + 4 \, a^{7} b^{4} x^{3} + a^{8} b^{3}\right )}}, \frac {84 \, a b^{5} x^{11} + 315 \, a^{2} b^{4} x^{8} + 432 \, a^{3} b^{3} x^{5} - 42 \, a^{4} b^{2} x^{2} + 84 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 14 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 28 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{2916 \, {\left (a^{4} b^{7} x^{12} + 4 \, a^{5} b^{6} x^{9} + 6 \, a^{6} b^{5} x^{6} + 4 \, a^{7} b^{4} x^{3} + a^{8} b^{3}\right )}}\right ] \]

[In]

integrate(x^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/2916*(84*a*b^5*x^11 + 315*a^2*b^4*x^8 + 432*a^3*b^3*x^5 - 42*a^4*b^2*x^2 + 42*sqrt(1/3)*(a*b^5*x^12 + 4*a^2
*b^4*x^9 + 6*a^3*b^3*x^6 + 4*a^4*b^2*x^3 + a^5*b)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a
*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 14
*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x
+ (-a*b^2)^(2/3)) - 28*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(2/3)*log(b*x - (
-a*b^2)^(1/3)))/(a^4*b^7*x^12 + 4*a^5*b^6*x^9 + 6*a^6*b^5*x^6 + 4*a^7*b^4*x^3 + a^8*b^3), 1/2916*(84*a*b^5*x^1
1 + 315*a^2*b^4*x^8 + 432*a^3*b^3*x^5 - 42*a^4*b^2*x^2 + 84*sqrt(1/3)*(a*b^5*x^12 + 4*a^2*b^4*x^9 + 6*a^3*b^3*
x^6 + 4*a^4*b^2*x^3 + a^5*b)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^
(1/3)/a)/b) + 14*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a
*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 28*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(-a*b^2)^(
2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^4*b^7*x^12 + 4*a^5*b^6*x^9 + 6*a^6*b^5*x^6 + 4*a^7*b^4*x^3 + a^8*b^3)]

Sympy [F]

\[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^{4}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(x**4/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**4/((a + b*x**3)**2)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.53 \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {28 \, b^{3} x^{11} + 105 \, a b^{2} x^{8} + 144 \, a^{2} b x^{5} - 14 \, a^{3} x^{2}}{972 \, {\left (a^{3} b^{5} x^{12} + 4 \, a^{4} b^{4} x^{9} + 6 \, a^{5} b^{3} x^{6} + 4 \, a^{6} b^{2} x^{3} + a^{7} b\right )}} + \frac {7 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {7 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

[In]

integrate(x^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/972*(28*b^3*x^11 + 105*a*b^2*x^8 + 144*a^2*b*x^5 - 14*a^3*x^2)/(a^3*b^5*x^12 + 4*a^4*b^4*x^9 + 6*a^5*b^3*x^6
+ 4*a^6*b^2*x^3 + a^7*b) + 7/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b^2*(a/b)^(
1/3)) + 7/1458*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b^2*(a/b)^(1/3)) - 7/729*log(x + (a/b)^(1/3))/(a^3*
b^2*(a/b)^(1/3))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.56 \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {7 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {7 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {7 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{4} b^{3} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {28 \, b^{3} x^{11} + 105 \, a b^{2} x^{8} + 144 \, a^{2} b x^{5} - 14 \, a^{3} x^{2}}{972 \, {\left (b x^{3} + a\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} \]

[In]

integrate(x^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

-7/1458*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^3*b*sgn(b*x^3 + a)) - 7/729*(-a/b)^(2/3)*lo
g(abs(x - (-a/b)^(1/3)))/(a^4*b*sgn(b*x^3 + a)) - 7/729*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b
)^(1/3))/(-a/b)^(1/3))/(a^4*b^3*sgn(b*x^3 + a)) + 1/972*(28*b^3*x^11 + 105*a*b^2*x^8 + 144*a^2*b*x^5 - 14*a^3*
x^2)/((b*x^3 + a)^4*a^3*b*sgn(b*x^3 + a))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^4}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]

[In]

int(x^4/(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2),x)

[Out]

int(x^4/(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2), x)