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

Optimal. Leaf size=251 \[ -\frac {663 a^{19/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1463 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {1326 a^4 \sqrt {a x+b \sqrt [3]{x}}}{1463 b^5 x^{2/3}}+\frac {3978 a^3 \sqrt {a x+b \sqrt [3]{x}}}{7315 b^4 x^{4/3}}-\frac {442 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1045 b^3 x^2}+\frac {34 a \sqrt {a x+b \sqrt [3]{x}}}{95 b^2 x^{8/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}} \]

[Out]

-6/19*(b*x^(1/3)+a*x)^(1/2)/b/x^(10/3)+34/95*a*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(8/3)-442/1045*a^2*(b*x^(1/3)+a*x)^
(1/2)/b^3/x^2+3978/7315*a^3*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(4/3)-1326/1463*a^4*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)-
663/1463*a^(19/4)*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)

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Rubi [A]  time = 0.35, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 2025, 2011, 329, 220} \[ -\frac {1326 a^4 \sqrt {a x+b \sqrt [3]{x}}}{1463 b^5 x^{2/3}}+\frac {3978 a^3 \sqrt {a x+b \sqrt [3]{x}}}{7315 b^4 x^{4/3}}-\frac {442 a^2 \sqrt {a x+b \sqrt [3]{x}}}{1045 b^3 x^2}-\frac {663 a^{19/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{1463 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {34 a \sqrt {a x+b \sqrt [3]{x}}}{95 b^2 x^{8/3}}-\frac {6 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[b*x^(1/3) + a*x]),x]

[Out]

(-6*Sqrt[b*x^(1/3) + a*x])/(19*b*x^(10/3)) + (34*a*Sqrt[b*x^(1/3) + a*x])/(95*b^2*x^(8/3)) - (442*a^2*Sqrt[b*x
^(1/3) + a*x])/(1045*b^3*x^2) + (3978*a^3*Sqrt[b*x^(1/3) + a*x])/(7315*b^4*x^(4/3)) - (1326*a^4*Sqrt[b*x^(1/3)
+ a*x])/(1463*b^5*x^(2/3)) - (663*a^(19/4)*(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])/(1463*b^(21/4)*Sqrt[b*x^(1/3) + a*x
])

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 329

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 2011

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 2018

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 2025

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*} \int \frac {1}{x^4 \sqrt {b \sqrt [3]{x}+a x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^{10} \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}-\frac {(51 a) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{19 b}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}+\frac {\left (221 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{95 b^2}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac {442 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1045 b^3 x^2}-\frac {\left (1989 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1045 b^3}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac {442 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac {3978 a^3 \sqrt {b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}+\frac {\left (1989 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1463 b^4}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac {442 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac {3978 a^3 \sqrt {b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac {1326 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac {\left (663 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1463 b^5}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac {442 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac {3978 a^3 \sqrt {b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac {1326 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac {\left (663 a^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{1463 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac {442 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac {3978 a^3 \sqrt {b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac {1326 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac {\left (1326 a^5 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{1463 b^5 \sqrt {b \sqrt [3]{x}+a x}}\\ &=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{19 b x^{10/3}}+\frac {34 a \sqrt {b \sqrt [3]{x}+a x}}{95 b^2 x^{8/3}}-\frac {442 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1045 b^3 x^2}+\frac {3978 a^3 \sqrt {b \sqrt [3]{x}+a x}}{7315 b^4 x^{4/3}}-\frac {1326 a^4 \sqrt {b \sqrt [3]{x}+a x}}{1463 b^5 x^{2/3}}-\frac {663 a^{19/4} \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 )}{1463 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 59, normalized size = 0.24 \[ -\frac {6 \sqrt {\frac {a x^{2/3}}{b}+1} \, _2F_1\left (-\frac {19}{4},\frac {1}{2};-\frac {15}{4};-\frac {a x^{2/3}}{b}\right )}{19 x^3 \sqrt {a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-19/4, 1/2, -15/4, -((a*x^(2/3))/b)])/(19*x^3*Sqrt[b*x^(1/3) + a
*x])

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fricas [F]  time = 1.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} x^{2} - a b x^{\frac {4}{3}} + b^{2} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {1}{3}}}}{a^{3} x^{7} + b^{3} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*x^2 - a*b*x^(4/3) + b^2*x^(2/3))*sqrt(a*x + b*x^(1/3))/(a^3*x^7 + b^3*x^5), x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(x)]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time: 8.266*((((
-8108100*b^7/154053900/b^8/x^(1/3)/x^(1/3)+9189180*b^6*a/154053900/b^8)/x^(1/3)/x^(1/3)-10859940*b^5*a^2/15405
3900/b^8)/x^(1/3)/x^(1/3)[phone]*b^4*a^3/154053900/b^8)/x^(1/3)/x^(1/3)-23271300*b^3*a^4/154053900/b^8)*sqrt
(a/x^(1/3)+b/x)+integrate(-69813900*b^3*a^5/154053900/b^8/3/((x^(1/6))^5*sqrt(a*(x^(1/3))^2+b)*sign(x)),x)

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maple [A]  time = 0.10, size = 179, normalized size = 0.71 \[ -\frac {6630 a^{5} x^{\frac {17}{3}}+3315 \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {a \,x^{\frac {1}{3}}}{\sqrt {-a b}}}\, a^{4} x^{\frac {16}{3}} \EllipticF \left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+2652 a^{4} b \,x^{5}-884 a^{3} b^{2} x^{\frac {13}{3}}+476 a^{2} b^{3} x^{\frac {11}{3}}-308 a \,b^{4} x^{3}+2310 b^{5} x^{\frac {7}{3}}}{7315 \sqrt {\left (a \,x^{\frac {2}{3}}+b \right ) x^{\frac {1}{3}}}\, b^{5} x^{\frac {16}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7315*(3315*a^4*(-a*b)^(1/2)*((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(a*x^(1/3)-(-a*b)^(1/2))/(-a*
b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*a*x^(1/3))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*
2^(1/2))*x^(16/3)+2652*a^4*b*x^5+6630*x^(17/3)*a^5+476*x^(11/3)*a^2*b^3-884*x^(13/3)*a^3*b^2-308*a*b^4*x^3+231
0*x^(7/3)*b^5)/b^5/((a*x^(2/3)+b)*x^(1/3))^(1/2)/x^(16/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,\sqrt {a\,x+b\,x^{1/3}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \sqrt {a x + b \sqrt [3]{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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