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

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 = 0.0172689, 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, Rules used = {385, 217, 206} \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}+\frac{x (b c-a d)}{a b \sqrt{a+b x^2}} \]

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)

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{(b c-a d) x}{a b \sqrt{a+b x^2}}+\frac{d \int \frac{1}{\sqrt{a+b x^2}} \, dx}{b}\\ &=\frac{(b c-a d) x}{a b \sqrt{a+b x^2}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{b}\\ &=\frac{(b c-a d) x}{a b \sqrt{a+b x^2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0575813, size = 70, normalized size = 1.3 \[ \frac{a^{3/2} d \sqrt{\frac{b x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{b} x (b c-a d)}{a b^{3/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*(b*c - a*d)*x + a^(3/2)*d*Sqrt[1 + (b*x^2)/a]*ArcSinh[(Sqrt[b]*x)/Sqrt[a]])/(a*b^(3/2)*Sqrt[a + b*x^2
])

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Maple [A]  time = 0.004, size = 54, normalized size = 1. \begin{align*} -{\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}}}}+{\frac{cx}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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 = 1.545, size = 367, normalized size = 6.8 \begin{align*} \left [\frac{2 \,{\left (b^{2} c - a b d\right )} \sqrt{b x^{2} + a} x +{\left (a b d x^{2} + a^{2} d\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right )}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, \frac{{\left (b^{2} c - a b d\right )} \sqrt{b x^{2} + a} x -{\left (a b d x^{2} + a^{2} d\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{a b^{3} x^{2} + a^{2} b^{2}}\right ] \end{align*}

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*(b^2*c - a*b*d)*sqrt(b*x^2 + a)*x + (a*b*d*x^2 + a^2*d)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(
b)*x - a))/(a*b^3*x^2 + a^2*b^2), ((b^2*c - a*b*d)*sqrt(b*x^2 + a)*x - (a*b*d*x^2 + a^2*d)*sqrt(-b)*arctan(sqr
t(-b)*x/sqrt(b*x^2 + a)))/(a*b^3*x^2 + a^2*b^2)]

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Sympy [A]  time = 3.70417, size = 60, normalized size = 1.11 \begin{align*} 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}}} \end{align*}

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 [A]  time = 1.1755, size = 68, normalized size = 1.26 \begin{align*} -\frac{d \log \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} \end{align*}

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