3.432 \(\int \frac {(a+b x^2)^{9/2}}{x^8} \, dx\)

Optimal. Leaf size=126 \[ \frac {9}{2} a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {9}{2} b^4 x \sqrt {a+b x^2}-\frac {3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5} \]

[Out]

-3*b^3*(b*x^2+a)^(3/2)/x-3/5*b^2*(b*x^2+a)^(5/2)/x^3-9/35*b*(b*x^2+a)^(7/2)/x^5-1/7*(b*x^2+a)^(9/2)/x^7+9/2*a*
b^(7/2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))+9/2*b^4*x*(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {3 b^3 \left (a+b x^2\right )^{3/2}}{x}+\frac {9}{2} b^4 x \sqrt {a+b x^2}+\frac {9}{2} a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^8,x]

[Out]

(9*b^4*x*Sqrt[a + b*x^2])/2 - (3*b^3*(a + b*x^2)^(3/2))/x - (3*b^2*(a + b*x^2)^(5/2))/(5*x^3) - (9*b*(a + b*x^
2)^(7/2))/(35*x^5) - (a + b*x^2)^(9/2)/(7*x^7) + (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/2

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 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^8} \, dx &=-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\frac {1}{7} (9 b) \int \frac {\left (a+b x^2\right )^{7/2}}{x^6} \, dx\\ &=-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\frac {1}{5} \left (9 b^2\right ) \int \frac {\left (a+b x^2\right )^{5/2}}{x^4} \, dx\\ &=-\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\left (3 b^3\right ) \int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\left (9 b^4\right ) \int \sqrt {a+b x^2} \, dx\\ &=\frac {9}{2} b^4 x \sqrt {a+b x^2}-\frac {3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\frac {1}{2} \left (9 a b^4\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {9}{2} b^4 x \sqrt {a+b x^2}-\frac {3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\frac {1}{2} \left (9 a b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {9}{2} b^4 x \sqrt {a+b x^2}-\frac {3 b^3 \left (a+b x^2\right )^{3/2}}{x}-\frac {3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac {9 b \left (a+b x^2\right )^{7/2}}{35 x^5}-\frac {\left (a+b x^2\right )^{9/2}}{7 x^7}+\frac {9}{2} a b^{7/2} \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 = 0.43 \[ -\frac {a^4 \sqrt {a+b x^2} \, _2F_1\left (-\frac {9}{2},-\frac {7}{2};-\frac {5}{2};-\frac {b x^2}{a}\right )}{7 x^7 \sqrt {\frac {b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^8,x]

[Out]

-1/7*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -7/2, -5/2, -((b*x^2)/a)])/(x^7*Sqrt[1 + (b*x^2)/a])

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fricas [A]  time = 1.17, size = 187, normalized size = 1.48 \[ \left [\frac {315 \, a b^{\frac {7}{2}} x^{7} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (35 \, b^{4} x^{8} - 388 \, a b^{3} x^{6} - 156 \, a^{2} b^{2} x^{4} - 58 \, a^{3} b x^{2} - 10 \, a^{4}\right )} \sqrt {b x^{2} + a}}{140 \, x^{7}}, -\frac {315 \, a \sqrt {-b} b^{3} x^{7} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (35 \, b^{4} x^{8} - 388 \, a b^{3} x^{6} - 156 \, a^{2} b^{2} x^{4} - 58 \, a^{3} b x^{2} - 10 \, a^{4}\right )} \sqrt {b x^{2} + a}}{70 \, x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^8,x, algorithm="fricas")

[Out]

[1/140*(315*a*b^(7/2)*x^7*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(35*b^4*x^8 - 388*a*b^3*x^6 - 15
6*a^2*b^2*x^4 - 58*a^3*b*x^2 - 10*a^4)*sqrt(b*x^2 + a))/x^7, -1/70*(315*a*sqrt(-b)*b^3*x^7*arctan(sqrt(-b)*x/s
qrt(b*x^2 + a)) - (35*b^4*x^8 - 388*a*b^3*x^6 - 156*a^2*b^2*x^4 - 58*a^3*b*x^2 - 10*a^4)*sqrt(b*x^2 + a))/x^7]

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giac [B]  time = 1.20, size = 240, normalized size = 1.90 \[ \frac {1}{2} \, \sqrt {b x^{2} + a} b^{4} x - \frac {9}{4} \, a b^{\frac {7}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {4 \, {\left (175 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a^{2} b^{\frac {7}{2}} - 700 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{3} b^{\frac {7}{2}} + 1575 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{4} b^{\frac {7}{2}} - 1820 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{5} b^{\frac {7}{2}} + 1337 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{6} b^{\frac {7}{2}} - 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{7} b^{\frac {7}{2}} + 97 \, a^{8} b^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^8,x, algorithm="giac")

