\(\int \frac {1}{x^4 (b x^{2/3}+a x)^{3/2}} \, dx\) [202]

Optimal result

Integrand size = 19, antiderivative size = 412 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {50702925 a^{12} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{27/2}} \]

[Out]

-50702925/2097152*a^12*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(27/2)+6/b/x^(11/3)/(b*x^(2/3)+a*x)^(1
/2)-25/4*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(13/3)+575/88*a*(b*x^(2/3)+a*x)^(1/2)/b^3/x^4-2415/352*a^2*(b*x^(2/3)+a*x
)^(1/2)/b^4/x^(11/3)+15295/2112*a^3*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(10/3)-260015/33792*a^4*(b*x^(2/3)+a*x)^(1/2)/
b^6/x^3+185725/22528*a^5*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(8/3)-2414425/270336*a^6*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(7/3
)+482885/49152*a^7*(b*x^(2/3)+a*x)^(1/2)/b^9/x^2-1448655/131072*a^8*(b*x^(2/3)+a*x)^(1/2)/b^10/x^(5/3)+3380195
/262144*a^9*(b*x^(2/3)+a*x)^(1/2)/b^11/x^(4/3)-16900975/1048576*a^10*(b*x^(2/3)+a*x)^(1/2)/b^12/x+50702925/209
7152*a^11*(b*x^(2/3)+a*x)^(1/2)/b^13/x^(2/3)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2048, 2050, 2054, 212} \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {50702925 a^{12} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{2097152 b^{27/2}}+\frac {50702925 a^{11} \sqrt {a x+b x^{2/3}}}{2097152 b^{13} x^{2/3}}-\frac {16900975 a^{10} \sqrt {a x+b x^{2/3}}}{1048576 b^{12} x}+\frac {3380195 a^9 \sqrt {a x+b x^{2/3}}}{262144 b^{11} x^{4/3}}-\frac {1448655 a^8 \sqrt {a x+b x^{2/3}}}{131072 b^{10} x^{5/3}}+\frac {482885 a^7 \sqrt {a x+b x^{2/3}}}{49152 b^9 x^2}-\frac {2414425 a^6 \sqrt {a x+b x^{2/3}}}{270336 b^8 x^{7/3}}+\frac {185725 a^5 \sqrt {a x+b x^{2/3}}}{22528 b^7 x^{8/3}}-\frac {260015 a^4 \sqrt {a x+b x^{2/3}}}{33792 b^6 x^3}+\frac {15295 a^3 \sqrt {a x+b x^{2/3}}}{2112 b^5 x^{10/3}}-\frac {2415 a^2 \sqrt {a x+b x^{2/3}}}{352 b^4 x^{11/3}}+\frac {575 a \sqrt {a x+b x^{2/3}}}{88 b^3 x^4}-\frac {25 \sqrt {a x+b x^{2/3}}}{4 b^2 x^{13/3}}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}} \]

[In]

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

[Out]

6/(b*x^(11/3)*Sqrt[b*x^(2/3) + a*x]) - (25*Sqrt[b*x^(2/3) + a*x])/(4*b^2*x^(13/3)) + (575*a*Sqrt[b*x^(2/3) + a
*x])/(88*b^3*x^4) - (2415*a^2*Sqrt[b*x^(2/3) + a*x])/(352*b^4*x^(11/3)) + (15295*a^3*Sqrt[b*x^(2/3) + a*x])/(2
112*b^5*x^(10/3)) - (260015*a^4*Sqrt[b*x^(2/3) + a*x])/(33792*b^6*x^3) + (185725*a^5*Sqrt[b*x^(2/3) + a*x])/(2
2528*b^7*x^(8/3)) - (2414425*a^6*Sqrt[b*x^(2/3) + a*x])/(270336*b^8*x^(7/3)) + (482885*a^7*Sqrt[b*x^(2/3) + a*
x])/(49152*b^9*x^2) - (1448655*a^8*Sqrt[b*x^(2/3) + a*x])/(131072*b^10*x^(5/3)) + (3380195*a^9*Sqrt[b*x^(2/3)
+ a*x])/(262144*b^11*x^(4/3)) - (16900975*a^10*Sqrt[b*x^(2/3) + a*x])/(1048576*b^12*x) + (50702925*a^11*Sqrt[b
*x^(2/3) + a*x])/(2097152*b^13*x^(2/3)) - (50702925*a^12*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(20
97152*b^(27/2))

