3.433 \(\int \frac {(a+b x^2)^{9/2}}{x^{10}} \, dx\)

Optimal. Leaf size=124 \[ b^{9/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b^4 \sqrt {a+b x^2}}{x}-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {277, 217, 206} \[ -\frac {b^4 \sqrt {a+b x^2}}{x}-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}+b^{9/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^10,x]

[Out]

-((b^4*Sqrt[a + b*x^2])/x) - (b^3*(a + b*x^2)^(3/2))/(3*x^3) - (b^2*(a + b*x^2)^(5/2))/(5*x^5) - (b*(a + b*x^2
)^(7/2))/(7*x^7) - (a + b*x^2)^(9/2)/(9*x^9) + b^(9/2)*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 )^{9/2}}{x^{10}} \, dx &=-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b \int \frac {\left (a+b x^2\right )^{7/2}}{x^8} \, dx\\ &=-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^2 \int \frac {\left (a+b x^2\right )^{5/2}}{x^6} \, dx\\ &=-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^3 \int \frac {\left (a+b x^2\right )^{3/2}}{x^4} \, dx\\ &=-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^4 \int \frac {\sqrt {a+b x^2}}{x^2} \, dx\\ &=-\frac {b^4 \sqrt {a+b x^2}}{x}-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^5 \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=-\frac {b^4 \sqrt {a+b x^2}}{x}-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^5 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=-\frac {b^4 \sqrt {a+b x^2}}{x}-\frac {b^3 \left (a+b x^2\right )^{3/2}}{3 x^3}-\frac {b^2 \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac {b \left (a+b x^2\right )^{7/2}}{7 x^7}-\frac {\left (a+b x^2\right )^{9/2}}{9 x^9}+b^{9/2} \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 = 54, normalized size = 0.44 \[ -\frac {a^4 \sqrt {a+b x^2} \, _2F_1\left (-\frac {9}{2},-\frac {9}{2};-\frac {7}{2};-\frac {b x^2}{a}\right )}{9 x^9 \sqrt {\frac {b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^10,x]

[Out]

-1/9*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -9/2, -7/2, -((b*x^2)/a)])/(x^9*Sqrt[1 + (b*x^2)/a])

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fricas [A]  time = 1.04, size = 184, normalized size = 1.48 \[ \left [\frac {315 \, b^{\frac {9}{2}} x^{9} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{630 \, x^{9}}, -\frac {315 \, \sqrt {-b} b^{4} x^{9} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (563 \, b^{4} x^{8} + 506 \, a b^{3} x^{6} + 408 \, a^{2} b^{2} x^{4} + 185 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{2} + a}}{315 \, x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^10,x, algorithm="fricas")

[Out]

[1/630*(315*b^(9/2)*x^9*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(563*b^4*x^8 + 506*a*b^3*x^6 + 408
*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4)*sqrt(b*x^2 + a))/x^9, -1/315*(315*sqrt(-b)*b^4*x^9*arctan(sqrt(-b)*x/sq
rt(b*x^2 + a)) + (563*b^4*x^8 + 506*a*b^3*x^6 + 408*a^2*b^2*x^4 + 185*a^3*b*x^2 + 35*a^4)*sqrt(b*x^2 + a))/x^9
]

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giac [B]  time = 1.12, size = 276, normalized size = 2.23 \[ -\frac {1}{2} \, b^{\frac {9}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {2 \, {\left (1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{16} a b^{\frac {9}{2}} - 6300 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} a^{2} b^{\frac {9}{2}} + 21000 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{3} b^{\frac {9}{2}} - 31500 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{4} b^{\frac {9}{2}} + 39438 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{5} b^{\frac {9}{2}} - 26292 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{6} b^{\frac {9}{2}} + 13968 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{7} b^{\frac {9}{2}} - 3492 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{8} b^{\frac {9}{2}} + 563 \, a^{9} b^{\frac {9}{2}}\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^10,x, algorithm="giac")

[Out]

