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

Optimal. Leaf size=266 \[ -\frac{1287 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^7 x^{2/3}}+\frac{429 a^6 \sqrt{a x+b x^{2/3}}}{8192 b^6 x}-\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{10240 b^5 x^{4/3}}+\frac{1287 a^4 \sqrt{a x+b x^{2/3}}}{35840 b^4 x^{5/3}}-\frac{143 a^3 \sqrt{a x+b x^{2/3}}}{4480 b^3 x^2}+\frac{13 a^2 \sqrt{a x+b x^{2/3}}}{448 b^2 x^{7/3}}+\frac{1287 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{16384 b^{15/2}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{112 b x^{8/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{8 x^3} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(8*x^3) - (3*a*Sqrt[b*x^(2/3) + a*x])/(112*b*x^(8/3)) + (13*a^2*Sqrt[b*x^(2/3) + a*
x])/(448*b^2*x^(7/3)) - (143*a^3*Sqrt[b*x^(2/3) + a*x])/(4480*b^3*x^2) + (1287*a^4*Sqrt[b*x^(2/3) + a*x])/(358
40*b^4*x^(5/3)) - (429*a^5*Sqrt[b*x^(2/3) + a*x])/(10240*b^5*x^(4/3)) + (429*a^6*Sqrt[b*x^(2/3) + a*x])/(8192*
b^6*x) - (1287*a^7*Sqrt[b*x^(2/3) + a*x])/(16384*b^7*x^(2/3)) + (1287*a^8*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(
2/3) + a*x]])/(16384*b^(15/2))

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Rubi [A]  time = 0.475003, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2025, 2029, 206} \[ -\frac{1287 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^7 x^{2/3}}+\frac{429 a^6 \sqrt{a x+b x^{2/3}}}{8192 b^6 x}-\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{10240 b^5 x^{4/3}}+\frac{1287 a^4 \sqrt{a x+b x^{2/3}}}{35840 b^4 x^{5/3}}-\frac{143 a^3 \sqrt{a x+b x^{2/3}}}{4480 b^3 x^2}+\frac{13 a^2 \sqrt{a x+b x^{2/3}}}{448 b^2 x^{7/3}}+\frac{1287 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{16384 b^{15/2}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{112 b x^{8/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{8 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^(2/3) + a*x]/x^4,x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(8*x^3) - (3*a*Sqrt[b*x^(2/3) + a*x])/(112*b*x^(8/3)) + (13*a^2*Sqrt[b*x^(2/3) + a*
x])/(448*b^2*x^(7/3)) - (143*a^3*Sqrt[b*x^(2/3) + a*x])/(4480*b^3*x^2) + (1287*a^4*Sqrt[b*x^(2/3) + a*x])/(358
40*b^4*x^(5/3)) - (429*a^5*Sqrt[b*x^(2/3) + a*x])/(10240*b^5*x^(4/3)) + (429*a^6*Sqrt[b*x^(2/3) + a*x])/(8192*
b^6*x) - (1287*a^7*Sqrt[b*x^(2/3) + a*x])/(16384*b^7*x^(2/3)) + (1287*a^8*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(
2/3) + a*x]])/(16384*b^(15/2))

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^4} \, dx &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}+\frac{1}{16} a \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}-\frac{\left (13 a^2\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{224 b}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}+\frac{\left (143 a^3\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{2688 b^2}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}-\frac{\left (429 a^4\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{8960 b^3}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}+\frac{1287 a^4 \sqrt{b x^{2/3}+a x}}{35840 b^4 x^{5/3}}+\frac{\left (429 a^5\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{10240 b^4}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}+\frac{1287 a^4 \sqrt{b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{10240 b^5 x^{4/3}}-\frac{\left (143 a^6\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{4096 b^5}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}+\frac{1287 a^4 \sqrt{b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac{429 a^6 \sqrt{b x^{2/3}+a x}}{8192 b^6 x}+\frac{\left (429 a^7\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{16384 b^6}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}+\frac{1287 a^4 \sqrt{b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac{429 a^6 \sqrt{b x^{2/3}+a x}}{8192 b^6 x}-\frac{1287 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^7 x^{2/3}}-\frac{\left (429 a^8\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{32768 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}+\frac{1287 a^4 \sqrt{b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac{429 a^6 \sqrt{b x^{2/3}+a x}}{8192 b^6 x}-\frac{1287 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^7 x^{2/3}}+\frac{\left (1287 a^8\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 )}{16384 b^7}\\ &=-\frac{3 \sqrt{b x^{2/3}+a x}}{8 x^3}-\frac{3 a \sqrt{b x^{2/3}+a x}}{112 b x^{8/3}}+\frac{13 a^2 \sqrt{b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac{143 a^3 \sqrt{b x^{2/3}+a x}}{4480 b^3 x^2}+\frac{1287 a^4 \sqrt{b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac{429 a^6 \sqrt{b x^{2/3}+a x}}{8192 b^6 x}-\frac{1287 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^7 x^{2/3}}+\frac{1287 a^8 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{16384 b^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0456889, size = 57, normalized size = 0.21 \[ -\frac{2 a^8 \left (a \sqrt [3]{x}+b\right ) \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{3}{2},9;\frac{5}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^9 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^8*(b + a*x^(1/3))*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[3/2, 9, 5/2, 1 + (a*x^(1/3))/b])/(b^9*x^(1/3))