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 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]

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 {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}+\frac {25 \int \frac {1}{x^{14/3} \sqrt {b x^{2/3}+a x}} \, dx}{b} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}-\frac {(575 a) \int \frac {1}{x^{13/3} \sqrt {b x^{2/3}+a x}} \, dx}{24 b^2} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}+\frac {\left (4025 a^2\right ) \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx}{176 b^3} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}-\frac {\left (15295 a^3\right ) \int \frac {1}{x^{11/3} \sqrt {b x^{2/3}+a x}} \, dx}{704 b^4} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}+\frac {\left (260015 a^4\right ) \int \frac {1}{x^{10/3} \sqrt {b x^{2/3}+a x}} \, dx}{12672 b^5} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}-\frac {\left (1300075 a^5\right ) \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx}{67584 b^6} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}+\frac {\left (2414425 a^6\right ) \int \frac {1}{x^{8/3} \sqrt {b x^{2/3}+a x}} \, dx}{135168 b^7} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}-\frac {\left (2414425 a^7\right ) \int \frac {1}{x^{7/3} \sqrt {b x^{2/3}+a x}} \, dx}{147456 b^8} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}+\frac {\left (482885 a^8\right ) \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx}{32768 b^9} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}-\frac {\left (3380195 a^9\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{262144 b^{10}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}+\frac {\left (16900975 a^{10}\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{1572864 b^{11}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}-\frac {\left (16900975 a^{11}\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{2097152 b^{12}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}+\frac {\left (16900975 a^{12}\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{4194304 b^{13}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {\left (50702925 a^{12}\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 )}{2097152 b^{13}} \\ & = \frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {50702925 a^{12} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{27/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6 a^{12} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},13,\frac {1}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{b^{13} \sqrt {b x^{2/3}+a x}} \]

[In]

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

[Out]

(6*a^12*x^(1/3)*Hypergeometric2F1[-1/2, 13, 1/2, 1 + (a*x^(1/3))/b])/(b^13*Sqrt[b*x^(2/3) + a*x])

Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.47

method	result	size
derivativedivides	\(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (17301504 b^{\frac {25}{2}}[phone] \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{12} x^{4}-1673196525 a^{12} x^{4} \sqrt {b}-19660800 b^{\frac {23}{2}} a \,x^{\frac {1}{3}}[phone] b^{\frac {21}{2}} a^{2} x^{\frac {2}{3}}-26378240 b^{\frac {19}{2}} a^{3} x [phone] b^{\frac {17}{2}} a^{4} x^{\frac {4}{3}}-38036480 b^{\frac {15}{2}} a^{5} x^{\frac {5}{3}}[phone] b^{\frac {13}{2}} a^{6} x^{2}-61809280 b^{\frac {11}{2}} a^{7} x^{\frac {7}{3}}[phone] b^{\frac {9}{2}} a^{8} x^{\frac {8}{3}}-127481640 b^{\frac {7}{2}} a^{9} x^{3}[phone] b^{\frac {5}{2}} a^{10} x^{\frac {10}{3}}-557732175 b^{\frac {3}{2}} a^{11} x^{\frac {11}{3}}\right )}{69206016 x^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {27}{2}}}\)	\(192\)
default	\(\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (1673196525 a^{12} x^{4} \sqrt {b}-17301504 b^{\frac {25}{2}}-1673196525 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{12} x^{4}[phone] b^{\frac {23}{2}} a \,x^{\frac {1}{3}}-22609920 b^{\frac {21}{2}} a^{2} x^{\frac {2}{3}}[phone] b^{\frac {19}{2}} a^{3} x -31324160 b^{\frac {17}{2}} a^{4} x^{\frac {4}{3}}[phone] b^{\frac {15}{2}} a^{5} x^{\frac {5}{3}}-47545600 b^{\frac {13}{2}} a^{6} x^{2}[phone] b^{\frac {11}{2}} a^{7} x^{\frac {7}{3}}-84987760 b^{\frac {9}{2}} a^{8} x^{\frac {8}{3}}[phone] b^{\frac {7}{2}} a^{9} x^{3}-223092870 b^{\frac {5}{2}} a^{10} x^{\frac {10}{3}}[phone] b^{\frac {3}{2}} a^{11} x^{\frac {11}{3}}\right )}{69206016 x^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {27}{2}}}\)	\(192\)

