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

Optimal. Leaf size=91 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2} \]

[Out]

(a^2*x*Sqrt[a + b*x^2])/(16*b) + (a*x^3*Sqrt[a + b*x^2])/8 + (x^3*(a + b*x^2)^(3
/2))/6 - (a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(3/2))

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Rubi [A]  time = 0.101538, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{3/2}}+\frac{a^2 x \sqrt{a+b x^2}}{16 b}+\frac{1}{8} a x^3 \sqrt{a+b x^2}+\frac{1}{6} x^3 \left (a+b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^2)^(3/2),x]

[Out]

(a^2*x*Sqrt[a + b*x^2])/(16*b) + (a*x^3*Sqrt[a + b*x^2])/8 + (x^3*(a + b*x^2)^(3
/2))/6 - (a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(3/2))

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Rubi in Sympy [A]  time = 12.8823, size = 78, normalized size = 0.86 \[ - \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{3}{2}}} + \frac{a^{2} x \sqrt{a + b x^{2}}}{16 b} + \frac{a x^{3} \sqrt{a + b x^{2}}}{8} + \frac{x^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(16*b**(3/2)) + a**2*x*sqrt(a + b*x**2)/
(16*b) + a*x**3*sqrt(a + b*x**2)/8 + x**3*(a + b*x**2)**(3/2)/6

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Mathematica [A]  time = 0.0564863, size = 75, normalized size = 0.82 \[ \sqrt{a+b x^2} \left (\frac{a^2 x}{16 b}+\frac{7 a x^3}{24}+\frac{b x^5}{6}\right )-\frac{a^3 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{16 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x^2]*((a^2*x)/(16*b) + (7*a*x^3)/24 + (b*x^5)/6) - (a^3*Log[b*x + Sqr
t[b]*Sqrt[a + b*x^2]])/(16*b^(3/2))

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Maple [A]  time = 0.008, size = 75, normalized size = 0.8 \[{\frac{x}{6\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{24\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}x}{16\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(3/2),x)

[Out]

1/6*x*(b*x^2+a)^(5/2)/b-1/24*a/b*x*(b*x^2+a)^(3/2)-1/16*a^2*x*(b*x^2+a)^(1/2)/b-
1/16*a^3/b^(3/2)*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255876, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{5} + 14 \, a b x^{3} + 3 \, a^{2} x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{96 \, b^{\frac{3}{2}}}, -\frac{3 \, a^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, b^{2} x^{5} + 14 \, a b x^{3} + 3 \, a^{2} x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{48 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*a^3*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(8*b^2*x^5 +
14*a*b*x^3 + 3*a^2*x)*sqrt(b*x^2 + a)*sqrt(b))/b^(3/2), -1/48*(3*a^3*arctan(sqr
t(-b)*x/sqrt(b*x^2 + a)) - (8*b^2*x^5 + 14*a*b*x^3 + 3*a^2*x)*sqrt(b*x^2 + a)*sq
rt(-b))/(sqrt(-b)*b)]

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Sympy [A]  time = 17.1265, size = 119, normalized size = 1.31 \[ \frac{a^{\frac{5}{2}} x}{16 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{11 \sqrt{a} b x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{3}{2}}} + \frac{b^{2} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*x/(16*b*sqrt(1 + b*x**2/a)) + 17*a**(3/2)*x**3/(48*sqrt(1 + b*x**2/a))
+ 11*sqrt(a)*b*x**5/(24*sqrt(1 + b*x**2/a)) - a**3*asinh(sqrt(b)*x/sqrt(a))/(16*
b**(3/2)) + b**2*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.21102, size = 85, normalized size = 0.93 \[ \frac{1}{48} \,{\left (2 \,{\left (4 \, b x^{2} + 7 \, a\right )} x^{2} + \frac{3 \, a^{2}}{b}\right )} \sqrt{b x^{2} + a} x + \frac{a^{3}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*b*x^2 + 7*a)*x^2 + 3*a^2/b)*sqrt(b*x^2 + a)*x + 1/16*a^3*ln(abs(-sqrt
(b)*x + sqrt(b*x^2 + a)))/b^(3/2)