3.399 \(\int \left (a+b x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=84 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2} \]

[Out]

(5*a^2*x*Sqrt[a + b*x^2])/16 + (5*a*x*(a + b*x^2)^(3/2))/24 + (x*(a + b*x^2)^(5/
2))/6 + (5*a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])

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Rubi [A]  time = 0.0609324, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{16} a^2 x \sqrt{a+b x^2}+\frac{5}{24} a x \left (a+b x^2\right )^{3/2}+\frac{1}{6} x \left (a+b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2),x]

[Out]

(5*a^2*x*Sqrt[a + b*x^2])/16 + (5*a*x*(a + b*x^2)^(3/2))/24 + (x*(a + b*x^2)^(5/
2))/6 + (5*a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*Sqrt[b])

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Rubi in Sympy [A]  time = 5.75486, size = 78, normalized size = 0.93 \[ \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 \sqrt{b}} + \frac{5 a^{2} x \sqrt{a + b x^{2}}}{16} + \frac{5 a x \left (a + b x^{2}\right )^{\frac{3}{2}}}{24} + \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2),x)

[Out]

5*a**3*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(16*sqrt(b)) + 5*a**2*x*sqrt(a + b*x**2
)/16 + 5*a*x*(a + b*x**2)**(3/2)/24 + x*(a + b*x**2)**(5/2)/6

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Mathematica [A]  time = 0.0768996, size = 71, normalized size = 0.85 \[ \frac{1}{48} \left (\frac{15 a^3 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{\sqrt{b}}+x \sqrt{a+b x^2} \left (33 a^2+26 a b x^2+8 b^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2),x]

[Out]

(x*Sqrt[a + b*x^2]*(33*a^2 + 26*a*b*x^2 + 8*b^2*x^4) + (15*a^3*Log[b*x + Sqrt[b]
*Sqrt[a + b*x^2]])/Sqrt[b])/48

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Maple [A]  time = 0.004, size = 66, normalized size = 0.8 \[{\frac{x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,ax}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257092, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{5} + 26 \, a b x^{3} + 33 \, a^{2} x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{96 \, \sqrt{b}}, \frac{15 \, a^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, b^{2} x^{5} + 26 \, a b x^{3} + 33 \, a^{2} x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{48 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

[1/96*(15*a^3*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(8*b^2*x^5
+ 26*a*b*x^3 + 33*a^2*x)*sqrt(b*x^2 + a)*sqrt(b))/sqrt(b), 1/48*(15*a^3*arctan(
sqrt(-b)*x/sqrt(b*x^2 + a)) + (8*b^2*x^5 + 26*a*b*x^3 + 33*a^2*x)*sqrt(b*x^2 + a
)*sqrt(-b))/sqrt(-b)]

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Sympy [A]  time = 14.4441, size = 97, normalized size = 1.15 \[ \frac{11 a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{16} + \frac{13 a^{\frac{3}{2}} b x^{3} \sqrt{1 + \frac{b x^{2}}{a}}}{24} + \frac{\sqrt{a} b^{2} x^{5} \sqrt{1 + \frac{b x^{2}}{a}}}{6} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2),x)

[Out]

11*a**(5/2)*x*sqrt(1 + b*x**2/a)/16 + 13*a**(3/2)*b*x**3*sqrt(1 + b*x**2/a)/24 +
sqrt(a)*b**2*x**5*sqrt(1 + b*x**2/a)/6 + 5*a**3*asinh(sqrt(b)*x/sqrt(a))/(16*sq
rt(b))

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GIAC/XCAS [A]  time = 0.212144, size = 85, normalized size = 1.01 \[ -\frac{5 \, a^{3}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{48} \,{\left (2 \,{\left (4 \, b^{2} x^{2} + 13 \, a b\right )} x^{2} + 33 \, a^{2}\right )} \sqrt{b x^{2} + a} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

-5/16*a^3*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/48*(2*(4*b^2*x^2 + 1
3*a*b)*x^2 + 33*a^2)*sqrt(b*x^2 + a)*x