3.378 \(\int x^4 \left (a+b x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=115 \[ \frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2} \]

[Out]

(-3*a^3*x*Sqrt[a + b*x^2])/(128*b^2) + (a^2*x^3*Sqrt[a + b*x^2])/(64*b) + (a*x^5
*Sqrt[a + b*x^2])/16 + (x^5*(a + b*x^2)^(3/2))/8 + (3*a^4*ArcTanh[(Sqrt[b]*x)/Sq
rt[a + b*x^2]])/(128*b^(5/2))

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Rubi [A]  time = 0.136401, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{3 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{5/2}}-\frac{3 a^3 x \sqrt{a+b x^2}}{128 b^2}+\frac{a^2 x^3 \sqrt{a+b x^2}}{64 b}+\frac{1}{8} x^5 \left (a+b x^2\right )^{3/2}+\frac{1}{16} a x^5 \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^2)^(3/2),x]

[Out]

(-3*a^3*x*Sqrt[a + b*x^2])/(128*b^2) + (a^2*x^3*Sqrt[a + b*x^2])/(64*b) + (a*x^5
*Sqrt[a + b*x^2])/16 + (x^5*(a + b*x^2)^(3/2))/8 + (3*a^4*ArcTanh[(Sqrt[b]*x)/Sq
rt[a + b*x^2]])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 16.9259, size = 104, normalized size = 0.9 \[ \frac{3 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{128 b^{\frac{5}{2}}} - \frac{3 a^{3} x \sqrt{a + b x^{2}}}{128 b^{2}} + \frac{a^{2} x^{3} \sqrt{a + b x^{2}}}{64 b} + \frac{a x^{5} \sqrt{a + b x^{2}}}{16} + \frac{x^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**(3/2),x)

[Out]

3*a**4*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(128*b**(5/2)) - 3*a**3*x*sqrt(a + b*x*
*2)/(128*b**2) + a**2*x**3*sqrt(a + b*x**2)/(64*b) + a*x**5*sqrt(a + b*x**2)/16
+ x**5*(a + b*x**2)**(3/2)/8

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Mathematica [A]  time = 0.0731491, size = 88, normalized size = 0.77 \[ \frac{3 a^4 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{128 b^{5/2}}+\sqrt{a+b x^2} \left (-\frac{3 a^3 x}{128 b^2}+\frac{a^2 x^3}{64 b}+\frac{3 a x^5}{16}+\frac{b x^7}{8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^2)^(3/2),x]

[Out]

Sqrt[a + b*x^2]*((-3*a^3*x)/(128*b^2) + (a^2*x^3)/(64*b) + (3*a*x^5)/16 + (b*x^7
)/8) + (3*a^4*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(128*b^(5/2))

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Maple [A]  time = 0.01, size = 95, normalized size = 0.8 \[{\frac{{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}x}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^(3/2),x)

[Out]

1/8*x^3*(b*x^2+a)^(5/2)/b-1/16*a/b^2*x*(b*x^2+a)^(5/2)+1/64*a^2/b^2*x*(b*x^2+a)^
(3/2)+3/128*a^3*x*(b*x^2+a)^(1/2)/b^2+3/128*a^4/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)^(3/2)*x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265435, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{4} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (16 \, b^{3} x^{7} + 24 \, a b^{2} x^{5} + 2 \, a^{2} b x^{3} - 3 \, a^{3} x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{256 \, b^{\frac{5}{2}}}, \frac{3 \, a^{4} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (16 \, b^{3} x^{7} + 24 \, a b^{2} x^{5} + 2 \, a^{2} b x^{3} - 3 \, a^{3} x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{128 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^4,x, algorithm="fricas")

[Out]

[1/256*(3*a^4*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(16*b^3*x^
7 + 24*a*b^2*x^5 + 2*a^2*b*x^3 - 3*a^3*x)*sqrt(b*x^2 + a)*sqrt(b))/b^(5/2), 1/12
8*(3*a^4*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (16*b^3*x^7 + 24*a*b^2*x^5 + 2*a^2
*b*x^3 - 3*a^3*x)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*b^2)]

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Sympy [A]  time = 26.197, size = 148, normalized size = 1.29 \[ - \frac{3 a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)**(3/2),x)

[Out]

-3*a**(7/2)*x/(128*b**2*sqrt(1 + b*x**2/a)) - a**(5/2)*x**3/(128*b*sqrt(1 + b*x*
*2/a)) + 13*a**(3/2)*x**5/(64*sqrt(1 + b*x**2/a)) + 5*sqrt(a)*b*x**7/(16*sqrt(1
+ b*x**2/a)) + 3*a**4*asinh(sqrt(b)*x/sqrt(a))/(128*b**(5/2)) + b**2*x**9/(8*sqr
t(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.215324, size = 103, normalized size = 0.9 \[ \frac{1}{128} \,{\left (2 \,{\left (4 \,{\left (2 \, b x^{2} + 3 \, a\right )} x^{2} + \frac{a^{2}}{b}\right )} x^{2} - \frac{3 \, a^{3}}{b^{2}}\right )} \sqrt{b x^{2} + a} x - \frac{3 \, a^{4}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^4,x, algorithm="giac")

[Out]

1/128*(2*(4*(2*b*x^2 + 3*a)*x^2 + a^2/b)*x^2 - 3*a^3/b^2)*sqrt(b*x^2 + a)*x - 3/
128*a^4*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)