3.973 \(\int \frac{1}{x^3 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=91 \[ \frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 a c x^2} \]

[Out]

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

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Rubi [A]  time = 0.268698, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 a c x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.4453, size = 78, normalized size = 0.86 \[ - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 a c x^{2}} + \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.322101, size = 192, normalized size = 2.11 \[ \frac{\frac{2 b d x^4 (a d+b c) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )-a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}-\left (a+b x^2\right ) \left (c+d x^2\right )}{2 a c x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]

[Out]

(-((a + b*x^2)*(c + d*x^2)) + (2*b*d*(b*c + a*d)*x^4*AppellF1[1, 1/2, 1/2, 2, -(
a/(b*x^2)), -(c/(d*x^2))])/(4*b*d*x^2*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^2)), -(c
/(d*x^2))] - b*c*AppellF1[2, 1/2, 3/2, 3, -(a/(b*x^2)), -(c/(d*x^2))] - a*d*Appe
llF1[2, 3/2, 1/2, 3, -(a/(b*x^2)), -(c/(d*x^2))]))/(2*a*c*x^2*Sqrt[a + b*x^2]*Sq
rt[c + d*x^2])

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Maple [B]  time = 0.03, size = 209, normalized size = 2.3 \[{\frac{1}{4\,ac{x}^{2}} \left ( \ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{2}ad+\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{2}bc-2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac} \right ) \sqrt{d{x}^{2}+c}\sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293475, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} x^{2} \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{a c}}{x^{4}}\right ) - 4 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{a c}}{8 \, \sqrt{a c} a c x^{2}}, \frac{{\left (b c + a d\right )} x^{2} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} a c}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-a c}}{4 \, \sqrt{-a c} a c x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*((b*c + a*d)*x^2*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x^2)*sqrt(b*x^2 +
a)*sqrt(d*x^2 + c) + ((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*a^2*c^2 + 8*(a*b*c
^2 + a^2*c*d)*x^2)*sqrt(a*c))/x^4) - 4*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a*c)
)/(sqrt(a*c)*a*c*x^2), 1/4*((b*c + a*d)*x^2*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)
*sqrt(-a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a*c)) - 2*sqrt(b*x^2 + a)*sqrt(d*x^
2 + c)*sqrt(-a*c))/(sqrt(-a*c)*a*c*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260639, size = 558, normalized size = 6.13 \[ \frac{\sqrt{b d} b^{4} d{\left (\frac{{\left (b c + a d\right )} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} a b^{3} c d} - \frac{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b c -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d +{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )} a b^{2} c d}\right )}}{2 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b*d)*b^4*d*((b*c + a*d)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*x^2 + a)*s
qrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-
a*b*c*d)*a*b^3*c*d) - 2*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2 - (sqrt(b*x^2 + a)*sq
rt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*b*c - (sqrt(b*x^2 + a)*sqrt(b
*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^2*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^
2*b^2*d^2 - 2*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)
)^2*b^2*c - 2*(sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)
)^2*a*b*d + (sqrt(b*x^2 + a)*sqrt(b*d) - sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d))^
4)*a*b^2*c*d))/abs(b)