3.184 \(\int \frac{(b x^{2/3}+a x)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=379 \[ \frac{12597 a^{11} \sqrt{a x+b x^{2/3}}}{2097152 b^{10} x^{2/3}}-\frac{4199 a^{10} \sqrt{a x+b x^{2/3}}}{1048576 b^9 x}+\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{1310720 b^8 x^{4/3}}-\frac{12597 a^8 \sqrt{a x+b x^{2/3}}}{4587520 b^7 x^{5/3}}+\frac{4199 a^7 \sqrt{a x+b x^{2/3}}}{1720320 b^6 x^2}-\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{1892352 b^5 x^{7/3}}+\frac{323 a^5 \sqrt{a x+b x^{2/3}}}{157696 b^4 x^{8/3}}-\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{168960 b^3 x^3}+\frac{19 a^3 \sqrt{a x+b x^{2/3}}}{10560 b^2 x^{10/3}}-\frac{12597 a^{12} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{2097152 b^{21/2}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{1760 b x^{11/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{88 x^4}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5} \]

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(88*x^4) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(1760*b*x^(11/3)) + (19*a^3*Sqrt[b*x^(2/
3) + a*x])/(10560*b^2*x^(10/3)) - (323*a^4*Sqrt[b*x^(2/3) + a*x])/(168960*b^3*x^3) + (323*a^5*Sqrt[b*x^(2/3) +
a*x])/(157696*b^4*x^(8/3)) - (4199*a^6*Sqrt[b*x^(2/3) + a*x])/(1892352*b^5*x^(7/3)) + (4199*a^7*Sqrt[b*x^(2/3
) + a*x])/(1720320*b^6*x^2) - (12597*a^8*Sqrt[b*x^(2/3) + a*x])/(4587520*b^7*x^(5/3)) + (4199*a^9*Sqrt[b*x^(2/
3) + a*x])/(1310720*b^8*x^(4/3)) - (4199*a^10*Sqrt[b*x^(2/3) + a*x])/(1048576*b^9*x) + (12597*a^11*Sqrt[b*x^(2
/3) + a*x])/(2097152*b^10*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(4*x^5) - (12597*a^12*ArcTanh[(Sqrt[b]*x^(1/3))/S
qrt[b*x^(2/3) + a*x]])/(2097152*b^(21/2))

________________________________________________________________________________________

Rubi [A]  time = 0.718415, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2025, 2029, 206} \[ \frac{12597 a^{11} \sqrt{a x+b x^{2/3}}}{2097152 b^{10} x^{2/3}}-\frac{4199 a^{10} \sqrt{a x+b x^{2/3}}}{1048576 b^9 x}+\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{1310720 b^8 x^{4/3}}-\frac{12597 a^8 \sqrt{a x+b x^{2/3}}}{4587520 b^7 x^{5/3}}+\frac{4199 a^7 \sqrt{a x+b x^{2/3}}}{1720320 b^6 x^2}-\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{1892352 b^5 x^{7/3}}+\frac{323 a^5 \sqrt{a x+b x^{2/3}}}{157696 b^4 x^{8/3}}-\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{168960 b^3 x^3}+\frac{19 a^3 \sqrt{a x+b x^{2/3}}}{10560 b^2 x^{10/3}}-\frac{12597 a^{12} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{2097152 b^{21/2}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{1760 b x^{11/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{88 x^4}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^(2/3) + a*x)^(3/2)/x^6,x]

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(88*x^4) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(1760*b*x^(11/3)) + (19*a^3*Sqrt[b*x^(2/
3) + a*x])/(10560*b^2*x^(10/3)) - (323*a^4*Sqrt[b*x^(2/3) + a*x])/(168960*b^3*x^3) + (323*a^5*Sqrt[b*x^(2/3) +
a*x])/(157696*b^4*x^(8/3)) - (4199*a^6*Sqrt[b*x^(2/3) + a*x])/(1892352*b^5*x^(7/3)) + (4199*a^7*Sqrt[b*x^(2/3
) + a*x])/(1720320*b^6*x^2) - (12597*a^8*Sqrt[b*x^(2/3) + a*x])/(4587520*b^7*x^(5/3)) + (4199*a^9*Sqrt[b*x^(2/
3) + a*x])/(1310720*b^8*x^(4/3)) - (4199*a^10*Sqrt[b*x^(2/3) + a*x])/(1048576*b^9*x) + (12597*a^11*Sqrt[b*x^(2
/3) + a*x])/(2097152*b^10*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(4*x^5) - (12597*a^12*ArcTanh[(Sqrt[b]*x^(1/3))/S
qrt[b*x^(2/3) + a*x]])/(2097152*b^(21/2))