[In]

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

[Out]

-1/69206016*(b+a*x^(1/3))*(17301504*b^(25/2)[phone]*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*(b+a*x^(1/3))^(1/
2)*a^12*x^4-1673196525*a^12*x^4*b^(1/2)-19660800*b^(23/2)*a*x^(1/3)[phone]*b^(21/2)*a^2*x^(2/3)-26378240*b^(
19/2)*a^3*x+31324160*b^(17/2)*a^4*x^(4/3)-38036480*b^(15/2)*a^5*x^(5/3)[phone]*b^(13/2)*a^6*x^2-61809280*b^(
11/2)*a^7*x^(7/3)[phone]*b^(9/2)*a^8*x^(8/3)-127481640*b^(7/2)*a^9*x^3+223092870*b^(5/2)*a^10*x^(10/3)-55773
2175*b^(3/2)*a^11*x^(11/3))/x^3/(b*x^(2/3)+a*x)^(3/2)/b^(27/2)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {50702925 \, a^{12} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{2097152 \, \sqrt {-b} b^{13}} + \frac {6 \, a^{12}}{\sqrt {a x^{\frac {1}{3}} + b} b^{13}} + \frac {1257960429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} a^{12} - 14537792973 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{12} b + 76667241519 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{12} b^{2} - 243717614415 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{12} b^{3} + 519393101810 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{12} b^{4} - 780150847218 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{12} b^{5} + 844265343246 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{12} b^{6} - 659969685518 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{12} b^{7} + 366679446705 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{12} b^{8} - 138840292305 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{12} b^{9} + 32660709939 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{12} b^{10} - 3724872723 \, \sqrt {a x^{\frac {1}{3}} + b} a^{12} b^{11}}{69206016 \, a^{12} b^{13} x^{4}} \]

[In]

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

[Out]

50702925/2097152*a^12*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^13) + 6*a^12/(sqrt(a*x^(1/3) + b)*b^13)
+ 1/69206016*(1257960429*(a*x^(1/3) + b)^(23/2)*a^12 - 14537792973*(a*x^(1/3) + b)^(21/2)*a^12*b + 7666724151
9*(a*x^(1/3) + b)^(19/2)*a^12*b^2 - 243717614415*(a*x^(1/3) + b)^(17/2)*a^12*b^3 + 519393101810*(a*x^(1/3) + b
)^(15/2)*a^12*b^4 - 780150847218*(a*x^(1/3) + b)^(13/2)*a^12*b^5 + 844265343246*(a*x^(1/3) + b)^(11/2)*a^12*b^
6 - 659969685518*(a*x^(1/3) + b)^(9/2)*a^12*b^7 + 366679446705*(a*x^(1/3) + b)^(7/2)*a^12*b^8 - 138840292305*(
a*x^(1/3) + b)^(5/2)*a^12*b^9 + 32660709939*(a*x^(1/3) + b)^(3/2)*a^12*b^10 - 3724872723*sqrt(a*x^(1/3) + b)*a
^12*b^11)/(a^12*b^13*x^4)

Mupad [F(-1)]

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

[In]

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

[Out]

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