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

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

[Out]

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

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Rubi [A]  time = 0.0536519, 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 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 \sqrt{c}}+\frac{5}{16} a^2 x \sqrt{a+c x^2}+\frac{5}{24} a x \left (a+c x^2\right )^{3/2}+\frac{1}{6} x \left (a+c x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.83583, size = 78, normalized size = 0.93 \[ \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{16 \sqrt{c}} + \frac{5 a^{2} x \sqrt{a + c x^{2}}}{16} + \frac{5 a x \left (a + c x^{2}\right )^{\frac{3}{2}}}{24} + \frac{x \left (a + c x^{2}\right )^{\frac{5}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0811618, size = 71, normalized size = 0.85 \[ \frac{1}{48} \left (\frac{15 a^3 \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}+x \sqrt{a+c x^2} \left (33 a^2+26 a c x^2+8 c^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 66, normalized size = 0.8 \[{\frac{x}{6} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,ax}{24} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{16}\sqrt{c{x}^{2}+a}}+{\frac{5\,{a}^{3}}{16}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248609, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8 \, c^{2} x^{5} + 26 \, a c x^{3} + 33 \, a^{2} x\right )} \sqrt{c x^{2} + a} \sqrt{c}}{96 \, \sqrt{c}}, \frac{15 \, a^{3} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, c^{2} x^{5} + 26 \, a c x^{3} + 33 \, a^{2} x\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{48 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.3432, size = 97, normalized size = 1.15 \[ \frac{11 a^{\frac{5}{2}} x \sqrt{1 + \frac{c x^{2}}{a}}}{16} + \frac{13 a^{\frac{3}{2}} c x^{3} \sqrt{1 + \frac{c x^{2}}{a}}}{24} + \frac{\sqrt{a} c^{2} x^{5} \sqrt{1 + \frac{c x^{2}}{a}}}{6} + \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215024, size = 85, normalized size = 1.01 \[ -\frac{5 \, a^{3}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, \sqrt{c}} + \frac{1}{48} \,{\left (2 \,{\left (4 \, c^{2} x^{2} + 13 \, a c\right )} x^{2} + 33 \, a^{2}\right )} \sqrt{c x^{2} + a} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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