3.175 \(\int \frac{\sqrt{b x^{2/3}+a x}}{x^5} \, dx\)

Optimal. Leaf size=354 \[ \frac{12597 a^{10} \sqrt{a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^9 x}+\frac{4199 a^8 \sqrt{a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac{12597 a^7 \sqrt{a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{215040 b^6 x^2}-\frac{4199 a^5 \sqrt{a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{21120 b^3 x^3}+\frac{19 a^2 \sqrt{a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac{12597 a^{11} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{262144 b^{21/2}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{220 b x^{11/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{11 x^4} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) - (3*a*Sqrt[b*x^(2/3) + a*x])/(220*b*x^(11/3)) + (19*a^2*Sqrt[b*x^(2/3) +
a*x])/(1320*b^2*x^(10/3)) - (323*a^3*Sqrt[b*x^(2/3) + a*x])/(21120*b^3*x^3) + (323*a^4*Sqrt[b*x^(2/3) + a*x])/
(19712*b^4*x^(8/3)) - (4199*a^5*Sqrt[b*x^(2/3) + a*x])/(236544*b^5*x^(7/3)) + (4199*a^6*Sqrt[b*x^(2/3) + a*x])
/(215040*b^6*x^2) - (12597*a^7*Sqrt[b*x^(2/3) + a*x])/(573440*b^7*x^(5/3)) + (4199*a^8*Sqrt[b*x^(2/3) + a*x])/
(163840*b^8*x^(4/3)) - (4199*a^9*Sqrt[b*x^(2/3) + a*x])/(131072*b^9*x) + (12597*a^10*Sqrt[b*x^(2/3) + a*x])/(2
62144*b^10*x^(2/3)) - (12597*a^11*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(262144*b^(21/2))

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Rubi [A]  time = 0.660036, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2025, 2029, 206} \[ \frac{12597 a^{10} \sqrt{a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^9 x}+\frac{4199 a^8 \sqrt{a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac{12597 a^7 \sqrt{a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{215040 b^6 x^2}-\frac{4199 a^5 \sqrt{a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{21120 b^3 x^3}+\frac{19 a^2 \sqrt{a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac{12597 a^{11} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{262144 b^{21/2}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{220 b x^{11/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{11 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^(2/3) + a*x]/x^5,x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) - (3*a*Sqrt[b*x^(2/3) + a*x])/(220*b*x^(11/3)) + (19*a^2*Sqrt[b*x^(2/3) +
a*x])/(1320*b^2*x^(10/3)) - (323*a^3*Sqrt[b*x^(2/3) + a*x])/(21120*b^3*x^3) + (323*a^4*Sqrt[b*x^(2/3) + a*x])/
(19712*b^4*x^(8/3)) - (4199*a^5*Sqrt[b*x^(2/3) + a*x])/(236544*b^5*x^(7/3)) + (4199*a^6*Sqrt[b*x^(2/3) + a*x])
/(215040*b^6*x^2) - (12597*a^7*Sqrt[b*x^(2/3) + a*x])/(573440*b^7*x^(5/3)) + (4199*a^8*Sqrt[b*x^(2/3) + a*x])/
(163840*b^8*x^(4/3)) - (4199*a^9*Sqrt[b*x^(2/3) + a*x])/(131072*b^9*x) + (12597*a^10*Sqrt[b*x^(2/3) + a*x])/(2
62144*b^10*x^(2/3)) - (12597*a^11*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(262144*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{\sqrt{b x^{2/3}+a x}}{x^5} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}+\frac{1}{22} a \int \frac{1}{x^4 \sqrt{b x^{2/3}+a x}} \, dx\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}-\frac{\left (19 a^2\right ) \int \frac{1}{x^{11/3} \sqrt{b x^{2/3}+a x}} \, dx}{440 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}+\frac{\left (323 a^3\right ) \int \frac{1}{x^{10/3} \sqrt{b x^{2/3}+a x}} \, dx}{7920 b^2}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}-\frac{\left (323 a^4\right ) \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx}{8448 b^3}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}+\frac{\left (4199 a^5\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{118272 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}-\frac{\left (4199 a^6\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{129024 b^5}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}+\frac{\left (4199 a^7\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{143360 b^6}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}-\frac{12597 a^7 \sqrt{b x^{2/3}+a x}}{573440 b^7 x^{5/3}}-\frac{\left (4199 a^8\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{163840 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}-\frac{12597 a^7 \sqrt{b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac{4199 a^8 \sqrt{b x^{2/3}+a x}}{163840 b^8 x^{4/3}}+\frac{\left (4199 a^9\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{196608 b^8}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}-\frac{12597 a^7 \sqrt{b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac{4199 a^8 \sqrt{b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^9 x}-\frac{\left (4199 a^{10}\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{262144 b^9}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}-\frac{12597 a^7 \sqrt{b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac{4199 a^8 \sqrt{b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^9 x}+\frac{12597 a^{10} \sqrt{b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}+\frac{\left (4199 a^{11}\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{524288 b^{10}}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}-\frac{12597 a^7 \sqrt{b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac{4199 a^8 \sqrt{b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^9 x}+\frac{12597 a^{10} \sqrt{b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac{\left (12597 a^{11}\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 )}{262144 b^{10}}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{11 x^4}-\frac{3 a \sqrt{b x^{2/3}+a x}}{220 b x^{11/3}}+\frac{19 a^2 \sqrt{b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac{323 a^3 \sqrt{b x^{2/3}+a x}}{21120 b^3 x^3}+\frac{323 a^4 \sqrt{b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac{4199 a^5 \sqrt{b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac{4199 a^6 \sqrt{b x^{2/3}+a x}}{215040 b^6 x^2}-\frac{12597 a^7 \sqrt{b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac{4199 a^8 \sqrt{b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac{4199 a^9 \sqrt{b x^{2/3}+a x}}{131072 b^9 x}+\frac{12597 a^{10} \sqrt{b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac{12597 a^{11} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{262144 b^{21/2}}\\ \end{align*}

