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

Optimal. Leaf size=61 \[ b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 217, 206} \begin {gather*} b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(3/2)/x^4,x]

[Out]

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

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 277

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{3/2}}{x^4} \, dx &=-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b \int \frac {\sqrt {a+b x^2}}{x^2} \, dx\\ &=-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^2 \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {b \sqrt {a+b x^2}}{x}-\frac {\left (a+b x^2\right )^{3/2}}{3 x^3}+b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 52, normalized size = 0.85 \begin {gather*} -\frac {a \sqrt {a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \sqrt {\frac {b x^2}{a}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(3/2)/x^4,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.10, size = 57, normalized size = 0.93 \begin {gather*} \frac {\left (-a-4 b x^2\right ) \sqrt {a+b x^2}}{3 x^3}-b^{3/2} \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, size = 112, normalized size = 1.84 \begin {gather*} \left [\frac {3 \, b^{\frac {3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (4 \, b x^{2} + a\right )} \sqrt {b x^{2} + a}}{6 \, x^{3}}, -\frac {3 \, \sqrt {-b} b x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (4 \, b x^{2} + a\right )} \sqrt {b x^{2} + a}}{3 \, x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.71, size = 114, normalized size = 1.87 \begin {gather*} -\frac {1}{2} \, b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {4 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {3}{2}} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {3}{2}} + 2 \, a^{3} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 92, normalized size = 1.51 \begin {gather*} b^{\frac {3}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {\sqrt {b \,x^{2}+a}\, b^{2} x}{a}+\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{2} x}{3 a^{2}}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b}{3 a^{2} x}-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{3 a \,x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/a/x^3*(b*x^2+a)^(5/2)-2/3/a^2*b/x*(b*x^2+a)^(5/2)+2/3/a^2*b^2*x*(b*x^2+a)^(3/2)+1/a*b^2*x*(b*x^2+a)^(1/2)
+b^(3/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.32, size = 66, normalized size = 1.08 \begin {gather*} \frac {\sqrt {b x^{2} + a} b^{2} x}{a} + b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{3 \, a x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2 + a)*b^2*x/a + b^(3/2)*arcsinh(b*x/sqrt(a*b)) - 2/3*(b*x^2 + a)^(3/2)*b/(a*x) - 1/3*(b*x^2 + a)^(5/
2)/(a*x^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{3/2}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(3/2)/x^4,x)

[Out]

int((a + b*x^2)^(3/2)/x^4, x)

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sympy [A]  time = 2.03, size = 78, normalized size = 1.28 \begin {gather*} - \frac {a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{3 x^{2}} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{3} - \frac {b^{\frac {3}{2}} \log {\left (\frac {a}{b x^{2}} \right )}}{2} + b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{b x^{2}} + 1} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*sqrt(b)*sqrt(a/(b*x**2) + 1)/(3*x**2) - 4*b**(3/2)*sqrt(a/(b*x**2) + 1)/3 - b**(3/2)*log(a/(b*x**2))/2 + b*
*(3/2)*log(sqrt(a/(b*x**2) + 1) + 1)

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