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

Optimal. Leaf size=78 \[ -6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )+\frac {6 b \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]

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Rubi [A]  time = 0.14, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2021, 2029, 206} \begin {gather*} -6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )+\frac {6 b \sqrt {a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^(2/3) + a*x)^(3/2)/x^2,x]

[Out]

(6*b*Sqrt[b*x^(2/3) + a*x])/x^(1/3) + (2*(b*x^(2/3) + a*x)^(3/2))/x - 6*b^(3/2)*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt
[b*x^(2/3) + a*x]]

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 2021

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 + n*p + 1)), x] + Dist[(a*(n - j)*p)/(c^j*(m + n*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]) && G
tQ[p, 0] && NeQ[m + n*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 {\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx &=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}+b \int \frac {\sqrt {b x^{2/3}+a x}}{x^{4/3}} \, dx\\ &=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}+b^2 \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx\\ &=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-\left (6 b^2\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 )\\ &=\frac {6 b \sqrt {b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 88, normalized size = 1.13 \begin {gather*} \frac {2 \sqrt {a x+b x^{2/3}} \left (\sqrt {a \sqrt [3]{x}+b} \left (a \sqrt [3]{x}+4 b\right )-3 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sqrt [3]{x}+b}}{\sqrt {b}}\right )\right )}{\sqrt [3]{x} \sqrt {a \sqrt [3]{x}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(Sqrt[b + a*x^(1/3)]*(4*b + a*x^(1/3)) - 3*b^(3/2)*ArcTanh[Sqrt[b + a*x^(1/3)]/Sqrt[b
]]))/(Sqrt[b + a*x^(1/3)]*x^(1/3))

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IntegrateAlgebraic [A]  time = 10.82, size = 90, normalized size = 1.15 \begin {gather*} \frac {\left (x^{2/3} \left (a \sqrt [3]{x}+b\right )\right )^{3/2} \left (2 \sqrt {a \sqrt [3]{x}+b} \left (a \sqrt [3]{x}+4 b\right )-6 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sqrt [3]{x}+b}}{\sqrt {b}}\right )\right )}{x \left (a \sqrt [3]{x}+b\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((b + a*x^(1/3))*x^(2/3))^(3/2)*(2*Sqrt[b + a*x^(1/3)]*(4*b + a*x^(1/3)) - 6*b^(3/2)*ArcTanh[Sqrt[b + a*x^(1/
3)]/Sqrt[b]]))/((b + a*x^(1/3))^(3/2)*x)

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

[Out]

Timed out

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giac [A]  time = 0.23, size = 83, normalized size = 1.06 \begin {gather*} \frac {6 \, b^{2} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} + 6 \, \sqrt {a x^{\frac {1}{3}} + b} b - \frac {2 \, {\left (3 \, b^{2} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b} b^{\frac {3}{2}}\right )}}{\sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b^2*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/sqrt(-b) + 2*(a*x^(1/3) + b)^(3/2) + 6*sqrt(a*x^(1/3) + b)*b - 2*(3
*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b)*b^(3/2))/sqrt(-b)

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maple [A]  time = 0.05, size = 69, normalized size = 0.88 \begin {gather*} -\frac {2 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (3 b^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {a \,x^{\frac {1}{3}}+b}}{\sqrt {b}}\right )-3 \sqrt {a \,x^{\frac {1}{3}}+b}\, b -\left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}}\right )}{\left (a \,x^{\frac {1}{3}}+b \right )^{\frac {3}{2}} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*x+b*x^(2/3))^(3/2)*(3*b^(3/2)*arctanh((a*x^(1/3)+b)^(1/2)/b^(1/2))-(a*x^(1/3)+b)^(3/2)-3*(a*x^(1/3)+b)^(
1/2)*b)/x/(a*x^(1/3)+b)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^(2/3))^(3/2)/x^2,x)

[Out]

int((a*x + b*x^(2/3))^(3/2)/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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