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

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]

(-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 = 0.25589, 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 \[ \frac{9 a^7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2048 b^{5/2}}-\frac{9 a^6 x \sqrt{a+b x^2}}{2048 b^2}+\frac{3 a^5 x^3 \sqrt{a+b x^2}}{1024 b}+\frac{3}{256} a^4 x^5 \sqrt{a+b x^2}+\frac{3}{128} a^3 x^5 \left (a+b x^2\right )^{3/2}+\frac{3}{80} a^2 x^5 \left (a+b x^2\right )^{5/2}+\frac{3}{56} a x^5 \left (a+b x^2\right )^{7/2}+\frac{1}{14} x^5 \left (a+b x^2\right )^{9/2} \]

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))

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Rubi in Sympy [A]  time = 31.0129, size = 168, normalized size = 0.94 \[ \frac{9 a^{7} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2048 b^{\frac{5}{2}}} - \frac{9 a^{6} x \sqrt{a + b x^{2}}}{2048 b^{2}} + \frac{3 a^{5} x^{3} \sqrt{a + b x^{2}}}{1024 b} + \frac{3 a^{4} x^{5} \sqrt{a + b x^{2}}}{256} + \frac{3 a^{3} x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}}{128} + \frac{3 a^{2} x^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}}{80} + \frac{3 a x^{5} \left (a + b x^{2}\right )^{\frac{7}{2}}}{56} + \frac{x^{5} \left (a + b x^{2}\right )^{\frac{9}{2}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**(9/2),x)

[Out]

9*a**7*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(2048*b**(5/2)) - 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

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Mathematica [A]  time = 0.0982156, size = 120, normalized size = 0.67 \[ \frac{315 a^7 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} x \sqrt{a+b x^2} \left (-315 a^6+210 a^5 b x^2+14168 a^4 b^2 x^4+39056 a^3 b^3 x^6+44928 a^2 b^4 x^8+24320 a b^5 x^{10}+5120 b^6 x^{12}\right )}{71680 b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^2)^(9/2),x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-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^7*L
og[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(71680*b^(5/2))

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Maple [A]  time = 0.012, size = 149, normalized size = 0.8 \[{\frac{{x}^{3}}{14\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}-{\frac{ax}{56\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}}+{\frac{{a}^{2}x}{560\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{9}{2}}}}+{\frac{9\,{a}^{3}x}{4480\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{4}x}{1280\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,{a}^{5}x}{1024\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{9\,{a}^{6}x}{2048\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,{a}^{7}}{2048}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

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)+3/10
24*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(x*b^(1/2)+(b*x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.490239, size = 1, normalized size = 0.01 \[ \left [\frac{315 \, a^{7} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (5120 \, b^{6} x^{13} + 24320 \, a b^{5} x^{11} + 44928 \, a^{2} b^{4} x^{9} + 39056 \, a^{3} b^{3} x^{7} + 14168 \, a^{4} b^{2} x^{5} + 210 \, a^{5} b x^{3} - 315 \, a^{6} x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{143360 \, b^{\frac{5}{2}}}, \frac{315 \, a^{7} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (5120 \, b^{6} x^{13} + 24320 \, a b^{5} x^{11} + 44928 \, a^{2} b^{4} x^{9} + 39056 \, a^{3} b^{3} x^{7} + 14168 \, a^{4} b^{2} x^{5} + 210 \, a^{5} b x^{3} - 315 \, a^{6} x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{71680 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/143360*(315*a^7*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(5120
*b^6*x^13 + 24320*a*b^5*x^11 + 44928*a^2*b^4*x^9 + 39056*a^3*b^3*x^7 + 14168*a^4
*b^2*x^5 + 210*a^5*b*x^3 - 315*a^6*x)*sqrt(b*x^2 + a)*sqrt(b))/b^(5/2), 1/71680*
(315*a^7*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (5120*b^6*x^13 + 24320*a*b^5*x^11
+ 44928*a^2*b^4*x^9 + 39056*a^3*b^3*x^7 + 14168*a^4*b^2*x^5 + 210*a^5*b*x^3 - 31
5*a^6*x)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 74.3441, size = 231, normalized size = 1.3 \[ - \frac{9 a^{\frac{13}{2}} x}{2048 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{11}{2}} x^{3}}{2048 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{1027 a^{\frac{9}{2}} x^{5}}{5120 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{6653 a^{\frac{7}{2}} b x^{7}}{8960 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5249 a^{\frac{5}{2}} b^{2} x^{9}}{4480 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{541 a^{\frac{3}{2}} b^{3} x^{11}}{560 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 \sqrt{a} b^{4} x^{13}}{56 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{9 a^{7} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2048 b^{\frac{5}{2}}} + \frac{b^{5} x^{15}}{14 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

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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GIAC/XCAS [A]  time = 0.213462, size = 161, normalized size = 0.9 \[ -\frac{9 \, a^{7}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2048 \, b^{\frac{5}{2}}} - \frac{1}{71680} \,{\left (\frac{315 \, a^{6}}{b^{2}} - 2 \,{\left (\frac{105 \, a^{5}}{b} + 4 \,{\left (1771 \, a^{4} + 2 \,{\left (2441 \, a^{3} b + 8 \,{\left (351 \, a^{2} b^{2} + 10 \,{\left (4 \, b^{4} x^{2} + 19 \, a b^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(9/2)*x^4,x, algorithm="giac")

[Out]

-9/2048*a^7*ln(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