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

Optimal. Leaf size=236 \[ -\frac {9009 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac {9009 a^5 \sqrt {a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac {3003 a^4 \sqrt {a x+b x^{2/3}}}{256 b^6 x}+\frac {3003 a^3 \sqrt {a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac {1287 a^2 \sqrt {a x+b x^{2/3}}}{160 b^4 x^{5/3}}+\frac {143 a \sqrt {a x+b x^{2/3}}}{20 b^3 x^2}-\frac {13 \sqrt {a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}} \]

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Rubi [A]  time = 0.41, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2023, 2025, 2029, 206} \begin {gather*} \frac {9009 a^5 \sqrt {a x+b x^{2/3}}}{512 b^7 x^{2/3}}-\frac {3003 a^4 \sqrt {a x+b x^{2/3}}}{256 b^6 x}+\frac {3003 a^3 \sqrt {a x+b x^{2/3}}}{320 b^5 x^{4/3}}-\frac {1287 a^2 \sqrt {a x+b x^{2/3}}}{160 b^4 x^{5/3}}-\frac {9009 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{512 b^{15/2}}+\frac {143 a \sqrt {a x+b x^{2/3}}}{20 b^3 x^2}-\frac {13 \sqrt {a x+b x^{2/3}}}{2 b^2 x^{7/3}}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

6/(b*x^(5/3)*Sqrt[b*x^(2/3) + a*x]) - (13*Sqrt[b*x^(2/3) + a*x])/(2*b^2*x^(7/3)) + (143*a*Sqrt[b*x^(2/3) + a*x
])/(20*b^3*x^2) - (1287*a^2*Sqrt[b*x^(2/3) + a*x])/(160*b^4*x^(5/3)) + (3003*a^3*Sqrt[b*x^(2/3) + a*x])/(320*b
^5*x^(4/3)) - (3003*a^4*Sqrt[b*x^(2/3) + a*x])/(256*b^6*x) + (9009*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^7*x^(2/3)
) - (9009*a^6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(512*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 2023

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 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^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}+\frac {13 \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{b}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}-\frac {(143 a) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{12 b^2}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}+\frac {\left (429 a^2\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{40 b^3}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}-\frac {\left (3003 a^3\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{320 b^4}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac {3003 a^3 \sqrt {b x^{2/3}+a x}}{320 b^5 x^{4/3}}+\frac {\left (1001 a^4\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{128 b^5}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac {3003 a^3 \sqrt {b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac {3003 a^4 \sqrt {b x^{2/3}+a x}}{256 b^6 x}-\frac {\left (3003 a^5\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{512 b^6}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac {3003 a^3 \sqrt {b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac {3003 a^4 \sqrt {b x^{2/3}+a x}}{256 b^6 x}+\frac {9009 a^5 \sqrt {b x^{2/3}+a x}}{512 b^7 x^{2/3}}+\frac {\left (3003 a^6\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{1024 b^7}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac {3003 a^3 \sqrt {b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac {3003 a^4 \sqrt {b x^{2/3}+a x}}{256 b^6 x}+\frac {9009 a^5 \sqrt {b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac {\left (9009 a^6\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 )}{512 b^7}\\ &=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac {3003 a^3 \sqrt {b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac {3003 a^4 \sqrt {b x^{2/3}+a x}}{256 b^6 x}+\frac {9009 a^5 \sqrt {b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac {9009 a^6 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{512 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^6 \sqrt [3]{x} \, _2F_1\left (-\frac {1}{2},7;\frac {1}{2};\frac {\sqrt [3]{x} a}{b}+1\right )}{b^7 \sqrt {a x+b x^{2/3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 12.49, size = 165, normalized size = 0.70 \begin {gather*} \frac {\sqrt [3]{x} \sqrt {a \sqrt [3]{x}+b} \left (\frac {45045 a^6 x^2+15015 a^5 b x^{5/3}-6006 a^4 b^2 x^{4/3}+3432 a^3 b^3 x-2288 a^2 b^4 x^{2/3}+1664 a b^5 \sqrt [3]{x}-1280 b^6}{2560 b^7 x^2 \sqrt {a \sqrt [3]{x}+b}}-\frac {9009 a^6 \tanh ^{-1}\left (\frac {\sqrt {a \sqrt [3]{x}+b}}{\sqrt {b}}\right )}{512 b^{15/2}}\right )}{\sqrt {x^{2/3} \left (a \sqrt [3]{x}+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b + a*x^(1/3)]*x^(1/3)*((-1280*b^6 + 1664*a*b^5*x^(1/3) - 2288*a^2*b^4*x^(2/3) + 3432*a^3*b^3*x - 6006*a
^4*b^2*x^(4/3) + 15015*a^5*b*x^(5/3) + 45045*a^6*x^2)/(2560*b^7*Sqrt[b + a*x^(1/3)]*x^2) - (9009*a^6*ArcTanh[S
qrt[b + a*x^(1/3)]/Sqrt[b]])/(512*b^(15/2))))/Sqrt[(b + a*x^(1/3))*x^(2/3)]

