3.939 \(\int \frac{\sqrt{a+b x^2}}{x^5 \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=143 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4} \]

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*a*c^2*x^2) - ((a + b*x^2)^(3/
2)*Sqrt[c + d*x^2])/(4*a*c*x^4) + ((b*c - a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(3/2)*c^(5/2))

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Rubi [A]  time = 0.376479, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (3 a d+b c)}{8 a c^2 x^2}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/(x^5*Sqrt[c + d*x^2]),x]

[Out]

((b*c + 3*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(8*a*c^2*x^2) - ((a + b*x^2)^(3/
2)*Sqrt[c + d*x^2])/(4*a*c*x^4) + ((b*c - a*d)*(b*c + 3*a*d)*ArcTanh[(Sqrt[c]*Sq
rt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(3/2)*c^(5/2))

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Rubi in Sympy [A]  time = 29.4279, size = 126, normalized size = 0.88 \[ - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{4 a c x^{4}} + \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (3 a d + b c\right )}{8 a c^{2} x^{2}} - \frac{\left (a d - b c\right ) \left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{2}}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{8 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/x**5/(d*x**2+c)**(1/2),x)

[Out]

-(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(4*a*c*x**4) + sqrt(a + b*x**2)*sqrt(c + d
*x**2)*(3*a*d + b*c)/(8*a*c**2*x**2) - (a*d - b*c)*(3*a*d + b*c)*atanh(sqrt(c)*s
qrt(a + b*x**2)/(sqrt(a)*sqrt(c + d*x**2)))/(8*a**(3/2)*c**(5/2))

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Mathematica [C]  time = 0.387784, size = 224, normalized size = 1.57 \[ \frac{\frac{2 b d x^6 \left (3 a^2 d^2-2 a b c d-b^2 c^2\right ) F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}{-4 b d x^2 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^2},-\frac{c}{d x^2}\right )}+\left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 a c+3 a d x^2-b c x^2\right )}{8 a c^2 x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[a + b*x^2]/(x^5*Sqrt[c + d*x^2]),x]

[Out]

((a + b*x^2)*(c + d*x^2)*(-2*a*c - b*c*x^2 + 3*a*d*x^2) + (2*b*d*(-(b^2*c^2) - 2
*a*b*c*d + 3*a^2*d^2)*x^6*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*AppellF1[2, 3/2, 1/2, 3, -(a/(
b*x^2)), -(c/(d*x^2))]))/(8*a*c^2*x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [B]  time = 0.04, size = 355, normalized size = 2.5 \[ -{\frac{1}{16\,{c}^{2}a{x}^{4}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}{a}^{2}{d}^{2}-2\,\ln \left ({\frac{ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac}{{x}^{2}}} \right ){x}^{4}abcd-\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+c{x}^{2}b+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}+2\,ac \right ) } \right ){x}^{4}{b}^{2}{c}^{2}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}da{x}^{2}\sqrt{ac}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}bc{x}^{2}\sqrt{ac}+4\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x^5/(d*x^2+c)^(1/2),x)

[Out]

-1/16*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c^2*(3*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^4*a^2*d^2-2*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^4*a*b*c*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
^4*b^2*c^2-6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*d*a*x^2*(a*c)^(1/2)+2*(b*d*x^4+
a*d*x^2+b*c*x^2+a*c)^(1/2)*b*c*x^2*(a*c)^(1/2)+4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(
1/2)*c*a*(a*c)^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^4/(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(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.334684, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} -{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{a c}}{x^{4}}\right ) + 4 \,{\left ({\left (b c - 3 \, a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{a c}}{32 \, \sqrt{a c} a c^{2} x^{4}}, \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{4} \arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} a c}\right ) - 2 \,{\left ({\left (b c - 3 \, a d\right )} x^{2} + 2 \, a c\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-a c}}{16 \, \sqrt{-a c} a c^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^5),x, algorithm="fricas")

[Out]

[-1/32*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*x^4*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*((b*c - 3*a*d)*x
^2 + 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a*c))/(sqrt(a*c)*a*c^2*x^4), 1/
16*((b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2)*x^4*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)*s
qrt(-a*c)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*a*c)) - 2*((b*c - 3*a*d)*x^2 + 2*a*c)
*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a*c))/(sqrt(-a*c)*a*c^2*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x^{2}}}{x^{5} \sqrt{c + d x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)/x**5/(d*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(a + b*x**2)/(x**5*sqrt(c + d*x**2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)/(sqrt(d*x^2 + c)*x^5),x, algorithm="giac")

[Out]

Exception raised: TypeError