3.430 \(\int \frac {(a+b x^2)^{9/2}}{x^4} \, dx\)

Optimal. Leaf size=128 \[ \frac {105}{16} a^3 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {105}{16} a^2 b^2 x \sqrt {a+b x^2}+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}+\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3} \]

[Out]

35/8*a*b^2*x*(b*x^2+a)^(3/2)+7/2*b^2*x*(b*x^2+a)^(5/2)-3*b*(b*x^2+a)^(7/2)/x-1/3*(b*x^2+a)^(9/2)/x^3+105/16*a^
3*b^(3/2)*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))+105/16*a^2*b^2*x*(b*x^2+a)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 128, 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} \[ \frac {105}{16} a^2 b^2 x \sqrt {a+b x^2}+\frac {105}{16} a^3 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}+\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2)/x^4,x]

[Out]

(105*a^2*b^2*x*Sqrt[a + b*x^2])/16 + (35*a*b^2*x*(a + b*x^2)^(3/2))/8 + (7*b^2*x*(a + b*x^2)^(5/2))/2 - (3*b*(
a + b*x^2)^(7/2))/x - (a + b*x^2)^(9/2)/(3*x^3) + (105*a^3*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/16

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^4} \, dx &=-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+(3 b) \int \frac {\left (a+b x^2\right )^{7/2}}{x^2} \, dx\\ &=-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\left (21 b^2\right ) \int \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\frac {1}{2} \left (35 a b^2\right ) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\frac {1}{8} \left (105 a^2 b^2\right ) \int \sqrt {a+b x^2} \, dx\\ &=\frac {105}{16} a^2 b^2 x \sqrt {a+b x^2}+\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\frac {1}{16} \left (105 a^3 b^2\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx\\ &=\frac {105}{16} a^2 b^2 x \sqrt {a+b x^2}+\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\frac {1}{16} \left (105 a^3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )\\ &=\frac {105}{16} a^2 b^2 x \sqrt {a+b x^2}+\frac {35}{8} a b^2 x \left (a+b x^2\right )^{3/2}+\frac {7}{2} b^2 x \left (a+b x^2\right )^{5/2}-\frac {3 b \left (a+b x^2\right )^{7/2}}{x}-\frac {\left (a+b x^2\right )^{9/2}}{3 x^3}+\frac {105}{16} a^3 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 54, normalized size = 0.42 \[ -\frac {a^4 \sqrt {a+b x^2} \, _2F_1\left (-\frac {9}{2},-\frac {3}{2};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \sqrt {\frac {b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2)/x^4,x]

[Out]

-1/3*(a^4*Sqrt[a + b*x^2]*Hypergeometric2F1[-9/2, -3/2, -1/2, -((b*x^2)/a)])/(x^3*Sqrt[1 + (b*x^2)/a])

________________________________________________________________________________________

fricas [A]  time = 1.20, size = 189, normalized size = 1.48 \[ \left [\frac {315 \, a^{3} b^{\frac {3}{2}} x^{3} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (8 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 165 \, a^{2} b^{2} x^{4} - 208 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt {b x^{2} + a}}{96 \, x^{3}}, -\frac {315 \, a^{3} \sqrt {-b} b x^{3} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (8 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 165 \, a^{2} b^{2} x^{4} - 208 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt {b x^{2} + a}}{48 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^4,x, algorithm="fricas")

[Out]

[1/96*(315*a^3*b^(3/2)*x^3*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(8*b^4*x^8 + 50*a*b^3*x^6 + 165
*a^2*b^2*x^4 - 208*a^3*b*x^2 - 16*a^4)*sqrt(b*x^2 + a))/x^3, -1/48*(315*a^3*sqrt(-b)*b*x^3*arctan(sqrt(-b)*x/s
qrt(b*x^2 + a)) - (8*b^4*x^8 + 50*a*b^3*x^6 + 165*a^2*b^2*x^4 - 208*a^3*b*x^2 - 16*a^4)*sqrt(b*x^2 + a))/x^3]

