\(\int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx\) [193]

Optimal result

Integrand size = 19, antiderivative size = 329 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac {138567 a^{10} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{21/2}} \]

[Out]

-138567/131072*a^10*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(21/2)-3/10*(b*x^(2/3)+a*x)^(1/2)/b/x^(11
/3)+19/60*a*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(10/3)-323/960*a^2*(b*x^(2/3)+a*x)^(1/2)/b^3/x^3+323/896*a^3*(b*x^(2/3
)+a*x)^(1/2)/b^4/x^(8/3)-4199/10752*a^4*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(7/3)+46189/107520*a^5*(b*x^(2/3)+a*x)^(1/
2)/b^6/x^2-138567/286720*a^6*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(5/3)+46189/81920*a^7*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(4/
3)-46189/65536*a^8*(b*x^(2/3)+a*x)^(1/2)/b^9/x+138567/131072*a^9*(b*x^(2/3)+a*x)^(1/2)/b^10/x^(2/3)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2050, 2054, 212} \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {138567 a^{10} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{131072 b^{21/2}}+\frac {138567 a^9 \sqrt {a x+b x^{2/3}}}{131072 b^{10} x^{2/3}}-\frac {46189 a^8 \sqrt {a x+b x^{2/3}}}{65536 b^9 x}+\frac {46189 a^7 \sqrt {a x+b x^{2/3}}}{81920 b^8 x^{4/3}}-\frac {138567 a^6 \sqrt {a x+b x^{2/3}}}{286720 b^7 x^{5/3}}+\frac {46189 a^5 \sqrt {a x+b x^{2/3}}}{107520 b^6 x^2}-\frac {4199 a^4 \sqrt {a x+b x^{2/3}}}{10752 b^5 x^{7/3}}+\frac {323 a^3 \sqrt {a x+b x^{2/3}}}{896 b^4 x^{8/3}}-\frac {323 a^2 \sqrt {a x+b x^{2/3}}}{960 b^3 x^3}+\frac {19 a \sqrt {a x+b x^{2/3}}}{60 b^2 x^{10/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}} \]

[In]

Int[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(10*b*x^(11/3)) + (19*a*Sqrt[b*x^(2/3) + a*x])/(60*b^2*x^(10/3)) - (323*a^2*Sqrt[b*
x^(2/3) + a*x])/(960*b^3*x^3) + (323*a^3*Sqrt[b*x^(2/3) + a*x])/(896*b^4*x^(8/3)) - (4199*a^4*Sqrt[b*x^(2/3) +
a*x])/(10752*b^5*x^(7/3)) + (46189*a^5*Sqrt[b*x^(2/3) + a*x])/(107520*b^6*x^2) - (138567*a^6*Sqrt[b*x^(2/3) +
a*x])/(286720*b^7*x^(5/3)) + (46189*a^7*Sqrt[b*x^(2/3) + a*x])/(81920*b^8*x^(4/3)) - (46189*a^8*Sqrt[b*x^(2/3
) + a*x])/(65536*b^9*x) + (138567*a^9*Sqrt[b*x^(2/3) + a*x])/(131072*b^10*x^(2/3)) - (138567*a^10*ArcTanh[(Sqr
t[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(131072*b^(21/2))

Rule 212

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

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]

Rule 2054

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

Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}-\frac {(19 a) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{20 b} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}+\frac {\left (323 a^2\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{360 b^2} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}-\frac {\left (323 a^3\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{384 b^3} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}+\frac {\left (4199 a^4\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{5376 b^4} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}-\frac {\left (46189 a^5\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{64512 b^5} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}+\frac {\left (46189 a^6\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{71680 b^6} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}-\frac {\left (46189 a^7\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{81920 b^7} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}+\frac {\left (46189 a^8\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{98304 b^8} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}-\frac {\left (46189 a^9\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{131072 b^9} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}+\frac {\left (46189 a^{10}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{262144 b^{10}} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac {\left (138567 a^{10}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{10}} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac {138567 a^{10} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{21/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-4128768 b^9+4358144 a b^8 \sqrt [3]{x}-4630528 a^2 b^7 x^{2/3}+4961280 a^3 b^6 x-5374720 a^4 b^5 x^{4/3}+5912192 a^5 b^4 x^{5/3}-6651216 a^6 b^3 x^2+7759752 a^7 b^2 x^{7/3}-9699690 a^8 b x^{8/3}[phone] a^9 x^3\right )}{13762560 b^{10} x^{11/3}}-\frac {138567 a^{10} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{21/2}} \]

[In]

Integrate[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-4128768*b^9 + 4358144*a*b^8*x^(1/3) - 4630528*a^2*b^7*x^(2/3) + 4961280*a^3*b^6*x - 5
374720*a^4*b^5*x^(4/3) + 5912192*a^5*b^4*x^(5/3) - 6651216*a^6*b^3*x^2 + 7759752*a^7*b^2*x^(7/3) - 9699690*a^8
*b*x^(8/3) + 14549535*a^9*x^3))/(13762560*b^10*x^(11/3)) - (138567*a^10*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/
3) + a*x]])/(131072*b^(21/2))

