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

Optimal. Leaf size=178 \[ -\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2} \]

[Out]

(-9*a^6*x*Sqrt[a + b*x^2])/(2048*b^2) + (3*a^5*x^3*Sqrt[a + b*x^2])/(1024*b) + (3*a^4*x^5*Sqrt[a + b*x^2])/256
 + (3*a^3*x^5*(a + b*x^2)^(3/2))/128 + (3*a^2*x^5*(a + b*x^2)^(5/2))/80 + (3*a*x^5*(a + b*x^2)^(7/2))/56 + (x^
5*(a + b*x^2)^(9/2))/14 + (9*a^7*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2048*b^(5/2))

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Rubi [A]  time = 0.0855914, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ -\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^(9/2),x]

[Out]

(-9*a^6*x*Sqrt[a + b*x^2])/(2048*b^2) + (3*a^5*x^3*Sqrt[a + b*x^2])/(1024*b) + (3*a^4*x^5*Sqrt[a + b*x^2])/256
 + (3*a^3*x^5*(a + b*x^2)^(3/2))/128 + (3*a^2*x^5*(a + b*x^2)^(5/2))/80 + (3*a*x^5*(a + b*x^2)^(7/2))/56 + (x^
5*(a + b*x^2)^(9/2))/14 + (9*a^7*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2048*b^(5/2))

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^4 \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{14} (9 a) \int x^4 \left (a+b x^2\right )^{7/2} \, dx\\ &=\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{8} \left (3 a^2\right ) \int x^4 \left (a+b x^2\right )^{5/2} \, dx\\ &=\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{16} \left (3 a^3\right ) \int x^4 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{128} \left (9 a^4\right ) \int x^4 \sqrt{a+b x^2} \, dx\\ &=\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{1}{256} \left (3 a^5\right ) \int \frac{x^4}{\sqrt{a+b x^2}} \, dx\\ &=\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}-\frac{\left (9 a^6\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{1024 b}\\ &=-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{\left (9 a^7\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{2048 b^2}\\ &=-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{\left (9 a^7\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{2048 b^2}\\ &=-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2}+\frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.184786, size = 127, normalized size = 0.71 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (44928 a^2 b^4 x^8+39056 a^3 b^3 x^6+14168 a^4 b^2 x^4+210 a^5 b x^2-315 a^6+24320 a b^5 x^{10}+5120 b^6 x^{12}\right )+\frac{315 a^{13/2} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{71680 b^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[a + b*x^2]*(Sqrt[b]*x*(-315*a^6 + 210*a^5*b*x^2 + 14168*a^4*b^2*x^4 + 39056*a^3*b^3*x^6 + 44928*a^2*b^4*
x^8 + 24320*a*b^5*x^10 + 5120*b^6*x^12) + (315*a^(13/2)*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[1 + (b*x^2)/a]))/(7
1680*b^(5/2))

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Maple [A]  time = 0.008, size = 149, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{14\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{ax}{56\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{a}^{2}x}{560\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{a}^{3}x}{4480\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{4}x}{1280\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{5}x}{1024\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{6}x}{2048\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,{a}^{7}}{2048}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^3*(b*x^2+a)^(11/2)/b-1/56/b^2*a*x*(b*x^2+a)^(11/2)+1/560/b^2*a^2*x*(b*x^2+a)^(9/2)+9/4480/b^2*a^3*x*(b*
x^2+a)^(7/2)+3/1280/b^2*a^4*x*(b*x^2+a)^(5/2)+3/1024/b^2*a^5*x*(b*x^2+a)^(3/2)+9/2048*a^6*x*(b*x^2+a)^(1/2)/b^
2+9/2048/b^(5/2)*a^7*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^4*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.20406, size = 599, normalized size = 3.37 \begin{align*} \left [\frac{315 \, a^{7} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt{b x^{2} + a}}{143360 \, b^{3}}, -\frac{315 \, a^{7} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt{b x^{2} + a}}{71680 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/143360*(315*a^7*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(5120*b^7*x^13 + 24320*a*b^6*x^
11 + 44928*a^2*b^5*x^9 + 39056*a^3*b^4*x^7 + 14168*a^4*b^3*x^5 + 210*a^5*b^2*x^3 - 315*a^6*b*x)*sqrt(b*x^2 + a
))/b^3, -1/71680*(315*a^7*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (5120*b^7*x^13 + 24320*a*b^6*x^11 + 44
928*a^2*b^5*x^9 + 39056*a^3*b^4*x^7 + 14168*a^4*b^3*x^5 + 210*a^5*b^2*x^3 - 315*a^6*b*x)*sqrt(b*x^2 + a))/b^3]

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Sympy [A]  time = 19.973, size = 231, normalized size = 1.3 \begin{align*} - \frac{9 a^{\frac{13}{2}} x}{2048 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{11}{2}} x^{3}}{2048 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1027 a^{\frac{9}{2}} x^{5}}{5120 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{6653 a^{\frac{7}{2}} b x^{7}}{8960 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5249 a^{\frac{5}{2}} b^{2} x^{9}}{4480 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{541 a^{\frac{3}{2}} b^{3} x^{11}}{560 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{4} x^{13}}{56 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{9 a^{7} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2048 b^{\frac{5}{2}}} + \frac{b^{5} x^{15}}{14 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*a**(13/2)*x/(2048*b**2*sqrt(1 + b*x**2/a)) - 3*a**(11/2)*x**3/(2048*b*sqrt(1 + b*x**2/a)) + 1027*a**(9/2)*x
**5/(5120*sqrt(1 + b*x**2/a)) + 6653*a**(7/2)*b*x**7/(8960*sqrt(1 + b*x**2/a)) + 5249*a**(5/2)*b**2*x**9/(4480
*sqrt(1 + b*x**2/a)) + 541*a**(3/2)*b**3*x**11/(560*sqrt(1 + b*x**2/a)) + 23*sqrt(a)*b**4*x**13/(56*sqrt(1 + b
*x**2/a)) + 9*a**7*asinh(sqrt(b)*x/sqrt(a))/(2048*b**(5/2)) + b**5*x**15/(14*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.98651, size = 161, normalized size = 0.9 \begin{align*} -\frac{9 \, a^{7} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2048 \, b^{\frac{5}{2}}} - \frac{1}{71680} \,{\left (\frac{315 \, a^{6}}{b^{2}} - 2 \,{\left (\frac{105 \, a^{5}}{b} + 4 \,{\left (1771 \, a^{4} + 2 \,{\left (2441 \, a^{3} b + 8 \,{\left (351 \, a^{2} b^{2} + 10 \,{\left (4 \, b^{4} x^{2} + 19 \, a b^{3}\right )} x^{2}\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^4*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-9/2048*a^7*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2) - 1/71680*(315*a^6/b^2 - 2*(105*a^5/b + 4*(1771*a^4
 + 2*(2441*a^3*b + 8*(351*a^2*b^2 + 10*(4*b^4*x^2 + 19*a*b^3)*x^2)*x^2)*x^2)*x^2)*x^2)*sqrt(b*x^2 + a)*x