Rule 2020

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a*x^j + b*
x^n)^p)/(c*(m + j*p + 1)), x] - Dist[(b*p*(n - j))/(c^n*(m + j*p + 1)), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 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]

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{\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx &=-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{1}{8} a \int \frac{\sqrt{b x^{2/3}+a x}}{x^5} \, dx\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{1}{176} a^2 \int \frac{1}{x^4 \sqrt{b x^{2/3}+a x}} \, dx\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{\left (19 a^3\right ) \int \frac{1}{x^{11/3} \sqrt{b x^{2/3}+a x}} \, dx}{3520 b}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{\left (323 a^4\right ) \int \frac{1}{x^{10/3} \sqrt{b x^{2/3}+a x}} \, dx}{63360 b^2}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{\left (323 a^5\right ) \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx}{67584 b^3}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{\left (4199 a^6\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{946176 b^4}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{\left (4199 a^7\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{1032192 b^5}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{\left (4199 a^8\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{1146880 b^6}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{12597 a^8 \sqrt{b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{\left (4199 a^9\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{1310720 b^7}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{12597 a^8 \sqrt{b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{\left (4199 a^{10}\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{1572864 b^8}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{12597 a^8 \sqrt{b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac{4199 a^{10} \sqrt{b x^{2/3}+a x}}{1048576 b^9 x}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{\left (4199 a^{11}\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{2097152 b^9}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{12597 a^8 \sqrt{b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac{4199 a^{10} \sqrt{b x^{2/3}+a x}}{1048576 b^9 x}+\frac{12597 a^{11} \sqrt{b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}+\frac{\left (4199 a^{12}\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{4194304 b^{10}}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{12597 a^8 \sqrt{b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac{4199 a^{10} \sqrt{b x^{2/3}+a x}}{1048576 b^9 x}+\frac{12597 a^{11} \sqrt{b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{\left (12597 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^{10}}\\ &=-\frac{3 a \sqrt{b x^{2/3}+a x}}{88 x^4}-\frac{3 a^2 \sqrt{b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac{19 a^3 \sqrt{b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{168960 b^3 x^3}+\frac{323 a^5 \sqrt{b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac{4199 a^7 \sqrt{b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac{12597 a^8 \sqrt{b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac{4199 a^{10} \sqrt{b x^{2/3}+a x}}{1048576 b^9 x}+\frac{12597 a^{11} \sqrt{b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac{12597 a^{12} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{2097152 b^{21/2}}\\ \end{align*}

Mathematica [C]  time = 0.0609208, size = 61, normalized size = 0.16 \[ -\frac{6 a^{12} \left (a \sqrt [3]{x}+b\right )^2 \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{5}{2},13;\frac{7}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{5 b^{13} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^(2/3) + a*x)^(3/2)/x^6,x]

[Out]

(-6*a^12*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[5/2, 13, 7/2, 1 + (a*x^(1/3))/b])/(5*b^13*x
^(1/3))

