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

Optimal. Leaf size=241 \[ \frac {1287 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{1024 b^{15/2}}-\frac {1287 a^6 \sqrt {a x+b x^{2/3}}}{1024 b^7 x^{2/3}}+\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{512 b^6 x}-\frac {429 a^4 \sqrt {a x+b x^{2/3}}}{640 b^5 x^{4/3}}+\frac {1287 a^3 \sqrt {a x+b x^{2/3}}}{2240 b^4 x^{5/3}}-\frac {143 a^2 \sqrt {a x+b x^{2/3}}}{280 b^3 x^2}+\frac {13 a \sqrt {a x+b x^{2/3}}}{28 b^2 x^{7/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}} \]

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Rubi [A]  time = 0.41, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2025, 2029, 206} \begin {gather*} -\frac {1287 a^6 \sqrt {a x+b x^{2/3}}}{1024 b^7 x^{2/3}}+\frac {429 a^5 \sqrt {a x+b x^{2/3}}}{512 b^6 x}-\frac {429 a^4 \sqrt {a x+b x^{2/3}}}{640 b^5 x^{4/3}}+\frac {1287 a^3 \sqrt {a x+b x^{2/3}}}{2240 b^4 x^{5/3}}-\frac {143 a^2 \sqrt {a x+b x^{2/3}}}{280 b^3 x^2}+\frac {1287 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{1024 b^{15/2}}+\frac {13 a \sqrt {a x+b x^{2/3}}}{28 b^2 x^{7/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(7*b*x^(8/3)) + (13*a*Sqrt[b*x^(2/3) + a*x])/(28*b^2*x^(7/3)) - (143*a^2*Sqrt[b*x^(
2/3) + a*x])/(280*b^3*x^2) + (1287*a^3*Sqrt[b*x^(2/3) + a*x])/(2240*b^4*x^(5/3)) - (429*a^4*Sqrt[b*x^(2/3) + a
*x])/(640*b^5*x^(4/3)) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^6*x) - (1287*a^6*Sqrt[b*x^(2/3) + a*x])/(1024*
b^7*x^(2/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))

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 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]

Rule 2029

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*} \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}-\frac {(13 a) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{14 b}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}+\frac {\left (143 a^2\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{168 b^2}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}-\frac {\left (429 a^3\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{560 b^3}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}+\frac {\left (429 a^4\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{640 b^4}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}-\frac {\left (143 a^5\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{256 b^5}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}+\frac {\left (429 a^6\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{1024 b^6}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}-\frac {\left (429 a^7\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{2048 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac {\left (1287 a^7\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^7}\\ &=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac {1287 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^{15/2}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 48, normalized size = 0.20 \begin {gather*} \frac {6 a^7 \sqrt {a x+b x^{2/3}} \, _2F_1\left (\frac {1}{2},8;\frac {3}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^8 \sqrt [3]{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a^7*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[1/2, 8, 3/2, 1 + (a*x^(1/3))/b])/(b^8*x^(1/3))

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IntegrateAlgebraic [A]  time = 0.22, size = 138, normalized size = 0.57 \begin {gather*} \frac {1287 a^7 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{1024 b^{15/2}}+\frac {\sqrt {a x+b x^{2/3}} \left (-45045 a^6 x^2+30030 a^5 b x^{5/3}-24024 a^4 b^2 x^{4/3}+20592 a^3 b^3 x-18304 a^2 b^4 x^{2/3}+16640 a b^5 \sqrt [3]{x}-15360 b^6\right )}{35840 b^7 x^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-15360*b^6 + 16640*a*b^5*x^(1/3) - 18304*a^2*b^4*x^(2/3) + 20592*a^3*b^3*x - 24024*a^4
*b^2*x^(4/3) + 30030*a^5*b*x^(5/3) - 45045*a^6*x^2))/(35840*b^7*x^(8/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))
/Sqrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.31, size = 160, normalized size = 0.66 \begin {gather*} -\frac {\frac {45045 \, a^{8} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{8} - 300300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{8} b + 849849 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{8} b^{2} - 1317888 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{8} b^{3} + 1200199 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{8} b^{4} - 631540 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{8} b^{5} + 169995 \, \sqrt {a x^{\frac {1}{3}} + b} a^{8} b^{6}}{a^{7} b^{7} x^{\frac {7}{3}}}}{35840 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35840*(45045*a^8*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(13/2)*a^8 -
300300*(a*x^(1/3) + b)^(11/2)*a^8*b + 849849*(a*x^(1/3) + b)^(9/2)*a^8*b^2 - 1317888*(a*x^(1/3) + b)^(7/2)*a^8
*b^3 + 1200199*(a*x^(1/3) + b)^(5/2)*a^8*b^4 - 631540*(a*x^(1/3) + b)^(3/2)*a^8*b^5 + 169995*sqrt(a*x^(1/3) +
b)*a^8*b^6)/(a^7*b^7*x^(7/3)))/a

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maple [A]  time = 0.05, size = 188, normalized size = 0.78 \begin {gather*} \frac {\sqrt {a \,x^{\frac {1}{3}}+b}\, \left (45045 a^{7} b \,x^{\frac {13}{3}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-45045 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{6} b^{\frac {3}{2}} x^{4}+30030 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{5} b^{\frac {5}{2}} x^{\frac {11}{3}}-24024 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{4} b^{\frac {7}{2}} x^{\frac {10}{3}}+20592 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{3} b^{\frac {9}{2}} x^{3}-18304 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{2} b^{\frac {11}{2}} x^{\frac {8}{3}}+16640 \sqrt {a \,x^{\frac {1}{3}}+b}\, a \,b^{\frac {13}{2}} x^{\frac {7}{3}}-15360 \sqrt {a \,x^{\frac {1}{3}}+b}\, b^{\frac {15}{2}} x^{2}\right )}{35840 \sqrt {a x +b \,x^{\frac {2}{3}}}\, b^{\frac {17}{2}} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35840*(a*x^(1/3)+b)^(1/2)*(45045*x^(13/3)*arctanh((a*x^(1/3)+b)^(1/2)/b^(1/2))*a^7*b+30030*x^(11/3)*(a*x^(1/
3)+b)^(1/2)*b^(5/2)*a^5-24024*x^(10/3)*(a*x^(1/3)+b)^(1/2)*b^(7/2)*a^4-18304*x^(8/3)*(a*x^(1/3)+b)^(1/2)*b^(11
/2)*a^2+16640*x^(7/3)*(a*x^(1/3)+b)^(1/2)*b^(13/2)*a-15360*(a*x^(1/3)+b)^(1/2)*b^(15/2)*x^2+20592*x^3*(a*x^(1/
3)+b)^(1/2)*b^(9/2)*a^3-45045*x^4*(a*x^(1/3)+b)^(1/2)*b^(3/2)*a^6)/x^4/(a*x+b*x^(2/3))^(1/2)/b^(17/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^3\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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