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

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

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Rubi [A]  time = 0.84, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 15, 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 {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}}} \end {gather*}

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 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^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*}

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Mathematica [C]  time = 0.09, size = 48, normalized size = 0.12 \begin {gather*} \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}}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 18.97, size = 239, normalized size = 0.58 \begin {gather*} \frac {\sqrt [3]{x} \sqrt {a \sqrt [3]{x}+b} \left (\frac {1673196525 a^{12} x^4+557732175 a^{11} b x^{11/3}-223092870 a^{10} b^2 x^{10/3}[phone] a^9 b^3 x^3-84987760 a^8 b^4 x^{8/3}[phone] a^7 b^5 x^{7/3}-47545600 a^6 b^6 x^2+38036480 a^5 b^7 x^{5/3}-31324160 a^4 b^8 x^{4/3}[phone] a^3 b^9 x-22609920 a^2 b^{10} x^{2/3}[phone] a b^{11} \sqrt [3]{x}-17301504 b^{12}}{69206016 b^{13} x^4 \sqrt {a \sqrt [3]{x}+b}}-\frac {50702925 a^{12} \tanh ^{-1}\left (\frac {\sqrt {a \sqrt [3]{x}+b}}{\sqrt {b}}\right )}{2097152 b^{27/2}}\right )}{\sqrt {x^{2/3} \left (a \sqrt [3]{x}+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b + a*x^(1/3)]*x^(1/3)*((-17301504*b^12 + 19660800*a*b^11*x^(1/3) - 22609920*a^2*b^10*x^(2/3) + 26378240
*a^3*b^9*x - 31324160*a^4*b^8*x^(4/3) + 38036480*a^5*b^7*x^(5/3) - 47545600*a^6*b^6*x^2 + 61809280*a^7*b^5*x^(
7/3) - 84987760*a^8*b^4*x^(8/3) + 127481640*a^9*b^3*x^3 - 223092870*a^10*b^2*x^(10/3) + 557732175*a^11*b*x^(11
/3) + 1673196525*a^12*x^4)/(69206016*b^13*Sqrt[b + a*x^(1/3)]*x^4) - (50702925*a^12*ArcTanh[Sqrt[b + a*x^(1/3)
]/Sqrt[b]])/(2097152*b^(27/2))))/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^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.45, size = 258, normalized size = 0.63 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.08, size = 192, normalized size = 0.47 \begin {gather*} -\frac {\left (a \,x^{\frac {1}{3}}+b \right ) \left (1673196525 \sqrt {a \,x^{\frac {1}{3}}+b}\, a^{12} x^{4} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-1673196525 a^{12} \sqrt {b}\, x^{4}-557732175 a^{11} b^{\frac {3}{2}} x^{\frac {11}{3}}[phone] a^{10} b^{\frac {5}{2}} x^{\frac {10}{3}}-127481640 a^{9} b^{\frac {7}{2}} x^{3}[phone] a^{8} b^{\frac {9}{2}} x^{\frac {8}{3}}-61809280 a^{7} b^{\frac {11}{2}} x^{\frac {7}{3}}[phone] a^{6} b^{\frac {13}{2}} x^{2}-38036480 a^{5} b^{\frac {15}{2}} x^{\frac {5}{3}}[phone] a^{4} b^{\frac {17}{2}} x^{\frac {4}{3}}-26378240 a^{3} b^{\frac {19}{2}} x [phone] a^{2} b^{\frac {21}{2}} x^{\frac {2}{3}}-19660800 a \,b^{\frac {23}{2}} x^{\frac {1}{3}}[phone] b^{\frac {25}{2}}\right )}{69206016 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} b^{\frac {27}{2}} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/69206016*(a*x^(1/3)+b)*(17301504*b^(25/2)-1673196525*x^4*a^12*b^(1/2)-19660800*b^(23/2)*x^(1/3)*a+22609920*
b^(21/2)*x^(2/3)*a^2+1673196525*(a*x^(1/3)+b)^(1/2)*arctanh((a*x^(1/3)+b)^(1/2)/b^(1/2))*x^4*a^12-26378240*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^10-55773
2175*b^(3/2)*x^(11/3)*a^11)/x^3/(a*x+b*x^(2/3))^(3/2)/b^(27/2)

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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^{4}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^4\,{\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^4*(a*x + b*x^(2/3))^(3/2)),x)

[Out]

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

[Out]

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

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