________________________________________________________________________________________

giac [A]  time = 1.15, size = 160, normalized size = 1.25 \[ -\frac {105}{32} \, a^{3} b^{\frac {3}{2}} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right ) + \frac {1}{48} \, {\left (165 \, a^{2} b^{2} + 2 \, {\left (4 \, b^{4} x^{2} + 25 \, a b^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x + \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {3}{2}} - 24 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {3}{2}} + 13 \, a^{6} b^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^4,x, algorithm="giac")

[Out]

-105/32*a^3*b^(3/2)*log((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 1/48*(165*a^2*b^2 + 2*(4*b^4*x^2 + 25*a*b^3)*x^2)*s
qrt(b*x^2 + a)*x + 2/3*(15*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(3/2) - 24*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^
5*b^(3/2) + 13*a^6*b^(3/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 146, normalized size = 1.14 \[ \frac {105 a^{3} b^{\frac {3}{2}} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{16}+\frac {105 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} x}{16}+\frac {35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a \,b^{2} x}{8}+\frac {7 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2} x}{2}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2} x}{a}+\frac {8 \left (b \,x^{2}+a \right )^{\frac {9}{2}} b^{2} x}{3 a^{2}}-\frac {8 \left (b \,x^{2}+a \right )^{\frac {11}{2}} b}{3 a^{2} x}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}}}{3 a \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(9/2)/x^4,x)

[Out]

-1/3/a/x^3*(b*x^2+a)^(11/2)-8/3/a^2*b/x*(b*x^2+a)^(11/2)+8/3/a^2*b^2*x*(b*x^2+a)^(9/2)+3/a*b^2*x*(b*x^2+a)^(7/
2)+7/2*b^2*x*(b*x^2+a)^(5/2)+35/8*a*b^2*x*(b*x^2+a)^(3/2)+105/16*a^2*b^2*x*(b*x^2+a)^(1/2)+105/16*a^3*b^(3/2)*
ln(b^(1/2)*x+(b*x^2+a)^(1/2))

________________________________________________________________________________________

maxima [A]  time = 1.42, size = 120, normalized size = 0.94 \[ \frac {7}{2} \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} x + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2} x}{a} + \frac {35}{8} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2} x + \frac {105}{16} \, \sqrt {b x^{2} + a} a^{2} b^{2} x + \frac {105}{16} \, a^{3} b^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b}{3 \, a x} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}}}{3 \, a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(9/2)/x^4,x, algorithm="maxima")

[Out]

7/2*(b*x^2 + a)^(5/2)*b^2*x + 3*(b*x^2 + a)^(7/2)*b^2*x/a + 35/8*(b*x^2 + a)^(3/2)*a*b^2*x + 105/16*sqrt(b*x^2
+ a)*a^2*b^2*x + 105/16*a^3*b^(3/2)*arcsinh(b*x/sqrt(a*b)) - 8/3*(b*x^2 + a)^(9/2)*b/(a*x) - 1/3*(b*x^2 + a)^
(11/2)/(a*x^3)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{9/2}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^(9/2)/x^4,x)

[Out]

int((a + b*x^2)^(9/2)/x^4, x)

________________________________________________________________________________________

sympy [A]  time = 6.84, size = 175, normalized size = 1.37 \[ - \frac {a^{\frac {9}{2}}}{3 x^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {14 a^{\frac {7}{2}} b}{3 x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {43 a^{\frac {5}{2}} b^{2} x}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {215 a^{\frac {3}{2}} b^{3} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 \sqrt {a} b^{4} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {105 a^{3} b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16} + \frac {b^{5} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(9/2)/x**4,x)

[Out]

-a**(9/2)/(3*x**3*sqrt(1 + b*x**2/a)) - 14*a**(7/2)*b/(3*x*sqrt(1 + b*x**2/a)) - 43*a**(5/2)*b**2*x/(48*sqrt(1
+ b*x**2/a)) + 215*a**(3/2)*b**3*x**3/(48*sqrt(1 + b*x**2/a)) + 29*sqrt(a)*b**4*x**5/(24*sqrt(1 + b*x**2/a))
+ 105*a**3*b**(3/2)*asinh(sqrt(b)*x/sqrt(a))/16 + b**5*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a))

________________________________________________________________________________________