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

Optimal. Leaf size=291 \[ -\frac{429 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac{143 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^6 x}-\frac{143 a^6 \sqrt{a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac{143 a^4 \sqrt{a x+b x^{2/3}}}{26880 b^3 x^2}+\frac{13 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^2 x^{7/3}}+\frac{429 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac{a^2 \sqrt{a x+b x^{2/3}}}{224 b x^{8/3}}-\frac{a \sqrt{a x+b x^{2/3}}}{16 x^3}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4} \]

[Out]

-(a*Sqrt[b*x^(2/3) + a*x])/(16*x^3) - (a^2*Sqrt[b*x^(2/3) + a*x])/(224*b*x^(8/3)) + (13*a^3*Sqrt[b*x^(2/3) + a
*x])/(2688*b^2*x^(7/3)) - (143*a^4*Sqrt[b*x^(2/3) + a*x])/(26880*b^3*x^2) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(7
1680*b^4*x^(5/3)) - (143*a^6*Sqrt[b*x^(2/3) + a*x])/(20480*b^5*x^(4/3)) + (143*a^7*Sqrt[b*x^(2/3) + a*x])/(163
84*b^6*x) - (429*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^7*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(3*x^4) + (429*a^9*A
rcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(32768*b^(15/2))

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Rubi [A]  time = 0.521562, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2020, 2025, 2029, 206} \[ -\frac{429 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^7 x^{2/3}}+\frac{143 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^6 x}-\frac{143 a^6 \sqrt{a x+b x^{2/3}}}{20480 b^5 x^{4/3}}+\frac{429 a^5 \sqrt{a x+b x^{2/3}}}{71680 b^4 x^{5/3}}-\frac{143 a^4 \sqrt{a x+b x^{2/3}}}{26880 b^3 x^2}+\frac{13 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^2 x^{7/3}}+\frac{429 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{15/2}}-\frac{a^2 \sqrt{a x+b x^{2/3}}}{224 b x^{8/3}}-\frac{a \sqrt{a x+b x^{2/3}}}{16 x^3}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^(2/3) + a*x)^(3/2)/x^5,x]

[Out]

-(a*Sqrt[b*x^(2/3) + a*x])/(16*x^3) - (a^2*Sqrt[b*x^(2/3) + a*x])/(224*b*x^(8/3)) + (13*a^3*Sqrt[b*x^(2/3) + a
*x])/(2688*b^2*x^(7/3)) - (143*a^4*Sqrt[b*x^(2/3) + a*x])/(26880*b^3*x^2) + (429*a^5*Sqrt[b*x^(2/3) + a*x])/(7
1680*b^4*x^(5/3)) - (143*a^6*Sqrt[b*x^(2/3) + a*x])/(20480*b^5*x^(4/3)) + (143*a^7*Sqrt[b*x^(2/3) + a*x])/(163
84*b^6*x) - (429*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^7*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(3*x^4) + (429*a^9*A
rcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(32768*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{\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx &=-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{1}{6} a \int \frac{\sqrt{b x^{2/3}+a x}}{x^4} \, dx\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{1}{96} a^2 \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac{\left (13 a^3\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{1344 b}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{\left (143 a^4\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{16128 b^2}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac{\left (143 a^5\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{17920 b^3}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{\left (143 a^6\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{20480 b^4}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac{143 a^6 \sqrt{b x^{2/3}+a x}}{20480 b^5 x^{4/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac{\left (143 a^7\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{24576 b^5}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac{143 a^6 \sqrt{b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac{143 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^6 x}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{\left (143 a^8\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{32768 b^6}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac{143 a^6 \sqrt{b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac{143 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^6 x}-\frac{429 a^8 \sqrt{b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}-\frac{\left (143 a^9\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{65536 b^7}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac{143 a^6 \sqrt{b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac{143 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^6 x}-\frac{429 a^8 \sqrt{b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{\left (429 a^9\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 )}{32768 b^7}\\ &=-\frac{a \sqrt{b x^{2/3}+a x}}{16 x^3}-\frac{a^2 \sqrt{b x^{2/3}+a x}}{224 b x^{8/3}}+\frac{13 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac{143 a^4 \sqrt{b x^{2/3}+a x}}{26880 b^3 x^2}+\frac{429 a^5 \sqrt{b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac{143 a^6 \sqrt{b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac{143 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^6 x}-\frac{429 a^8 \sqrt{b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac{\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac{429 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{32768 b^{15/2}}\\ \end{align*}