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Maple [A]  time = 0.016, size = 223, normalized size = 0.6 \begin{align*}{\frac{1}{2422210560\,{x}^{5}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( 14549535\, \left ( b+a\sqrt [3]{x} \right ) ^{23/2}{b}^{21/2}-169744575\, \left ( b+a\sqrt [3]{x} \right ) ^{21/2}{b}^{23/2}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{19/2}{b}^{{\frac{25}{2}}}-2913648309\, \left ( b+a\sqrt [3]{x} \right ) ^{17/2}{b}^{{\frac{27}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{15/2}{b}^{{\frac{29}{2}}}-9643633350\, \left ( b+a\sqrt [3]{x} \right ) ^{13/2}{b}^{{\frac{31}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{11/2}{b}^{{\frac{33}{2}}}-8598579770\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{{\frac{35}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{{\frac{37}{2}}}-2001671595\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{{\frac{39}{2}}}-169744575\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{{\frac{41}{2}}}[phone]\,\sqrt{b+a\sqrt [3]{x}}{b}^{{\frac{43}{2}}}-14549535\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{10}{a}^{12}{x}^{4} \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{41}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^(2/3)+a*x)^(3/2)/x^6,x)

[Out]

1/2422210560*(b*x^(2/3)+a*x)^(3/2)*(14549535*(b+a*x^(1/3))^(23/2)*b^(21/2)-169744575*(b+a*x^(1/3))^(21/2)*b^(2
3/2)[phone]*(b+a*x^(1/3))^(19/2)*b^(25/2)-2913648309*(b+a*x^(1/3))^(17/2)*b^(27/2)[phone]*(b+a*x^(1/3))
^(15/2)*b^(29/2)-9643633350*(b+a*x^(1/3))^(13/2)*b^(31/2)[phone]*(b+a*x^(1/3))^(11/2)*b^(33/2)-8598579770
*(b+a*x^(1/3))^(9/2)*b^(35/2)[phone]*(b+a*x^(1/3))^(7/2)*b^(37/2)-2001671595*(b+a*x^(1/3))^(5/2)*b^(39/2)-
169744575*(b+a*x^(1/3))^(3/2)*b^(41/2)[phone]*(b+a*x^(1/3))^(1/2)*b^(43/2)-14549535*arctanh((b+a*x^(1/3))^(1
/2)/b^(1/2))*b^10*a^12*x^4)/x^5/(b+a*x^(1/3))^(3/2)/b^(41/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + b x^{\frac{2}{3}}\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(2/3))^(3/2)/x^6, 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((b*x^(2/3)+a*x)^(3/2)/x^6,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((b*x**(2/3)+a*x)**(3/2)/x**6,x)

[Out]

Timed out

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Giac [A]  time = 1.44672, size = 331, normalized size = 0.87 \begin{align*} \frac{\frac{14549535 \, a^{13} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{10}} + \frac{14549535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} a^{13} - 169744575 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{13} b + 904981077 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{13} b^{2} - 2913648309 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{13} b^{3} + 6303782342 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{13} b^{4} - 9643633350 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{13} b^{5} + 10677769530 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{13} b^{6} - 8598579770 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{13} b^{7} + 4975837515 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{13} b^{8} - 2001671595 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{13} b^{9} - 169744575 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{13} b^{10} + 14549535 \, \sqrt{a x^{\frac{1}{3}} + b} a^{13} b^{11}}{a^{12} b^{10} x^{4}}}{2422210560 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(3/2)/x^6,x, algorithm="giac")

[Out]

1/2422210560*(14549535*a^13*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) + (14549535*(a*x^(1/3) + b)^(
23/2)*a^13 - 169744575*(a*x^(1/3) + b)^(21/2)*a^13*b + 904981077*(a*x^(1/3) + b)^(19/2)*a^13*b^2 - 2913648309*
(a*x^(1/3) + b)^(17/2)*a^13*b^3 + 6303782342*(a*x^(1/3) + b)^(15/2)*a^13*b^4 - 9643633350*(a*x^(1/3) + b)^(13/
2)*a^13*b^5 + 10677769530*(a*x^(1/3) + b)^(11/2)*a^13*b^6 - 8598579770*(a*x^(1/3) + b)^(9/2)*a^13*b^7 + 497583
7515*(a*x^(1/3) + b)^(7/2)*a^13*b^8 - 2001671595*(a*x^(1/3) + b)^(5/2)*a^13*b^9 - 169744575*(a*x^(1/3) + b)^(3
/2)*a^13*b^10 + 14549535*sqrt(a*x^(1/3) + b)*a^13*b^11)/(a^12*b^10*x^4))/a