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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.83999, antiderivative size = 412, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2029, 206} \[ \frac{50702925 a^{11} \sqrt{a x+b x^{2/3}}}{2097152 b^{13} x^{2/3}}-\frac{16900975 a^{10} \sqrt{a x+b x^{2/3}}}{1048576 b^{12} x}+\frac{3380195 a^9 \sqrt{a x+b x^{2/3}}}{262144 b^{11} x^{4/3}}-\frac{1448655 a^8 \sqrt{a x+b x^{2/3}}}{131072 b^{10} x^{5/3}}+\frac{482885 a^7 \sqrt{a x+b x^{2/3}}}{49152 b^9 x^2}-\frac{2414425 a^6 \sqrt{a x+b x^{2/3}}}{270336 b^8 x^{7/3}}+\frac{185725 a^5 \sqrt{a x+b x^{2/3}}}{22528 b^7 x^{8/3}}-\frac{260015 a^4 \sqrt{a x+b x^{2/3}}}{33792 b^6 x^3}+\frac{15295 a^3 \sqrt{a x+b x^{2/3}}}{2112 b^5 x^{10/3}}-\frac{2415 a^2 \sqrt{a x+b x^{2/3}}}{352 b^4 x^{11/3}}-\frac{50702925 a^{12} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{2097152 b^{27/2}}+\frac{575 a \sqrt{a x+b x^{2/3}}}{88 b^3 x^4}-\frac{25 \sqrt{a x+b x^{2/3}}}{4 b^2 x^{13/3}}+\frac{6}{b x^{11/3} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[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 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]

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])

Rubi steps

\begin{align*} \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 \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 ) \operatorname{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]  time = 0.0589611, size = 48, normalized size = 0.12 \[ \frac{6 a^{12} \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},13;\frac{1}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^{13} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.025, size = 192, normalized size = 0.5 \begin{align*} -{\frac{1}{69206016\,{x}^{3}} \left ( b+a\sqrt [3]{x} \right ) \left ( 17301504\,{b}^{{\frac{25}{2}}}[phone]\,\sqrt{b+a\sqrt [3]{x}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{4}{a}^{12}-19660800\,{b}^{23/2}\sqrt [3]{x}a+22609920\,{b}^{21/2}{x}^{2/3}{a}^{2}-26378240\,{b}^{19/2}x{a}^{3}[phone]\,{b}^{17/2}{x}^{4/3}{a}^{4}-38036480\,{b}^{15/2}{x}^{5/3}{a}^{5}[phone]\,{b}^{13/2}{x}^{2}{a}^{6}-61809280\,{b}^{11/2}{x}^{7/3}{a}^{7}[phone]\,{b}^{9/2}{x}^{8/3}{a}^{8}-127481640\,{b}^{7/2}{x}^{3}{a}^{9}[phone]\,{b}^{5/2}{x}^{10/3}{a}^{10}-557732175\,{b}^{3/2}{x}^{11/3}{a}^{11}-1673196525\,{x}^{4}{a}^{12}\sqrt{b} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{27}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/69206016*(b+a*x^(1/3))*(17301504*b^(25/2)[phone]*(b+a*x^(1/3))^(1/2)*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2
))*x^4*a^12-19660800*b^(23/2)*x^(1/3)*a+22609920*b^(21/2)*x^(2/3)*a^2-26378240*b^(19/2)*x*a^3+31324160*b^(17/2
)*x^(4/3)*a^4-38036480*b^(15/2)*x^(5/3)*a^5+47545600*b^(13/2)*x^2*a^6-61809280*b^(11/2)*x^(7/3)*a^7+84987760*b
^(9/2)*x^(8/3)*a^8-127481640*b^(7/2)*x^3*a^9+223092870*b^(5/2)*x^(10/3)*a^10-557732175*b^(3/2)*x^(11/3)*a^11-1
673196525*x^4*a^12*b^(1/2))/x^3/(b*x^(2/3)+a*x)^(3/2)/b^(27/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b x^{\frac{2}{3}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.51867, size = 348, normalized size = 0.84 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)