3.92 \(\int \frac{c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{2 c x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]

[Out]

(2*c*x)/(3*a^2*Sqrt[a + b*x^2]) + (x*(c + d*x^2))/(3*a*(a + b*x^2)^(3/2))

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Rubi [A]  time = 0.036464, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 c x}{3 a^2 \sqrt{a+b x^2}}+\frac{x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/(a + b*x^2)^(5/2),x]

[Out]

(2*c*x)/(3*a^2*Sqrt[a + b*x^2]) + (x*(c + d*x^2))/(3*a*(a + b*x^2)^(3/2))

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Rubi in Sympy [A]  time = 6.51894, size = 41, normalized size = 0.87 \[ \frac{x \left (c + d x^{2}\right )}{3 a \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{2 c x}{3 a^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x**2)/(3*a*(a + b*x**2)**(3/2)) + 2*c*x/(3*a**2*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.0429686, size = 37, normalized size = 0.79 \[ \frac{x \left (3 a c+a d x^2+2 b c x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)/(a + b*x^2)^(5/2),x]

[Out]

(x*(3*a*c + 2*b*c*x^2 + a*d*x^2))/(3*a^2*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.005, size = 34, normalized size = 0.7 \[{\frac{x \left ( ad{x}^{2}+2\,c{x}^{2}b+3\,ac \right ) }{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(b*x^2+a)^(5/2),x)

[Out]

1/3*x*(a*d*x^2+2*b*c*x^2+3*a*c)/(b*x^2+a)^(3/2)/a^2

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Maxima [A]  time = 1.34891, size = 92, normalized size = 1.96 \[ \frac{2 \, c x}{3 \, \sqrt{b x^{2} + a} a^{2}} + \frac{c x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{d x}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} + \frac{d x}{3 \, \sqrt{b x^{2} + a} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*c*x/(sqrt(b*x^2 + a)*a^2) + 1/3*c*x/((b*x^2 + a)^(3/2)*a) - 1/3*d*x/((b*x^2
+ a)^(3/2)*b) + 1/3*d*x/(sqrt(b*x^2 + a)*a*b)

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Fricas [A]  time = 0.213794, size = 73, normalized size = 1.55 \[ \frac{{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{b x^{2} + a}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((2*b*c + a*d)*x^3 + 3*a*c*x)*sqrt(b*x^2 + a)/(a^2*b^2*x^4 + 2*a^3*b*x^2 + a
^4)

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Sympy [A]  time = 36.3887, size = 144, normalized size = 3.06 \[ c \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{5}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + \frac{d x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 3 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)/(b*x**2+a)**(5/2),x)

[Out]

c*(3*a*x/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a))
+ 2*b*x**3/(3*a**(7/2)*sqrt(1 + b*x**2/a) + 3*a**(5/2)*b*x**2*sqrt(1 + b*x**2/a)
)) + d*x**3/(3*a**(5/2)*sqrt(1 + b*x**2/a) + 3*a**(3/2)*b*x**2*sqrt(1 + b*x**2/a
))

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GIAC/XCAS [A]  time = 0.225743, size = 54, normalized size = 1.15 \[ \frac{x{\left (\frac{3 \, c}{a} + \frac{{\left (2 \, b^{2} c + a b d\right )} x^{2}}{a^{2} b}\right )}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x*(3*c/a + (2*b^2*c + a*b*d)*x^2/(a^2*b))/(b*x^2 + a)^(3/2)