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

Optimal. Leaf size=108 \[ a^{9/2} \left (-\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+a^4 \sqrt {a+b x^2}+\frac {1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2} \]

[Out]

1/3*a^3*(b*x^2+a)^(3/2)+1/5*a^2*(b*x^2+a)^(5/2)+1/7*a*(b*x^2+a)^(7/2)+1/9*(b*x^2+a)^(9/2)-a^(9/2)*arctanh((b*x
^2+a)^(1/2)/a^(1/2))+a^4*(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ a^4 \sqrt {a+b x^2}+\frac {1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+a^{9/2} \left (-\tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x,x]

[Out]

a^4*Sqrt[a + b*x^2] + (a^3*(a + b*x^2)^(3/2))/3 + (a^2*(a + b*x^2)^(5/2))/5 + (a*(a + b*x^2)^(7/2))/7 + (a + b
*x^2)^(9/2)/9 - a^(9/2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^{9/2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{9/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{9} \left (a+b x^2\right )^{9/2}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {(a+b x)^{7/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2}+\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2}+\frac {1}{2} a^4 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right )\\ &=a^4 \sqrt {a+b x^2}+\frac {1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2}+\frac {1}{2} a^5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )\\ &=a^4 \sqrt {a+b x^2}+\frac {1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2}+\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=a^4 \sqrt {a+b x^2}+\frac {1}{3} a^3 \left (a+b x^2\right )^{3/2}+\frac {1}{5} a^2 \left (a+b x^2\right )^{5/2}+\frac {1}{7} a \left (a+b x^2\right )^{7/2}+\frac {1}{9} \left (a+b x^2\right )^{9/2}-a^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 84, normalized size = 0.78 \[ \frac {1}{315} \sqrt {a+b x^2} \left (563 a^4+506 a^3 b x^2+408 a^2 b^2 x^4+185 a b^3 x^6+35 b^4 x^8\right )-a^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x,x]

[Out]

(Sqrt[a + b*x^2]*(563*a^4 + 506*a^3*b*x^2 + 408*a^2*b^2*x^4 + 185*a*b^3*x^6 + 35*b^4*x^8))/315 - a^(9/2)*ArcTa
nh[Sqrt[a + b*x^2]/Sqrt[a]]

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fricas [A]  time = 0.97, size = 170, normalized size = 1.57 \[ \left [\frac {1}{2} \, a^{\frac {9}{2}} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + \frac {1}{315} \, {\left (35 \, b^{4} x^{8} + 185 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 506 \, a^{3} b x^{2} + 563 \, a^{4}\right )} \sqrt {b x^{2} + a}, \sqrt {-a} a^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + \frac {1}{315} \, {\left (35 \, b^{4} x^{8} + 185 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 506 \, a^{3} b x^{2} + 563 \, a^{4}\right )} \sqrt {b x^{2} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*a^(9/2)*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 1/315*(35*b^4*x^8 + 185*a*b^3*x^6 + 408*a^2
*b^2*x^4 + 506*a^3*b*x^2 + 563*a^4)*sqrt(b*x^2 + a), sqrt(-a)*a^4*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + 1/315*(35
*b^4*x^8 + 185*a*b^3*x^6 + 408*a^2*b^2*x^4 + 506*a^3*b*x^2 + 563*a^4)*sqrt(b*x^2 + a)]

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giac [A]  time = 1.17, size = 90, normalized size = 0.83 \[ \frac {a^{5} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {1}{9} \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} + \frac {1}{7} \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a + \frac {1}{5} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} + \sqrt {b x^{2} + a} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^5*arctan(sqrt(b*x^2 + a)/sqrt(-a))/sqrt(-a) + 1/9*(b*x^2 + a)^(9/2) + 1/7*(b*x^2 + a)^(7/2)*a + 1/5*(b*x^2 +
a)^(5/2)*a^2 + 1/3*(b*x^2 + a)^(3/2)*a^3 + sqrt(b*x^2 + a)*a^4

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maple [A]  time = 0.00, size = 94, normalized size = 0.87 \[ -a^{\frac {9}{2}} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )+\sqrt {b \,x^{2}+a}\, a^{4}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}{3}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2}}{5}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} a}{7}+\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x,x)

[Out]

1/9*(b*x^2+a)^(9/2)+1/7*a*(b*x^2+a)^(7/2)+1/5*a^2*(b*x^2+a)^(5/2)+1/3*a^3*(b*x^2+a)^(3/2)-a^(9/2)*ln((2*a+2*(b
*x^2+a)^(1/2)*a^(1/2))/x)+a^4*(b*x^2+a)^(1/2)

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maxima [A]  time = 1.40, size = 82, normalized size = 0.76 \[ -a^{\frac {9}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{9} \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} + \frac {1}{7} \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a + \frac {1}{5} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} + \sqrt {b x^{2} + a} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^(9/2)*arcsinh(a/(sqrt(a*b)*abs(x))) + 1/9*(b*x^2 + a)^(9/2) + 1/7*(b*x^2 + a)^(7/2)*a + 1/5*(b*x^2 + a)^(5/
2)*a^2 + 1/3*(b*x^2 + a)^(3/2)*a^3 + sqrt(b*x^2 + a)*a^4

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mupad [B]  time = 5.31, size = 87, normalized size = 0.81 \[ \frac {a\,{\left (b\,x^2+a\right )}^{7/2}}{7}+\frac {{\left (b\,x^2+a\right )}^{9/2}}{9}+a^4\,\sqrt {b\,x^2+a}+\frac {a^3\,{\left (b\,x^2+a\right )}^{3/2}}{3}+\frac {a^2\,{\left (b\,x^2+a\right )}^{5/2}}{5}+a^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x,x)

[Out]

a^(9/2)*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*1i + (a*(a + b*x^2)^(7/2))/7 + (a + b*x^2)^(9/2)/9 + a^4*(a + b*x
^2)^(1/2) + (a^3*(a + b*x^2)^(3/2))/3 + (a^2*(a + b*x^2)^(5/2))/5

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sympy [A]  time = 10.48, size = 160, normalized size = 1.48 \[ \frac {563 a^{\frac {9}{2}} \sqrt {1 + \frac {b x^{2}}{a}}}{315} + \frac {a^{\frac {9}{2}} \log {\left (\frac {b x^{2}}{a} \right )}}{2} - a^{\frac {9}{2}} \log {\left (\sqrt {1 + \frac {b x^{2}}{a}} + 1 \right )} + \frac {506 a^{\frac {7}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}{315} + \frac {136 a^{\frac {5}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}}}{105} + \frac {37 a^{\frac {3}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}{63} + \frac {\sqrt {a} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

563*a**(9/2)*sqrt(1 + b*x**2/a)/315 + a**(9/2)*log(b*x**2/a)/2 - a**(9/2)*log(sqrt(1 + b*x**2/a) + 1) + 506*a*
*(7/2)*b*x**2*sqrt(1 + b*x**2/a)/315 + 136*a**(5/2)*b**2*x**4*sqrt(1 + b*x**2/a)/105 + 37*a**(3/2)*b**3*x**6*s
qrt(1 + b*x**2/a)/63 + sqrt(a)*b**4*x**8*sqrt(1 + b*x**2/a)/9

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