Mathematica [C]  time = 0.0490546, size = 57, normalized size = 0.16 \[ \frac{2 a^{11} \left (a \sqrt [3]{x}+b\right ) \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{3}{2},12;\frac{5}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^{12} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^(2/3) + a*x]/x^5,x]

[Out]

(2*a^11*(b + a*x^(1/3))*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[3/2, 12, 5/2, 1 + (a*x^(1/3))/b])/(b^12*x^(1/3
))

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Maple [A]  time = 0.015, size = 209, normalized size = 0.6 \begin{align*} -{\frac{1}{302776320\,{x}^{4}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( -14549535\, \left ( b+a\sqrt [3]{x} \right ) ^{21/2}{b}^{21/2}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{19/2}{b}^{23/2}-749786037\, \left ( b+a\sqrt [3]{x} \right ) ^{17/2}{b}^{{\frac{25}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{15/2}{b}^{{\frac{27}{2}}}-4139920070\, \left ( b+a\sqrt [3]{x} \right ) ^{13/2}{b}^{{\frac{29}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{11/2}{b}^{{\frac{31}{2}}}-5174056250\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{{\frac{33}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{{\frac{35}{2}}}-1551313995\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{{\frac{37}{2}}}[phone]\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{10}{a}^{11}{x}^{11/3}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{{\frac{39}{2}}}[phone]\,\sqrt{b+a\sqrt [3]{x}}{b}^{{\frac{41}{2}}} \right ){\frac{1}{\sqrt{b+a\sqrt [3]{x}}}}{b}^{-{\frac{41}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^(2/3)+a*x)^(1/2)/x^5,x)

[Out]

-1/302776320*(b*x^(2/3)+a*x)^(1/2)*(-14549535*(b+a*x^(1/3))^(21/2)*b^(21/2)[phone]*(b+a*x^(1/3))^(19/2)*b^(
23/2)-749786037*(b+a*x^(1/3))^(17/2)*b^(25/2)[phone]*(b+a*x^(1/3))^(15/2)*b^(27/2)-4139920070*(b+a*x^(1/3)
)^(13/2)*b^(29/2)[phone]*(b+a*x^(1/3))^(11/2)*b^(31/2)-5174056250*(b+a*x^(1/3))^(9/2)*b^(33/2)[phone]*
(b+a*x^(1/3))^(7/2)*b^(35/2)-1551313995*(b+a*x^(1/3))^(5/2)*b^(37/2)[phone]*arctanh((b+a*x^(1/3))^(1/2)/b^(1
/2))*b^10*a^11*x^(11/3)[phone]*(b+a*x^(1/3))^(3/2)*b^(39/2)[phone]*(b+a*x^(1/3))^(1/2)*b^(41/2))/x^4/(b+a
*x^(1/3))^(1/2)/b^(41/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x + b x^{\frac{2}{3}}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b*x^(2/3))/x^5, 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)^(1/2)/x^5,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)**(1/2)/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.39868, size = 308, normalized size = 0.87 \begin{align*} \frac{\frac{14549535 \, a^{12} \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{21}{2}} a^{12} - 155195040 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{12} b + 749786037 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{12} b^{2} - 2163862272 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{12} b^{3} + 4139920070 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{12} b^{4} - 5503713280 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{12} b^{5} + 5174056250 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{12} b^{6} - 3424523520 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{12} b^{7} + 1551313995 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{12} b^{8} - 450357600 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{12} b^{9} - 14549535 \, \sqrt{a x^{\frac{1}{3}} + b} a^{12} b^{10}}{a^{11} b^{10} x^{\frac{11}{3}}}}{302776320 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^(2/3)+a*x)^(1/2)/x^5,x, algorithm="giac")

[Out]

1/302776320*(14549535*a^12*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) + (14549535*(a*x^(1/3) + b)^(2
1/2)*a^12 - 155195040*(a*x^(1/3) + b)^(19/2)*a^12*b + 749786037*(a*x^(1/3) + b)^(17/2)*a^12*b^2 - 2163862272*(
a*x^(1/3) + b)^(15/2)*a^12*b^3 + 4139920070*(a*x^(1/3) + b)^(13/2)*a^12*b^4 - 5503713280*(a*x^(1/3) + b)^(11/2
)*a^12*b^5 + 5174056250*(a*x^(1/3) + b)^(9/2)*a^12*b^6 - 3424523520*(a*x^(1/3) + b)^(7/2)*a^12*b^7 + 155131399
5*(a*x^(1/3) + b)^(5/2)*a^12*b^8 - 450357600*(a*x^(1/3) + b)^(3/2)*a^12*b^9 - 14549535*sqrt(a*x^(1/3) + b)*a^1
2*b^10)/(a^11*b^10*x^(11/3)))/a