[Out]

1/2*sqrt(b*x^2 + a)*b^4*x - 9/4*a*b^(7/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 4/35*(175*(sqrt(b)*x - sqrt(b
*x^2 + a))^12*a^2*b^(7/2) - 700*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(7/2) + 1575*(sqrt(b)*x - sqrt(b*x^2 +
a))^8*a^4*b^(7/2) - 1820*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^5*b^(7/2) + 1337*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^
6*b^(7/2) - 504*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^7*b^(7/2) + 97*a^8*b^(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2
- a)^7

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maple [A]  time = 0.02, size = 186, normalized size = 1.48 \[ \frac {9 a \,b^{\frac {7}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2}+\frac {9 \sqrt {b \,x^{2}+a}\, b^{4} x}{2}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{4} x}{a}+\frac {12 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{4} x}{5 a^{2}}+\frac {72 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{4} x}{35 a^{3}}+\frac {64 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{4} x}{35 a^{4}}-\frac {64 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{3}}{35 a^{4} x}-\frac {8 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b^{2}}{35 a^{3} x^{3}}-\frac {4 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{35 a^{2} x^{5}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{7 a \,x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^8,x)

[Out]

-1/7/a/x^7*(b*x^2+a)^(11/2)-4/35/a^2*b/x^5*(b*x^2+a)^(11/2)-8/35/a^3*b^2/x^3*(b*x^2+a)^(11/2)-64/35/a^4*b^3/x*
(b*x^2+a)^(11/2)+64/35/a^4*b^4*x*(b*x^2+a)^(9/2)+72/35/a^3*b^4*x*(b*x^2+a)^(7/2)+12/5/a^2*b^4*x*(b*x^2+a)^(5/2
)+3/a*b^4*x*(b*x^2+a)^(3/2)+9/2*b^4*x*(b*x^2+a)^(1/2)+9/2*a*b^(7/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.47, size = 160, normalized size = 1.27 \[ \frac {9}{2} \, \sqrt {b x^{2} + a} b^{4} x + \frac {72 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4} x}{35 \, a^{3}} + \frac {12 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4} x}{5 \, a^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4} x}{a} + \frac {9}{2} \, a b^{\frac {7}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {64 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b^{3}}{35 \, a^{3} x} - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b^{2}}{35 \, a^{3} x^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} b}{35 \, a^{2} x^{5}} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{7 \, a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^8,x, algorithm="maxima")

[Out]

9/2*sqrt(b*x^2 + a)*b^4*x + 72/35*(b*x^2 + a)^(7/2)*b^4*x/a^3 + 12/5*(b*x^2 + a)^(5/2)*b^4*x/a^2 + 3*(b*x^2 +
a)^(3/2)*b^4*x/a + 9/2*a*b^(7/2)*arcsinh(b*x/sqrt(a*b)) - 64/35*(b*x^2 + a)^(9/2)*b^3/(a^3*x) - 8/35*(b*x^2 +
a)^(11/2)*b^2/(a^3*x^3) - 4/35*(b*x^2 + a)^(11/2)*b/(a^2*x^5) - 1/7*(b*x^2 + a)^(11/2)/(a*x^7)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^8} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^8,x)

[Out]

int((a + b*x^2)^(9/2)/x^8, x)

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sympy [A]  time = 8.79, size = 167, normalized size = 1.33 \[ - \frac {a^{4} \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{7 x^{6}} - \frac {29 a^{3} b^{\frac {3}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 x^{4}} - \frac {78 a^{2} b^{\frac {5}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 x^{2}} - \frac {194 a b^{\frac {7}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35} - \frac {9 a b^{\frac {7}{2}} \log {\left (\frac {a}{b x^{2}} \right )}}{4} + \frac {9 a b^{\frac {7}{2}} \log {\left (\sqrt {\frac {a}{b x^{2}} + 1} + 1 \right )}}{2} + \frac {b^{\frac {9}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**8,x)

[Out]

-a**4*sqrt(b)*sqrt(a/(b*x**2) + 1)/(7*x**6) - 29*a**3*b**(3/2)*sqrt(a/(b*x**2) + 1)/(35*x**4) - 78*a**2*b**(5/
2)*sqrt(a/(b*x**2) + 1)/(35*x**2) - 194*a*b**(7/2)*sqrt(a/(b*x**2) + 1)/35 - 9*a*b**(7/2)*log(a/(b*x**2))/4 +
9*a*b**(7/2)*log(sqrt(a/(b*x**2) + 1) + 1)/2 + b**(9/2)*x**2*sqrt(a/(b*x**2) + 1)/2

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