\(\int \frac {1}{x^3 (b \sqrt [3]{x}+a x)^{3/2}} \, dx\) [165]

Optimal result

Integrand size = 19, antiderivative size = 246 \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {663 a^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{154 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]

[Out]

3/b/x^(7/3)/(b*x^(1/3)+a*x)^(1/2)-17/5*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(8/3)+221/55*a*(b*x^(1/3)+a*x)^(1/2)/b^3/x^
2-1989/385*a^2*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(4/3)+663/77*a^3*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)+663/154*a^(15/4)
*x^(1/6)*(cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))^2)^(1/2)/cos(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)))*EllipticF(sin
(2*arctan(a^(1/4)*x^(1/6)/b^(1/4))),1/2*2^(1/2))*(x^(1/3)*a^(1/2)+b^(1/2))*((b+a*x^(2/3))/(x^(1/3)*a^(1/2)+b^(
1/2))^2)^(1/2)/b^(21/4)/(b*x^(1/3)+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2048, 2050, 2036, 335, 226} \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {663 a^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{154 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {663 a^3 \sqrt {a x+b \sqrt [3]{x}}}{77 b^5 x^{2/3}}-\frac {1989 a^2 \sqrt {a x+b \sqrt [3]{x}}}{385 b^4 x^{4/3}}+\frac {221 a \sqrt {a x+b \sqrt [3]{x}}}{55 b^3 x^2}-\frac {17 \sqrt {a x+b \sqrt [3]{x}}}{5 b^2 x^{8/3}}+\frac {3}{b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}} \]

[In]

Int[1/(x^3*(b*x^(1/3) + a*x)^(3/2)),x]

[Out]

3/(b*x^(7/3)*Sqrt[b*x^(1/3) + a*x]) - (17*Sqrt[b*x^(1/3) + a*x])/(5*b^2*x^(8/3)) + (221*a*Sqrt[b*x^(1/3) + a*x
])/(55*b^3*x^2) - (1989*a^2*Sqrt[b*x^(1/3) + a*x])/(385*b^4*x^(4/3)) + (663*a^3*Sqrt[b*x^(1/3) + a*x])/(77*b^5
*x^(2/3)) + (663*a^(15/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1
/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(154*b^(21/4)*Sqrt[b*x^(1/3) + a*x])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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 2043

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)
/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

Rule 2048

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] + Dist[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))
, Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n]
&& (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]

Rule 2050

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*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {1}{x^7 \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {51 \text {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}-\frac {(221 a) \text {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^2} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}+\frac {\left (1989 a^2\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 b^3} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}-\frac {\left (1989 a^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 b^4} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 b^5} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{154 b^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{77 b^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {663 a^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {15}{4},\frac {3}{2},-\frac {11}{4},-\frac {a x^{2/3}}{b}\right )}{5 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}} \]

[In]

Integrate[1/(x^3*(b*x^(1/3) + a*x)^(3/2)),x]

[Out]

(-2*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-15/4, 3/2, -11/4, -((a*x^(2/3))/b)])/(5*b*x^(7/3)*Sqrt[b*x^(1/3
) + a*x])

Maple [A] (verified)

Time = 5.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{5 b^{2} x^{\frac {8}{3}}}+\frac {56 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{55 b^{3} x^{2}}-\frac {834 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 b^{4} x^{\frac {4}{3}}}+\frac {432 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 b^{5} x^{\frac {2}{3}}}+\frac {3 x^{\frac {1}{3}} a^{4}}{b^{5} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {663 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{154 b^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(231\)
default	\(\frac {3315 x^{\frac {14}{3}} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3}-884 x^{\frac {11}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{2}+2652 x^{\frac {13}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b +476 x^{3} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{3}+2310 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{5} a^{4}+4320 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{4} x^{5}-308 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{4}}{770 b^{5} x^{5} \left (b +a \,x^{\frac {2}{3}}\right )}\)	\(261\)

[In]

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

[Out]

-2/5*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(8/3)+56/55*a*(b*x^(1/3)+a*x)^(1/2)/b^3/x^2-834/385*a^2*(b*x^(1/3)+a*x)^(1/2)
/b^4/x^(4/3)+432/77*a^3*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)+3*x^(1/3)*a^4/b^5/((x^(2/3)+b/a)*x^(1/3)*a)^(1/2)+66
3/154*a^3/b^5*(-a*b)^(1/2)*((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)-1/a*(-a*b)^(1/2))*a/
(-a*b)^(1/2))^(1/2)*(-x^(1/3)*a/(-a*b)^(1/2))^(1/2)/(b*x^(1/3)+a*x)^(1/2)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2)
)*a/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))

Fricas [F]

\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

integral((a^4*x^3 + 3*a^2*b^2*x^(5/3) - 2*a*b^3*x - (2*a^3*b*x^2 - b^4)*x^(1/3))*sqrt(a*x + b*x^(1/3))/(a^6*x^
8 + 2*a^3*b^3*x^6 + b^6*x^4), x)

Sympy [F]

\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

integrate(1/((a*x + b*x^(1/3))^(3/2)*x^3), x)

Giac [F]

\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]

[In]

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

[Out]

integrate(1/((a*x + b*x^(1/3))^(3/2)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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