-1/2*b^(9/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/315*(1575*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a*b^(9/2) - 6
300*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^2*b^(9/2) + 21000*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a^3*b^(9/2) - 31500*
(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(9/2) + 39438*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*b^(9/2) - 26292*(sqrt
(b)*x - sqrt(b*x^2 + a))^6*a^6*b^(9/2) + 13968*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^7*b^(9/2) - 3492*(sqrt(b)*x -
sqrt(b*x^2 + a))^2*a^8*b^(9/2) + 563*a^9*b^(9/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9

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maple [B]  time = 0.03, size = 206, normalized size = 1.66 \[ b^{\frac {9}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )+\frac {\sqrt {b \,x^{2}+a}\, b^{5} x}{a}+\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5} x}{3 a^{2}}+\frac {8 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{5} x}{15 a^{3}}+\frac {16 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5} x}{35 a^{4}}+\frac {128 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{5} x}{315 a^{5}}-\frac {128 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{4}}{315 a^{5} x}-\frac {16 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{3}}{315 a^{4} x^{3}}-\frac {8 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{2}}{315 a^{3} x^{5}}-\frac {2 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{63 a^{2} x^{7}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{9 a \,x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^10,x)

[Out]

-1/9/a/x^9*(b*x^2+a)^(11/2)-2/63/a^2*b/x^7*(b*x^2+a)^(11/2)-8/315/a^3*b^2/x^5*(b*x^2+a)^(11/2)-16/315/a^4*b^3/
x^3*(b*x^2+a)^(11/2)-128/315/a^5*b^4/x*(b*x^2+a)^(11/2)+128/315/a^5*b^5*x*(b*x^2+a)^(9/2)+16/35/a^4*b^5*x*(b*x
^2+a)^(7/2)+8/15/a^3*b^5*x*(b*x^2+a)^(5/2)+2/3/a^2*b^5*x*(b*x^2+a)^(3/2)+1/a*b^5*x*(b*x^2+a)^(1/2)+b^(9/2)*ln(
b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.51, size = 180, normalized size = 1.45 \[ \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5} x}{35 \, a^{4}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5} x}{15 \, a^{3}} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5} x}{3 \, a^{2}} + \frac {\sqrt {b x^{2} + a} b^{5} x}{a} + b^{\frac {9}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {128 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{4}}{315 \, a^{4} x} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{3}}{315 \, a^{4} x^{3}} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{315 \, a^{3} x^{5}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{63 \, a^{2} x^{7}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{9 \, a x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^10,x, algorithm="maxima")

[Out]

16/35*(b*x^2 + a)^(7/2)*b^5*x/a^4 + 8/15*(b*x^2 + a)^(5/2)*b^5*x/a^3 + 2/3*(b*x^2 + a)^(3/2)*b^5*x/a^2 + sqrt(
b*x^2 + a)*b^5*x/a + b^(9/2)*arcsinh(b*x/sqrt(a*b)) - 128/315*(b*x^2 + a)^(9/2)*b^4/(a^4*x) - 16/315*(b*x^2 +
a)^(11/2)*b^3/(a^4*x^3) - 8/315*(b*x^2 + a)^(11/2)*b^2/(a^3*x^5) - 2/63*(b*x^2 + a)^(11/2)*b/(a^2*x^7) - 1/9*(
b*x^2 + a)^(11/2)/(a*x^9)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^{10}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^10,x)

[Out]

int((a + b*x^2)^(9/2)/x^10, x)

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sympy [A]  time = 9.86, size = 160, normalized size = 1.29 \[ - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{9 x^{8}} - \frac {37 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{63 x^{6}} - \frac {136 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{105 x^{4}} - \frac {506 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{315 x^{2}} - \frac {563 b^{\frac {9}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{315} - \frac {b^{\frac {9}{2}} \log {\left (\frac {a}{b x^{2}} \right )}}{2} + b^{\frac {9}{2}} \log {\left (\sqrt {\frac {a}{b x^{2}} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**10,x)

[Out]

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

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