3.402 \(\int \frac{(a+b x^2)^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=82 \[ -\frac{b^2 \sqrt{a+b x^2}}{x}+b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5} \]

[Out]

-((b^2*Sqrt[a + b*x^2])/x) - (b*(a + b*x^2)^(3/2))/(3*x^3) - (a + b*x^2)^(5/2)/(5*x^5) + b^(5/2)*ArcTanh[(Sqrt
[b]*x)/Sqrt[a + b*x^2]]

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Rubi [A]  time = 0.0280164, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 217, 206} \[ -\frac{b^2 \sqrt{a+b x^2}}{x}+b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(5/2)/x^6,x]

[Out]

-((b^2*Sqrt[a + b*x^2])/x) - (b*(a + b*x^2)^(3/2))/(3*x^3) - (a + b*x^2)^(5/2)/(5*x^5) + b^(5/2)*ArcTanh[(Sqrt
[b]*x)/Sqrt[a + b*x^2]]

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]

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 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 (a+b x^2\right )^{5/2}}{x^6} \, dx &=-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}+b \int \frac{\left (a+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}+b^2 \int \frac{\sqrt{a+b x^2}}{x^2} \, dx\\ &=-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}+b^3 \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}+b^3 \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}+b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0093549, size = 54, normalized size = 0.66 \[ -\frac{a^2 \sqrt{a+b x^2} \, _2F_1\left (-\frac{5}{2},-\frac{5}{2};-\frac{3}{2};-\frac{b x^2}{a}\right )}{5 x^5 \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(5/2)/x^6,x]

[Out]

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

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Maple [A]  time = 0.007, size = 130, normalized size = 1.6 \begin{align*} -{\frac{1}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,b}{15\,{a}^{2}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{b}^{2}}{15\,{a}^{3}x} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{b}^{3}x}{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{3}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{5}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(5/2)/x^6,x)

[Out]

-1/5/a/x^5*(b*x^2+a)^(7/2)-2/15*b/a^2/x^3*(b*x^2+a)^(7/2)-8/15*b^2/a^3/x*(b*x^2+a)^(7/2)+8/15*b^3/a^3*x*(b*x^2
+a)^(5/2)+2/3*b^3/a^2*x*(b*x^2+a)^(3/2)+b^3/a*x*(b*x^2+a)^(1/2)+b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58854, size = 342, normalized size = 4.17 \begin{align*} \left [\frac{15 \, b^{\frac{5}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (23 \, b^{2} x^{4} + 11 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{30 \, x^{5}}, -\frac{15 \, \sqrt{-b} b^{2} x^{5} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (23 \, b^{2} x^{4} + 11 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{15 \, x^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/30*(15*b^(5/2)*x^5*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(23*b^2*x^4 + 11*a*b*x^2 + 3*a^2)*sq
rt(b*x^2 + a))/x^5, -1/15*(15*sqrt(-b)*b^2*x^5*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (23*b^2*x^4 + 11*a*b*x^2 +
3*a^2)*sqrt(b*x^2 + a))/x^5]

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Sympy [A]  time = 3.64533, size = 105, normalized size = 1.28 \begin{align*} - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{11 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 x^{2}} - \frac{23 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15} - \frac{b^{\frac{5}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{5}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(5/2)/x**6,x)

[Out]

-a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 11*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(15*x**2) - 23*b**(5/2)*sqrt(
a/(b*x**2) + 1)/15 - b**(5/2)*log(a/(b*x**2))/2 + b**(5/2)*log(sqrt(a/(b*x**2) + 1) + 1)

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Giac [B]  time = 2.73945, size = 227, normalized size = 2.77 \begin{align*} -\frac{1}{2} \, b^{\frac{5}{2}} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{5}{2}} - 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{5}{2}} + 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{5}{2}} - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{5}{2}} + 23 \, a^{5} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(5/2)/x^6,x, algorithm="giac")

[Out]

-1/2*b^(5/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/15*(45*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a*b^(5/2) - 90*(s
qrt(b)*x - sqrt(b*x^2 + a))^6*a^2*b^(5/2) + 140*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*b^(5/2) - 70*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^4*b^(5/2) + 23*a^5*b^(5/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5