3.84 \(\int \frac{c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}+\frac{x (b c-a d)}{a b \sqrt{a+b x^2}} \]

[Out]

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

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Rubi [A]  time = 0.050372, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}+\frac{x (b c-a d)}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)/(a + b*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.09783, size = 46, normalized size = 0.85 \[ \frac{d \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{3}{2}}} - \frac{x \left (a d - b c\right )}{a b \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0630773, size = 58, normalized size = 1.07 \[ \frac{d \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{3/2}}-\frac{x (a d-b c)}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 54, normalized size = 1. \[{\frac{cx}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{dx}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)/(b*x^2+a)^(3/2),x)

[Out]

c*x/a/(b*x^2+a)^(1/2)-d*x/b/(b*x^2+a)^(1/2)+d/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((d*x^2 + c)/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219841, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{b x^{2} + a}{\left (b c - a d\right )} \sqrt{b} x +{\left (a b d x^{2} + a^{2} d\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{b}}, \frac{\sqrt{b x^{2} + a}{\left (b c - a d\right )} \sqrt{-b} x +{\left (a b d x^{2} + a^{2} d\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{{\left (a b^{2} x^{2} + a^{2} b\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(b*x^2 + a)*(b*c - a*d)*sqrt(b)*x + (a*b*d*x^2 + a^2*d)*log(-2*sqrt(
b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((a*b^2*x^2 + a^2*b)*sqrt(b)), (sqrt(b*
x^2 + a)*(b*c - a*d)*sqrt(-b)*x + (a*b*d*x^2 + a^2*d)*arctan(sqrt(-b)*x/sqrt(b*x
^2 + a)))/((a*b^2*x^2 + a^2*b)*sqrt(-b))]

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Sympy [A]  time = 11.1484, size = 60, normalized size = 1.11 \[ d \left (\frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{x}{\sqrt{a} b \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + \frac{c x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22948, size = 68, normalized size = 1.26 \[ -\frac{d{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{3}{2}}} + \frac{{\left (b c - a d\right )} x}{\sqrt{b x^{2} + a} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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