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

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

[Out]

3/128*a^3*x^5*(b*x^2+a)^(3/2)+3/80*a^2*x^5*(b*x^2+a)^(5/2)+3/56*a*x^5*(b*x^2+a)^(7/2)+1/14*x^5*(b*x^2+a)^(9/2)
+9/2048*a^7*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(5/2)-9/2048*a^6*x*(b*x^2+a)^(1/2)/b^2+3/1024*a^5*x^3*(b*x^2+
a)^(1/2)/b+3/256*a^4*x^5*(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 178, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.19, size = 127, normalized size = 0.71 \[ \frac {\sqrt {a+b x^2} \left (\frac {315 a^{13/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}+\sqrt {b} x \left (-315 a^6+210 a^5 b x^2+14168 a^4 b^2 x^4+39056 a^3 b^3 x^6+44928 a^2 b^4 x^8+24320 a b^5 x^{10}+5120 b^6 x^{12}\right )\right )}{71680 b^{5/2}} \]

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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fricas [A]  time = 1.06, size = 234, normalized size = 1.31 \[ \left [\frac {315 \, a^{7} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt {b x^{2} + a}}{143360 \, b^{3}}, -\frac {315 \, a^{7} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (5120 \, b^{7} x^{13} + 24320 \, a b^{6} x^{11} + 44928 \, a^{2} b^{5} x^{9} + 39056 \, a^{3} b^{4} x^{7} + 14168 \, a^{4} b^{3} x^{5} + 210 \, a^{5} b^{2} x^{3} - 315 \, a^{6} b x\right )} \sqrt {b x^{2} + a}}{71680 \, b^{3}}\right ] \]

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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giac [A]  time = 1.17, size = 119, normalized size = 0.67 \[ -\frac {9 \, a^{7} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2048 \, b^{\frac {5}{2}}} - \frac {1}{71680} \, {\left (\frac {315 \, a^{6}}{b^{2}} - 2 \, {\left (\frac {105 \, a^{5}}{b} + 4 \, {\left (1771 \, a^{4} + 2 \, {\left (2441 \, a^{3} b + 8 \, {\left (351 \, a^{2} b^{2} + 10 \, {\left (4 \, b^{4} x^{2} + 19 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} x \]

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

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maple [A]  time = 0.01, size = 149, normalized size = 0.84 \[ \frac {9 a^{7} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2048 b^{\frac {5}{2}}}+\frac {9 \sqrt {b \,x^{2}+a}\, a^{6} x}{2048 b^{2}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{5} x}{1024 b^{2}}+\frac {3 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{4} x}{1280 b^{2}}+\frac {9 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} x}{4480 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} x^{3}}{14 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {9}{2}} a^{2} x}{560 b^{2}}-\frac {\left (b \,x^{2}+a \right )^{\frac {11}{2}} a x}{56 b^{2}} \]

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*a/b^2*x*(b*x^2+a)^(11/2)+1/560*a^2/b^2*x*(b*x^2+a)^(9/2)+9/4480*a^3/b^2*x*(b*
x^2+a)^(7/2)+3/1280*a^4/b^2*x*(b*x^2+a)^(5/2)+3/1024*a^5/b^2*x*(b*x^2+a)^(3/2)+9/2048*a^6*x*(b*x^2+a)^(1/2)/b^
2+9/2048*a^7/b^(5/2)*ln(b^(1/2)*x+(b*x^2+a)^(1/2))

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maxima [A]  time = 1.40, size = 141, normalized size = 0.79 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} x^{3}}{14 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {11}{2}} a x}{56 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {9}{2}} a^{2} x}{560 \, b^{2}} + \frac {9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x}{4480 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4} x}{1280 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5} x}{1024 \, b^{2}} + \frac {9 \, \sqrt {b x^{2} + a} a^{6} x}{2048 \, b^{2}} + \frac {9 \, a^{7} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2048 \, b^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(b*x^2 + a)^(11/2)*x^3/b - 1/56*(b*x^2 + a)^(11/2)*a*x/b^2 + 1/560*(b*x^2 + a)^(9/2)*a^2*x/b^2 + 9/4480*(
b*x^2 + a)^(7/2)*a^3*x/b^2 + 3/1280*(b*x^2 + a)^(5/2)*a^4*x/b^2 + 3/1024*(b*x^2 + a)^(3/2)*a^5*x/b^2 + 9/2048*
sqrt(b*x^2 + a)*a^6*x/b^2 + 9/2048*a^7*arcsinh(b*x/sqrt(a*b))/b^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^(9/2),x)

[Out]

int(x^4*(a + b*x^2)^(9/2), x)

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sympy [A]  time = 20.00, size = 231, normalized size = 1.30 \[ - \frac {9 a^{\frac {13}{2}} x}{2048 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {11}{2}} x^{3}}{2048 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {1027 a^{\frac {9}{2}} x^{5}}{5120 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {6653 a^{\frac {7}{2}} b x^{7}}{8960 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5249 a^{\frac {5}{2}} b^{2} x^{9}}{4480 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {541 a^{\frac {3}{2}} b^{3} x^{11}}{560 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{4} x^{13}}{56 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {9 a^{7} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2048 b^{\frac {5}{2}}} + \frac {b^{5} x^{15}}{14 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]

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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