3.979 \(\int \frac{1}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=307 \[ -\frac{b \sqrt{d} \sqrt{a+b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{3 a^2 c^2 x}-\frac{2 d x \sqrt{a+b x^2} (a d+b c)}{3 a^2 c^2 \sqrt{c+d x^2}}+\frac{2 \sqrt{d} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3} \]

[Out]

(-2*d*(b*c + a*d)*x*Sqrt[a + b*x^2])/(3*a^2*c^2*Sqrt[c + d*x^2]) - (Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*a*c*x^
3) + (2*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*a^2*c^2*x) + (2*Sqrt[d]*(b*c + a*d)*Sqrt[a + b*x^2]*El
lipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*c^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sq
rt[c + d*x^2]) - (b*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*Sq
rt[c]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.2505, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {480, 583, 531, 418, 492, 411} \[ \frac{2 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{3 a^2 c^2 x}-\frac{2 d x \sqrt{a+b x^2} (a d+b c)}{3 a^2 c^2 \sqrt{c+d x^2}}+\frac{2 \sqrt{d} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

(-2*d*(b*c + a*d)*x*Sqrt[a + b*x^2])/(3*a^2*c^2*Sqrt[c + d*x^2]) - (Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*a*c*x^
3) + (2*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*a^2*c^2*x) + (2*Sqrt[d]*(b*c + a*d)*Sqrt[a + b*x^2]*El
lipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*c^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sq
rt[c + d*x^2]) - (b*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*Sq
rt[c]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

Rule 480

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*e*(m + 1)), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{1}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx &=-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3}+\frac{\int \frac{-2 (b c+a d)-b d x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a c}\\ &=-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3}+\frac{2 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a^2 c^2 x}-\frac{\int \frac{a b c d+2 b d (b c+a d) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3}+\frac{2 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a^2 c^2 x}-\frac{(b d) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a c}-\frac{(2 b d (b c+a d)) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{3 a^2 c^2}\\ &=-\frac{2 d (b c+a d) x \sqrt{a+b x^2}}{3 a^2 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3}+\frac{2 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a^2 c^2 x}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{(2 d (b c+a d)) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=-\frac{2 d (b c+a d) x \sqrt{a+b x^2}}{3 a^2 c^2 \sqrt{c+d x^2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a c x^3}+\frac{2 (b c+a d) \sqrt{a+b x^2} \sqrt{c+d x^2}}{3 a^2 c^2 x}+\frac{2 \sqrt{d} (b c+a d) \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 c^{3/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{b \sqrt{d} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 a^2 \sqrt{c} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.34465, size = 229, normalized size = 0.75 \[ \frac{-i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+2 b c) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{b}{a}}\right ),\frac{a d}{b c}\right )+\sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c+2 a d x^2+2 b c x^2\right )+2 i b c x^3 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 a^2 c^2 x^3 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

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

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Maple [A]  time = 0.022, size = 435, normalized size = 1.4 \begin{align*}{\frac{1}{3\,{a}^{2}{c}^{2}{x}^{3} \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) } \left ( 2\,\sqrt{-{\frac{b}{a}}}{x}^{6}ab{d}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{6}{b}^{2}cd+bd\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}ac+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}abcd-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){x}^{3}{b}^{2}{c}^{2}+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{a}^{2}{d}^{2}+3\,\sqrt{-{\frac{b}{a}}}{x}^{4}abcd+2\,\sqrt{-{\frac{b}{a}}}{x}^{4}{b}^{2}{c}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{2}{a}^{2}cd+\sqrt{-{\frac{b}{a}}}{x}^{2}ab{c}^{2}-\sqrt{-{\frac{b}{a}}}{a}^{2}{c}^{2} \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*(-b/a)^(1/2)*x^6*a*b*d^2+2*(-b/a)^(1/2)*x^6*b^2*c*d+b*d*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti
cF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*a*c+2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),
(a*d/b/c)^(1/2))*x^3*b^2*c^2-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2
))*x^3*a*b*c*d-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*x^3*b^2*c^2
+2*(-b/a)^(1/2)*x^4*a^2*d^2+3*(-b/a)^(1/2)*x^4*a*b*c*d+2*(-b/a)^(1/2)*x^4*b^2*c^2+(-b/a)^(1/2)*x^2*a^2*c*d+(-b
/a)^(1/2)*x^2*a*b*c^2-(-b/a)^(1/2)*a^2*c^2)*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/x^3/a^2/(-b/a)^(1/2)/c^2/(b*d*x^4+
a*d*x^2+b*c*x^2+a*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{b d x^{8} +{\left (b c + a d\right )} x^{6} + a c x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(b*d*x^8 + (b*c + a*d)*x^6 + a*c*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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(1/x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

Integral(1/(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{1}{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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