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

Optimal. Leaf size=122 \[ \frac{63}{256} a^4 x \sqrt{a+b x^2}+\frac{21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{63 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b}}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2} \]

[Out]

(63*a^4*x*Sqrt[a + b*x^2])/256 + (21*a^3*x*(a + b*x^2)^(3/2))/128 + (21*a^2*x*(a + b*x^2)^(5/2))/160 + (9*a*x*
(a + b*x^2)^(7/2))/80 + (x*(a + b*x^2)^(9/2))/10 + (63*a^5*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*Sqrt[b])

________________________________________________________________________________________

Rubi [A]  time = 0.0421447, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {195, 217, 206} \[ \frac{63}{256} a^4 x \sqrt{a+b x^2}+\frac{21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{63 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b}}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(9/2),x]

[Out]

(63*a^4*x*Sqrt[a + b*x^2])/256 + (21*a^3*x*(a + b*x^2)^(3/2))/128 + (21*a^2*x*(a + b*x^2)^(5/2))/160 + (9*a*x*
(a + b*x^2)^(7/2))/80 + (x*(a + b*x^2)^(9/2))/10 + (63*a^5*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(256*Sqrt[b])

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 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{1}{10} (9 a) \int \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{1}{80} \left (63 a^2\right ) \int \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{1}{32} \left (21 a^3\right ) \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{1}{128} \left (63 a^4\right ) \int \sqrt{a+b x^2} \, dx\\ &=\frac{63}{256} a^4 x \sqrt{a+b x^2}+\frac{21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{1}{256} \left (63 a^5\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{63}{256} a^4 x \sqrt{a+b x^2}+\frac{21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{1}{256} \left (63 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{63}{256} a^4 x \sqrt{a+b x^2}+\frac{21}{128} a^3 x \left (a+b x^2\right )^{3/2}+\frac{21}{160} a^2 x \left (a+b x^2\right )^{5/2}+\frac{9}{80} a x \left (a+b x^2\right )^{7/2}+\frac{1}{10} x \left (a+b x^2\right )^{9/2}+\frac{63 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.131381, size = 98, normalized size = 0.8 \[ \frac{\sqrt{a+b x^2} \left (1368 a^2 b^2 x^5+1490 a^3 b x^3+\frac{315 a^{9/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{\frac{b x^2}{a}+1}}+965 a^4 x+656 a b^3 x^7+128 b^4 x^9\right )}{1280} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[a + b*x^2]*(965*a^4*x + 1490*a^3*b*x^3 + 1368*a^2*b^2*x^5 + 656*a*b^3*x^7 + 128*b^4*x^9 + (315*a^(9/2)*A
rcSinh[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*Sqrt[1 + (b*x^2)/a])))/1280

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 96, normalized size = 0.8 \begin{align*}{\frac{x}{10} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,ax}{80} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{21\,{a}^{2}x}{160} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{21\,{a}^{3}x}{128} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{63\,{a}^{4}x}{256}\sqrt{b{x}^{2}+a}}+{\frac{63\,{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x*(b*x^2+a)^(9/2)+9/80*a*x*(b*x^2+a)^(7/2)+21/160*a^2*x*(b*x^2+a)^(5/2)+21/128*a^3*x*(b*x^2+a)^(3/2)+63/2
56*a^4*x*(b*x^2+a)^(1/2)+63/256*a^5/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

________________________________________________________________________________________

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, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.9904, size = 468, normalized size = 3.84 \begin{align*} \left [\frac{315 \, a^{5} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (128 \, b^{5} x^{9} + 656 \, a b^{4} x^{7} + 1368 \, a^{2} b^{3} x^{5} + 1490 \, a^{3} b^{2} x^{3} + 965 \, a^{4} b x\right )} \sqrt{b x^{2} + a}}{2560 \, b}, -\frac{315 \, a^{5} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (128 \, b^{5} x^{9} + 656 \, a b^{4} x^{7} + 1368 \, a^{2} b^{3} x^{5} + 1490 \, a^{3} b^{2} x^{3} + 965 \, a^{4} b x\right )} \sqrt{b x^{2} + a}}{1280 \, b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2560*(315*a^5*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(128*b^5*x^9 + 656*a*b^4*x^7 + 13
68*a^2*b^3*x^5 + 1490*a^3*b^2*x^3 + 965*a^4*b*x)*sqrt(b*x^2 + a))/b, -1/1280*(315*a^5*sqrt(-b)*arctan(sqrt(-b)
*x/sqrt(b*x^2 + a)) - (128*b^5*x^9 + 656*a*b^4*x^7 + 1368*a^2*b^3*x^5 + 1490*a^3*b^2*x^3 + 965*a^4*b*x)*sqrt(b
*x^2 + a))/b]

________________________________________________________________________________________

Sympy [A]  time = 9.49426, size = 151, normalized size = 1.24 \begin{align*} \frac{193 a^{\frac{9}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{256} + \frac{149 a^{\frac{7}{2}} b x^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{128} + \frac{171 a^{\frac{5}{2}} b^{2} x^{5} \sqrt{1 + \frac{b x^{2}}{a}}}{160} + \frac{41 a^{\frac{3}{2}} b^{3} x^{7} \sqrt{1 + \frac{b x^{2}}{a}}}{80} + \frac{\sqrt{a} b^{4} x^{9} \sqrt{1 + \frac{b x^{2}}{a}}}{10} + \frac{63 a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

193*a**(9/2)*x*sqrt(1 + b*x**2/a)/256 + 149*a**(7/2)*b*x**3*sqrt(1 + b*x**2/a)/128 + 171*a**(5/2)*b**2*x**5*sq
rt(1 + b*x**2/a)/160 + 41*a**(3/2)*b**3*x**7*sqrt(1 + b*x**2/a)/80 + sqrt(a)*b**4*x**9*sqrt(1 + b*x**2/a)/10 +
63*a**5*asinh(sqrt(b)*x/sqrt(a))/(256*sqrt(b))

________________________________________________________________________________________

Giac [A]  time = 1.48273, size = 123, normalized size = 1.01 \begin{align*} -\frac{63 \, a^{5} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, \sqrt{b}} + \frac{1}{1280} \,{\left (965 \, a^{4} + 2 \,{\left (745 \, a^{3} b + 4 \,{\left (171 \, a^{2} b^{2} + 2 \,{\left (8 \, b^{4} x^{2} + 41 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-63/256*a^5*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/1280*(965*a^4 + 2*(745*a^3*b + 4*(171*a^2*b^2 +
2*(8*b^4*x^2 + 41*a*b^3)*x^2)*x^2)*x^2)*sqrt(b*x^2 + a)*x