3.589 \(\int \frac{a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=24 \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]

[Out]

x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]

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Rubi [A]  time = 0.0525376, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022, Rules used = {1590} \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + 2*(b*c + a*d)*x^2 + 3*b*d*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

\begin{align*} \int \frac{a c+2 (b c+a d) x^2+3 b d x^4}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx &=x \sqrt{a+b x^2} \sqrt{c+d x^2}\\ \end{align*}

Mathematica [A]  time = 0.148485, size = 24, normalized size = 1. \[ x \sqrt{a+b x^2} \sqrt{c+d x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + 2*(b*c + a*d)*x^2 + 3*b*d*x^4)/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]

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Maple [A]  time = 0.009, size = 21, normalized size = 0.9 \begin{align*} x\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10455, size = 27, normalized size = 1.12 \begin{align*} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+2*(a*d+b*c)*x^2+3*b*d*x^4)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x

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Fricas [A]  time = 1.61799, size = 47, normalized size = 1.96 \begin{align*} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+2*(a*d+b*c)*x^2+3*b*d*x^4)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a c + 2 a d x^{2} + 2 b c x^{2} + 3 b d x^{4}}{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+2*(a*d+b*c)*x**2+3*b*d*x**4)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral((a*c + 2*a*d*x**2 + 2*b*c*x**2 + 3*b*d*x**4)/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 \, b d x^{4} + 2 \,{\left (b c + a d\right )} x^{2} + a c}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+2*(a*d+b*c)*x^2+3*b*d*x^4)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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