Maple [A] (verified)

Time = 10.48 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (4128768 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {21}{2}}-4358144 b^{\frac {19}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a \,x^{\frac {1}{3}}+4630528 b^{\frac {17}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{2} x^{\frac {2}{3}}-4961280 b^{\frac {15}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{3} x +5374720 b^{\frac {13}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{4} x^{\frac {4}{3}}-5912192 b^{\frac {11}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{5} x^{\frac {5}{3}}+6651216 b^{\frac {9}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{6} x^{2}-7759752 b^{\frac {7}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{7} x^{\frac {7}{3}}+9699690 b^{\frac {5}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{8} x^{\frac {8}{3}}-14549535 b^{\frac {3}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{9} x^{3}[phone] \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{10} b \,x^{\frac {10}{3}}\right )}{13762560 x^{3} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {23}{2}}}\)	\(243\)
default	\(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (14549535 x^{\frac {19}{3}} \operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{10} b +9699690 x^{\frac {17}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {5}{2}} a^{8}-7759752 x^{\frac {16}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {7}{2}} a^{7}-5912192 x^{\frac {14}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {11}{2}} a^{5}+5374720 x^{\frac {13}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {13}{2}} a^{4}+4630528 x^{\frac {11}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {17}{2}} a^{2}-4358144 x^{\frac {10}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {19}{2}} a +4128768 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {21}{2}} x^{3}-4961280 x^{4} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {15}{2}} a^{3}+6651216 x^{5} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {9}{2}} a^{6}-14549535 x^{6} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}} a^{9}\right )}{13762560 x^{6} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {23}{2}}}\)	\(248\)

[In]

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

[Out]

-1/13762560*(b+a*x^(1/3))^(1/2)*(4128768*(b+a*x^(1/3))^(1/2)*b^(21/2)-4358144*b^(19/2)*(b+a*x^(1/3))^(1/2)*a*x
^(1/3)+4630528*b^(17/2)*(b+a*x^(1/3))^(1/2)*a^2*x^(2/3)-4961280*b^(15/2)*(b+a*x^(1/3))^(1/2)*a^3*x+5374720*b^(
13/2)*(b+a*x^(1/3))^(1/2)*a^4*x^(4/3)-5912192*b^(11/2)*(b+a*x^(1/3))^(1/2)*a^5*x^(5/3)+6651216*b^(9/2)*(b+a*x^
(1/3))^(1/2)*a^6*x^2-7759752*b^(7/2)*(b+a*x^(1/3))^(1/2)*a^7*x^(7/3)+9699690*b^(5/2)*(b+a*x^(1/3))^(1/2)*a^8*x
^(8/3)-14549535*b^(3/2)*(b+a*x^(1/3))^(1/2)*a^9*x^3+14549535*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*a^10*b*x^(10
/3))/x^3/(b*x^(2/3)+a*x)^(1/2)/b^(23/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^{4} \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]

[In]

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

[Out]

Integral(1/(x**4*sqrt(a*x + b*x**(2/3))), x)

Maxima [F]

\[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x^{4}} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(a*x + b*x^(2/3))*x^4), x)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\frac {14549535 \, a^{11} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{11} - 140645505 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{11} b + 609140532 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{11} b^{2} - 1554721740 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{11} b^{3} + 2585198330 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{11} b^{4} - 2918514950 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{11} b^{5} + 2255541300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{11} b^{6} - 1168982220 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{11} b^{7} + 382331775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{11} b^{8} - 68025825 \, \sqrt {a x^{\frac {1}{3}} + b} a^{11} b^{9}}{a^{10} b^{10} x^{\frac {10}{3}}}}{13762560 \, a} \]

[In]

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

[Out]

1/13762560*(14549535*a^11*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) + (14549535*(a*x^(1/3) + b)^(19
/2)*a^11 - 140645505*(a*x^(1/3) + b)^(17/2)*a^11*b + 609140532*(a*x^(1/3) + b)^(15/2)*a^11*b^2 - 1554721740*(a
*x^(1/3) + b)^(13/2)*a^11*b^3 + 2585198330*(a*x^(1/3) + b)^(11/2)*a^11*b^4 - 2918514950*(a*x^(1/3) + b)^(9/2)*
a^11*b^5 + 2255541300*(a*x^(1/3) + b)^(7/2)*a^11*b^6 - 1168982220*(a*x^(1/3) + b)^(5/2)*a^11*b^7 + 382331775*(
a*x^(1/3) + b)^(3/2)*a^11*b^8 - 68025825*sqrt(a*x^(1/3) + b)*a^11*b^9)/(a^10*b^10*x^(10/3)))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^4\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \]

[In]

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

[Out]

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