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

Optimal. Leaf size=154 \[ -\frac{21 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2}}+\frac{21 a^5 x \sqrt{a+b x^2}}{1024 b}+\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2} \]

[Out]

(21*a^5*x*Sqrt[a + b*x^2])/(1024*b) + (21*a^4*x^3*Sqrt[a + b*x^2])/512 + (7*a^3*x^3*(a + b*x^2)^(3/2))/128 + (
21*a^2*x^3*(a + b*x^2)^(5/2))/320 + (3*a*x^3*(a + b*x^2)^(7/2))/40 + (x^3*(a + b*x^2)^(9/2))/12 - (21*a^6*ArcT
anh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(3/2))

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Rubi [A]  time = 0.0702928, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ -\frac{21 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2}}+\frac{21 a^5 x \sqrt{a+b x^2}}{1024 b}+\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^(9/2),x]

[Out]

(21*a^5*x*Sqrt[a + b*x^2])/(1024*b) + (21*a^4*x^3*Sqrt[a + b*x^2])/512 + (7*a^3*x^3*(a + b*x^2)^(3/2))/128 + (
21*a^2*x^3*(a + b*x^2)^(5/2))/320 + (3*a*x^3*(a + b*x^2)^(7/2))/40 + (x^3*(a + b*x^2)^(9/2))/12 - (21*a^6*ArcT
anh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(1024*b^(3/2))

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 x^2 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}+\frac{1}{4} (3 a) \int x^2 \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}+\frac{1}{40} \left (21 a^2\right ) \int x^2 \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}+\frac{1}{64} \left (21 a^3\right ) \int x^2 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}+\frac{1}{128} \left (21 a^4\right ) \int x^2 \sqrt{a+b x^2} \, dx\\ &=\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}+\frac{1}{512} \left (21 a^5\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx\\ &=\frac{21 a^5 x \sqrt{a+b x^2}}{1024 b}+\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}-\frac{\left (21 a^6\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{1024 b}\\ &=\frac{21 a^5 x \sqrt{a+b x^2}}{1024 b}+\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}-\frac{\left (21 a^6\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{1024 b}\\ &=\frac{21 a^5 x \sqrt{a+b x^2}}{1024 b}+\frac{21}{512} a^4 x^3 \sqrt{a+b x^2}+\frac{7}{128} a^3 x^3 \left (a+b x^2\right )^{3/2}+\frac{21}{320} a^2 x^3 \left (a+b x^2\right )^{5/2}+\frac{3}{40} a x^3 \left (a+b x^2\right )^{7/2}+\frac{1}{12} x^3 \left (a+b x^2\right )^{9/2}-\frac{21 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.167707, size = 116, normalized size = 0.75 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (12144 a^2 b^3 x^6+11432 a^3 b^2 x^4+4910 a^4 b x^2+315 a^5+6272 a b^4 x^8+1280 b^5 x^{10}\right )-\frac{315 a^{11/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{15360 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(315*a^5 + 4910*a^4*b*x^2 + 11432*a^3*b^2*x^4 + 12144*a^2*b^3*x^6 + 6272*a*b^4*x^8
+ 1280*b^5*x^10) - (315*a^(11/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(15360*b^(3/2))

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Maple [A]  time = 0.007, size = 129, normalized size = 0.8 \begin{align*}{\frac{x}{12\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{ax}{120\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}-{\frac{3\,{a}^{2}x}{320\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{7\,{a}^{3}x}{640\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{a}^{4}x}{512\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{21\,{a}^{5}x}{1024\,b}\sqrt{b{x}^{2}+a}}-{\frac{21\,{a}^{6}}{1024}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x*(b*x^2+a)^(11/2)/b-1/120/b*a*x*(b*x^2+a)^(9/2)-3/320/b*a^2*x*(b*x^2+a)^(7/2)-7/640/b*a^3*x*(b*x^2+a)^(5
/2)-7/512/b*a^4*x*(b*x^2+a)^(3/2)-21/1024*a^5*x*(b*x^2+a)^(1/2)/b-21/1024/b^(3/2)*a^6*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(x^2*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.11596, size = 540, normalized size = 3.51 \begin{align*} \left [\frac{315 \, a^{6} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (1280 \, b^{6} x^{11} + 6272 \, a b^{5} x^{9} + 12144 \, a^{2} b^{4} x^{7} + 11432 \, a^{3} b^{3} x^{5} + 4910 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x\right )} \sqrt{b x^{2} + a}}{30720 \, b^{2}}, \frac{315 \, a^{6} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (1280 \, b^{6} x^{11} + 6272 \, a b^{5} x^{9} + 12144 \, a^{2} b^{4} x^{7} + 11432 \, a^{3} b^{3} x^{5} + 4910 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x\right )} \sqrt{b x^{2} + a}}{15360 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30720*(315*a^6*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(1280*b^6*x^11 + 6272*a*b^5*x^9
+ 12144*a^2*b^4*x^7 + 11432*a^3*b^3*x^5 + 4910*a^4*b^2*x^3 + 315*a^5*b*x)*sqrt(b*x^2 + a))/b^2, 1/15360*(315*a
^6*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (1280*b^6*x^11 + 6272*a*b^5*x^9 + 12144*a^2*b^4*x^7 + 11432*a
^3*b^3*x^5 + 4910*a^4*b^2*x^3 + 315*a^5*b*x)*sqrt(b*x^2 + a))/b^2]

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Sympy [A]  time = 13.8789, size = 204, normalized size = 1.32 \begin{align*} \frac{21 a^{\frac{11}{2}} x}{1024 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1045 a^{\frac{9}{2}} x^{3}}{3072 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{8171 a^{\frac{7}{2}} b x^{5}}{7680 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2947 a^{\frac{5}{2}} b^{2} x^{7}}{1920 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1151 a^{\frac{3}{2}} b^{3} x^{9}}{960 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{59 \sqrt{a} b^{4} x^{11}}{120 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{21 a^{6} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{1024 b^{\frac{3}{2}}} + \frac{b^{5} x^{13}}{12 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21*a**(11/2)*x/(1024*b*sqrt(1 + b*x**2/a)) + 1045*a**(9/2)*x**3/(3072*sqrt(1 + b*x**2/a)) + 8171*a**(7/2)*b*x*
*5/(7680*sqrt(1 + b*x**2/a)) + 2947*a**(5/2)*b**2*x**7/(1920*sqrt(1 + b*x**2/a)) + 1151*a**(3/2)*b**3*x**9/(96
0*sqrt(1 + b*x**2/a)) + 59*sqrt(a)*b**4*x**11/(120*sqrt(1 + b*x**2/a)) - 21*a**6*asinh(sqrt(b)*x/sqrt(a))/(102
4*b**(3/2)) + b**5*x**13/(12*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.47423, size = 142, normalized size = 0.92 \begin{align*} \frac{21 \, a^{6} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{1024 \, b^{\frac{3}{2}}} + \frac{1}{15360} \,{\left (\frac{315 \, a^{5}}{b} + 2 \,{\left (2455 \, a^{4} + 4 \,{\left (1429 \, a^{3} b + 2 \,{\left (759 \, a^{2} b^{2} + 8 \,{\left (10 \, b^{4} x^{2} + 49 \, a b^{3}\right )} x^{2}\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(x^2*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

21/1024*a^6*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(3/2) + 1/15360*(315*a^5/b + 2*(2455*a^4 + 4*(1429*a^3*b
+ 2*(759*a^2*b^2 + 8*(10*b^4*x^2 + 49*a*b^3)*x^2)*x^2)*x^2)*x^2)*sqrt(b*x^2 + a)*x