Mathematica [C]  time = 0.0501321, size = 61, normalized size = 0.21 \[ \frac{6 a^9 \left (a \sqrt [3]{x}+b\right )^2 \sqrt{a x+b x^{2/3}} \, _2F_1\left (\frac{5}{2},10;\frac{7}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{5 b^{10} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a^9*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[5/2, 10, 7/2, 1 + (a*x^(1/3))/b])/(5*b^10*x^(
1/3))

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Maple [A]  time = 0.014, size = 181, normalized size = 0.6 \begin{align*}{\frac{1}{3440640\,{x}^{4}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( 45045\,{b}^{{\frac{31}{2}}}\sqrt{b+a\sqrt [3]{x}}-390390\,{b}^{{\frac{29}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{3/2}-2633274\,{b}^{{\frac{27}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{5/2}+4349826\,{b}^{{\frac{25}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{7/2}-4685824\,{b}^{23/2} \left ( b+a\sqrt [3]{x} \right ) ^{9/2}+3317886\,{b}^{21/2} \left ( b+a\sqrt [3]{x} \right ) ^{11/2}-1495494\,{b}^{19/2} \left ( b+a\sqrt [3]{x} \right ) ^{13/2}+390390\,{b}^{17/2} \left ( b+a\sqrt [3]{x} \right ) ^{15/2}-45045\,{b}^{15/2} \left ( b+a\sqrt [3]{x} \right ) ^{17/2}+45045\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{7}{a}^{9}{x}^{3} \right ){b}^{-{\frac{29}{2}}} \left ( b+a\sqrt [3]{x} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3440640*(b*x^(2/3)+a*x)^(3/2)*(45045*b^(31/2)*(b+a*x^(1/3))^(1/2)-390390*b^(29/2)*(b+a*x^(1/3))^(3/2)-263327
4*b^(27/2)*(b+a*x^(1/3))^(5/2)+4349826*b^(25/2)*(b+a*x^(1/3))^(7/2)-4685824*b^(23/2)*(b+a*x^(1/3))^(9/2)+33178
86*b^(21/2)*(b+a*x^(1/3))^(11/2)-1495494*b^(19/2)*(b+a*x^(1/3))^(13/2)+390390*b^(17/2)*(b+a*x^(1/3))^(15/2)-45
045*b^(15/2)*(b+a*x^(1/3))^(17/2)+45045*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*b^7*a^9*x^3)/x^4/(b+a*x^(1/3))^(3
/2)/b^(29/2)

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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^{5}}\,{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^5,x, algorithm="maxima")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.372, size = 262, normalized size = 0.9 \begin{align*} -\frac{\frac{45045 \, a^{10} \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{17}{2}} a^{10} - 390390 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{10} b + 1495494 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{10} b^{2} - 3317886 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{10} b^{3} + 4685824 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{10} b^{4} - 4349826 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{10} b^{5} + 2633274 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{10} b^{6} + 390390 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{10} b^{7} - 45045 \, \sqrt{a x^{\frac{1}{3}} + b} a^{10} b^{8}}{a^{9} b^{7} x^{3}}}{3440640 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3440640*(45045*a^10*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(17/2)*a^1
0 - 390390*(a*x^(1/3) + b)^(15/2)*a^10*b + 1495494*(a*x^(1/3) + b)^(13/2)*a^10*b^2 - 3317886*(a*x^(1/3) + b)^(
11/2)*a^10*b^3 + 4685824*(a*x^(1/3) + b)^(9/2)*a^10*b^4 - 4349826*(a*x^(1/3) + b)^(7/2)*a^10*b^5 + 2633274*(a*
x^(1/3) + b)^(5/2)*a^10*b^6 + 390390*(a*x^(1/3) + b)^(3/2)*a^10*b^7 - 45045*sqrt(a*x^(1/3) + b)*a^10*b^8)/(a^9
*b^7*x^3))/a