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Maple [A]  time = 0.013, size = 167, normalized size = 0.6 \begin{align*} -{\frac{1}{573440\,{x}^{3}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( 45045\,{b}^{15/2} \left ( b+a\sqrt [3]{x} \right ) ^{15/2}-345345\,{b}^{17/2} \left ( b+a\sqrt [3]{x} \right ) ^{13/2}+1150149\,{b}^{19/2} \left ( b+a\sqrt [3]{x} \right ) ^{11/2}-2167737\,{b}^{21/2} \left ( b+a\sqrt [3]{x} \right ) ^{9/2}+2518087\,{b}^{23/2} \left ( b+a\sqrt [3]{x} \right ) ^{7/2}-1831739\,{b}^{{\frac{25}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{5/2}+801535\,{b}^{{\frac{27}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{3/2}-45045\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{7}{a}^{8}{x}^{8/3}+45045\,{b}^{{\frac{29}{2}}}\sqrt{b+a\sqrt [3]{x}} \right ){b}^{-{\frac{29}{2}}}{\frac{1}{\sqrt{b+a\sqrt [3]{x}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/573440*(b*x^(2/3)+a*x)^(1/2)*(45045*b^(15/2)*(b+a*x^(1/3))^(15/2)-345345*b^(17/2)*(b+a*x^(1/3))^(13/2)+1150
149*b^(19/2)*(b+a*x^(1/3))^(11/2)-2167737*b^(21/2)*(b+a*x^(1/3))^(9/2)+2518087*b^(23/2)*(b+a*x^(1/3))^(7/2)-18
31739*b^(25/2)*(b+a*x^(1/3))^(5/2)+801535*b^(27/2)*(b+a*x^(1/3))^(3/2)-45045*arctanh((b+a*x^(1/3))^(1/2)/b^(1/
2))*b^7*a^8*x^(8/3)+45045*b^(29/2)*(b+a*x^(1/3))^(1/2))/x^3/(b+a*x^(1/3))^(1/2)/b^(29/2)

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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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x + b x^{\frac{2}{3}}}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2688, size = 239, normalized size = 0.9 \begin{align*} -\frac{\frac{45045 \, a^{9} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{7}} + \frac{45045 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{9} - 345345 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{9} b + 1150149 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{9} b^{2} - 2167737 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{9} b^{3} + 2518087 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{9} b^{4} - 1831739 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{9} b^{5} + 801535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{9} b^{6} + 45045 \, \sqrt{a x^{\frac{1}{3}} + b} a^{9} b^{7}}{a^{8} b^{7} x^{\frac{8}{3}}}}{573440 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/573440*(45045*a^9*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(15/2)*a^9 -
345345*(a*x^(1/3) + b)^(13/2)*a^9*b + 1150149*(a*x^(1/3) + b)^(11/2)*a^9*b^2 - 2167737*(a*x^(1/3) + b)^(9/2)*
a^9*b^3 + 2518087*(a*x^(1/3) + b)^(7/2)*a^9*b^4 - 1831739*(a*x^(1/3) + b)^(5/2)*a^9*b^5 + 801535*(a*x^(1/3) +
b)^(3/2)*a^9*b^6 + 45045*sqrt(a*x^(1/3) + b)*a^9*b^7)/(a^8*b^7*x^(8/3)))/a