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

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

[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

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Rubi [A]  time = 0.0509176, antiderivative size = 126, 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} \[ -\frac{3 b^2 \left (a+b x^2\right )^{5/2}}{5 x^3}-\frac{3 b^3 \left (a+b x^2\right )^{3/2}}{x}+\frac{9}{2} b^4 x \sqrt{a+b x^2}+\frac{9}{2} a b^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{\left (a+b x^2\right )^{9/2}}{7 x^7}-\frac{9 b \left (a+b x^2\right )^{7/2}}{35 x^5} \]

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 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^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*}

Mathematica [C]  time = 0.0102338, 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]

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

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Maple [A]  time = 0.014, size = 186, normalized size = 1.5 \begin{align*} -{\frac{1}{7\,a{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{4\,b}{35\,{a}^{2}{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{8\,{b}^{2}}{35\,{a}^{3}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{64\,{b}^{3}}{35\,{a}^{4}x} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{64\,{b}^{4}x}{35\,{a}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{72\,{b}^{4}x}{35\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{12\,{b}^{4}x}{5\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+3\,{\frac{{b}^{4}x \left ( b{x}^{2}+a \right ) ^{3/2}}{a}}+{\frac{9\,{b}^{4}x}{2}\sqrt{b{x}^{2}+a}}+{\frac{9\,a}{2}{b}^{{\frac{7}{2}}}\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^8,x)

[Out]

-1/7/a/x^7*(b*x^2+a)^(11/2)-4/35*b/a^2/x^5*(b*x^2+a)^(11/2)-8/35*b^2/a^3/x^3*(b*x^2+a)^(11/2)-64/35*b^3/a^4/x*
(b*x^2+a)^(11/2)+64/35*b^4/a^4*x*(b*x^2+a)^(9/2)+72/35*b^4/a^3*x*(b*x^2+a)^(7/2)+12/5*b^4/a^2*x*(b*x^2+a)^(5/2
)+3*b^4/a*x*(b*x^2+a)^(3/2)+9/2*b^4*x*(b*x^2+a)^(1/2)+9/2*b^(7/2)*a*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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.1316, size = 451, normalized size = 3.58 \begin{align*} \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 ] \end{align*}

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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Sympy [A]  time = 9.03291, size = 167, normalized size = 1.33 \begin{align*} - \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} \end{align*}

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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Giac [B]  time = 1.93647, size = 324, normalized size = 2.57 \begin{align*} \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}} \end{align*}

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