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

[Out]

Timed out

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giac [A]  time = 0.39, size = 156, normalized size = 0.66 \begin {gather*} \frac {9009 \, a^{6} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{512 \, \sqrt {-b} b^{7}} + \frac {6 \, a^{6}}{\sqrt {a x^{\frac {1}{3}} + b} b^{7}} + \frac {29685 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{6} - 163095 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{6} b + 364194 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{6} b^{2} - 416094 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{6} b^{3} + 246505 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{6} b^{4} - 62475 \, \sqrt {a x^{\frac {1}{3}} + b} a^{6} b^{5}}{2560 \, a^{6} b^{7} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9009/512*a^6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + 6*a^6/(sqrt(a*x^(1/3) + b)*b^7) + 1/2560*(2
9685*(a*x^(1/3) + b)^(11/2)*a^6 - 163095*(a*x^(1/3) + b)^(9/2)*a^6*b + 364194*(a*x^(1/3) + b)^(7/2)*a^6*b^2 -
416094*(a*x^(1/3) + b)^(5/2)*a^6*b^3 + 246505*(a*x^(1/3) + b)^(3/2)*a^6*b^4 - 62475*sqrt(a*x^(1/3) + b)*a^6*b^
5)/(a^6*b^7*x^2)

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maple [A]  time = 0.06, size = 126, normalized size = 0.53 \begin {gather*} -\frac {\left (a \,x^{\frac {1}{3}}+b \right ) \left (45045 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{6} x^{2} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-45045 a^{6} \sqrt {b}\, x^{2}-15015 a^{5} b^{\frac {3}{2}} x^{\frac {5}{3}}+6006 a^{4} b^{\frac {5}{2}} x^{\frac {4}{3}}-3432 a^{3} b^{\frac {7}{2}} x +2288 a^{2} b^{\frac {9}{2}} x^{\frac {2}{3}}-1664 a \,b^{\frac {11}{2}} x^{\frac {1}{3}}+1280 b^{\frac {13}{2}}\right )}{2560 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} b^{\frac {15}{2}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2560*(a*x^(1/3)+b)*(1280*b^(13/2)+45045*(a*x^(1/3)+b)^(1/2)*arctanh((a*x^(1/3)+b)^(1/2)/b^(1/2))*x^2*a^6+60
06*x^(4/3)*b^(5/2)*a^4-3432*x*b^(7/2)*a^3+2288*x^(2/3)*b^(9/2)*a^2-1664*x^(1/3)*b^(11/2)*a-45045*x^2*a^6*b^(1/
2)-15015*x^(5/3)*b^(3/2)*a^5)/x/(a*x+b*x^(2/3))^(3/2)/b^(15/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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