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

Optimal. Leaf size=34 \[ -\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0268717, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {444, 37} \[ -\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0099777, size = 33, normalized size = 0.97 \[ \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 0.9 \begin{align*}{\frac{1}{ad-bc}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8887, size = 97, normalized size = 2.85 \begin{align*} -\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.18841, size = 95, normalized size = 2.79 \begin{align*} -\frac{2 \, \sqrt{b d} b}{{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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