3.429 \(\int \frac{(a+b x^2)^{9/2}}{x^2} \, dx\)

Optimal. Leaf size=123 \[ \frac{105}{64} a^2 b x \left (a+b x^2\right )^{3/2}+\frac{315}{128} a^3 b x \sqrt{a+b x^2}+\frac{315}{128} a^4 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}+\frac{21}{16} a b x \left (a+b x^2\right )^{5/2} \]

[Out]

(315*a^3*b*x*Sqrt[a + b*x^2])/128 + (105*a^2*b*x*(a + b*x^2)^(3/2))/64 + (21*a*b*x*(a + b*x^2)^(5/2))/16 + (9*
b*x*(a + b*x^2)^(7/2))/8 - (a + b*x^2)^(9/2)/x + (315*a^4*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/128

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Rubi [A]  time = 0.0439185, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {277, 195, 217, 206} \[ \frac{105}{64} a^2 b x \left (a+b x^2\right )^{3/2}+\frac{315}{128} a^3 b x \sqrt{a+b x^2}+\frac{315}{128} a^4 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}+\frac{21}{16} a b x \left (a+b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^2,x]

[Out]

(315*a^3*b*x*Sqrt[a + b*x^2])/128 + (105*a^2*b*x*(a + b*x^2)^(3/2))/64 + (21*a*b*x*(a + b*x^2)^(5/2))/16 + (9*
b*x*(a + b*x^2)^(7/2))/8 - (a + b*x^2)^(9/2)/x + (315*a^4*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/128

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 )^{9/2}}{x^2} \, dx &=-\frac{\left (a+b x^2\right )^{9/2}}{x}+(9 b) \int \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac{9}{8} b x \left (a+b x^2\right )^{7/2}-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{1}{8} (63 a b) \int \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac{21}{16} a b x \left (a+b x^2\right )^{5/2}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{1}{16} \left (105 a^2 b\right ) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{105}{64} a^2 b x \left (a+b x^2\right )^{3/2}+\frac{21}{16} a b x \left (a+b x^2\right )^{5/2}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{1}{64} \left (315 a^3 b\right ) \int \sqrt{a+b x^2} \, dx\\ &=\frac{315}{128} a^3 b x \sqrt{a+b x^2}+\frac{105}{64} a^2 b x \left (a+b x^2\right )^{3/2}+\frac{21}{16} a b x \left (a+b x^2\right )^{5/2}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{1}{128} \left (315 a^4 b\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{315}{128} a^3 b x \sqrt{a+b x^2}+\frac{105}{64} a^2 b x \left (a+b x^2\right )^{3/2}+\frac{21}{16} a b x \left (a+b x^2\right )^{5/2}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{1}{128} \left (315 a^4 b\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{315}{128} a^3 b x \sqrt{a+b x^2}+\frac{105}{64} a^2 b x \left (a+b x^2\right )^{3/2}+\frac{21}{16} a b x \left (a+b x^2\right )^{5/2}+\frac{9}{8} b x \left (a+b x^2\right )^{7/2}-\frac{\left (a+b x^2\right )^{9/2}}{x}+\frac{315}{128} a^4 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 0.0096428, size = 52, normalized size = 0.42 \[ -\frac{a^4 \sqrt{a+b x^2} \, _2F_1\left (-\frac{9}{2},-\frac{1}{2};\frac{1}{2};-\frac{b x^2}{a}\right )}{x \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^2,x]

[Out]

-((a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -1/2, 1/2, -((b*x^2)/a)])/(x*Sqrt[1 + (b*x^2)/a]))

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Maple [A]  time = 0.004, size = 117, normalized size = 1. \begin{align*} -{\frac{1}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{bx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,bx}{8} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{21\,abx}{16} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{105\,{a}^{2}bx}{64} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{315\,{a}^{3}bx}{128}\sqrt{b{x}^{2}+a}}+{\frac{315\,{a}^{4}}{128}\sqrt{b}\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)^(9/2)/x^2,x)

[Out]

-1/a/x*(b*x^2+a)^(11/2)+b/a*x*(b*x^2+a)^(9/2)+9/8*b*x*(b*x^2+a)^(7/2)+21/16*a*b*x*(b*x^2+a)^(5/2)+105/64*a^2*b
*x*(b*x^2+a)^(3/2)+315/128*a^3*b*x*(b*x^2+a)^(1/2)+315/128*b^(1/2)*a^4*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)^(9/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67496, size = 444, normalized size = 3.61 \begin{align*} \left [\frac{315 \, a^{4} \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (16 \, b^{4} x^{8} + 88 \, a b^{3} x^{6} + 210 \, a^{2} b^{2} x^{4} + 325 \, a^{3} b x^{2} - 128 \, a^{4}\right )} \sqrt{b x^{2} + a}}{256 \, x}, -\frac{315 \, a^{4} \sqrt{-b} x \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (16 \, b^{4} x^{8} + 88 \, a b^{3} x^{6} + 210 \, a^{2} b^{2} x^{4} + 325 \, a^{3} b x^{2} - 128 \, a^{4}\right )} \sqrt{b x^{2} + a}}{128 \, x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^2,x, algorithm="fricas")

[Out]

[1/256*(315*a^4*sqrt(b)*x*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(16*b^4*x^8 + 88*a*b^3*x^6 + 210
*a^2*b^2*x^4 + 325*a^3*b*x^2 - 128*a^4)*sqrt(b*x^2 + a))/x, -1/128*(315*a^4*sqrt(-b)*x*arctan(sqrt(-b)*x/sqrt(
b*x^2 + a)) - (16*b^4*x^8 + 88*a*b^3*x^6 + 210*a^2*b^2*x^4 + 325*a^3*b*x^2 - 128*a^4)*sqrt(b*x^2 + a))/x]

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Sympy [A]  time = 8.28635, size = 173, normalized size = 1.41 \begin{align*} - \frac{a^{\frac{9}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{197 a^{\frac{7}{2}} b x}{128 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{535 a^{\frac{5}{2}} b^{2} x^{3}}{128 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{149 a^{\frac{3}{2}} b^{3} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 \sqrt{a} b^{4} x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{315 a^{4} \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128} + \frac{b^{5} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**2,x)

[Out]

-a**(9/2)/(x*sqrt(1 + b*x**2/a)) + 197*a**(7/2)*b*x/(128*sqrt(1 + b*x**2/a)) + 535*a**(5/2)*b**2*x**3/(128*sqr
t(1 + b*x**2/a)) + 149*a**(3/2)*b**3*x**5/(64*sqrt(1 + b*x**2/a)) + 13*sqrt(a)*b**4*x**7/(16*sqrt(1 + b*x**2/a
)) + 315*a**4*sqrt(b)*asinh(sqrt(b)*x/sqrt(a))/128 + b**5*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.69715, size = 155, normalized size = 1.26 \begin{align*} -\frac{315}{256} \, a^{4} \sqrt{b} \log \left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \, a^{5} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} + \frac{1}{128} \,{\left (325 \, a^{3} b + 2 \,{\left (105 \, a^{2} b^{2} + 4 \,{\left (2 \, b^{4} x^{2} + 11 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^2,x, algorithm="giac")

[Out]

-315/256*a^4*sqrt(b)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2*a^5*sqrt(b)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a
) + 1/128*(325*a^3*b + 2*(105*a^2*b^2 + 4*(2*b^4*x^2 + 11*a*b^3)*x^2)*x^2)*sqrt(b*x^2 + a)*x