3.362 \(\int \frac{1}{(a+b x) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b^2 c}{a^2 d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c+d x^2} \sqrt{a^2 f+b^2 e}}{\sqrt{e+f x^2} \sqrt{a^2 d+b^2 c}}\right )}{\sqrt{a^2 d+b^2 c} \sqrt{a^2 f+b^2 e}} \]

[Out]

-((b*ArcTanh[(Sqrt[b^2*e + a^2*f]*Sqrt[c + d*x^2])/(Sqrt[b^2*c + a^2*d]*Sqrt[e +
f*x^2])])/(Sqrt[b^2*c + a^2*d]*Sqrt[b^2*e + a^2*f])) + (Sqrt[-c]*Sqrt[1 + (d*x^
2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b^2*c)/(a^2*d)), ArcSin[(Sqrt[d]*x)/Sqrt
[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Rubi [A]  time = 1.22223, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (-\frac{b^2 c}{a^2 d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{a \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c+d x^2} \sqrt{a^2 f+b^2 e}}{\sqrt{e+f x^2} \sqrt{a^2 d+b^2 c}}\right )}{\sqrt{a^2 d+b^2 c} \sqrt{a^2 f+b^2 e}} \]

Antiderivative was successfully verified.

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

[Out]

-((b*ArcTanh[(Sqrt[b^2*e + a^2*f]*Sqrt[c + d*x^2])/(Sqrt[b^2*c + a^2*d]*Sqrt[e +
f*x^2])])/(Sqrt[b^2*c + a^2*d]*Sqrt[b^2*e + a^2*f])) + (Sqrt[-c]*Sqrt[1 + (d*x^
2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b^2*c)/(a^2*d)), ArcSin[(Sqrt[d]*x)/Sqrt
[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])

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Rubi in Sympy [A]  time = 115.082, size = 264, normalized size = 1.38 \[ \frac{a \sqrt{c} \sqrt{d} \sqrt{e + f x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a^{2} d + b^{2} c\right )} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}} \sqrt{a^{2} f + b^{2} e}}{\sqrt{e + f x^{2}} \sqrt{a^{2} d + b^{2} c}} \right )}}{\sqrt{a^{2} d + b^{2} c} \sqrt{a^{2} f + b^{2} e}} + \frac{b^{2} c^{\frac{3}{2}} \sqrt{e + f x^{2}} \Pi \left (1 + \frac{b^{2} c}{a^{2} d}; \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{c f}{d e} + 1\right )}{a \sqrt{d} e \sqrt{\frac{c \left (e + f x^{2}\right )}{e \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}} \left (a^{2} d + b^{2} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*sqrt(c)*sqrt(d)*sqrt(e + f*x**2)*elliptic_f(atan(sqrt(d)*x/sqrt(c)), -c*f/(d*e
) + 1)/(e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(a**2*d + b**2*
c)) - b*atanh(sqrt(c + d*x**2)*sqrt(a**2*f + b**2*e)/(sqrt(e + f*x**2)*sqrt(a**2
*d + b**2*c)))/(sqrt(a**2*d + b**2*c)*sqrt(a**2*f + b**2*e)) + b**2*c**(3/2)*sqr
t(e + f*x**2)*elliptic_pi(1 + b**2*c/(a**2*d), atan(sqrt(d)*x/sqrt(c)), -c*f/(d*
e) + 1)/(a*sqrt(d)*e*sqrt(c*(e + f*x**2)/(e*(c + d*x**2)))*sqrt(c + d*x**2)*(a**
2*d + b**2*c))

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Mathematica [A]  time = 0.800918, size = 0, normalized size = 0. \[ \int \frac{1}{(a+b x) \sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [B]  time = 0.094, size = 352, normalized size = 1.8 \[ -{\frac{1}{2\,ab \left ( df{x}^{4}+c{x}^{2}f+{x}^{2}de+ce \right ) } \left ( -2\,\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{f{x}^{2}+e}{e}}}\sqrt{{\frac{{a}^{4}df+{a}^{2}{b}^{2}cf+{a}^{2}{b}^{2}de+{b}^{4}ce}{{b}^{4}}}}{\it EllipticPi} \left ( x\sqrt{-{\frac{d}{c}}},-{\frac{{b}^{2}c}{{a}^{2}d}},{1\sqrt{-{\frac{f}{e}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \right ) b+\sqrt{df{x}^{4}+c{x}^{2}f+{x}^{2}de+ce}{\it Artanh} \left ({\frac{2\,{a}^{2}df{x}^{2}+{b}^{2}cf{x}^{2}+{b}^{2}de{x}^{2}+{a}^{2}cf+{a}^{2}de+2\,{b}^{2}ce}{2\,{b}^{2}}{\frac{1}{\sqrt{{\frac{{a}^{4}df+{a}^{2}{b}^{2}cf+{a}^{2}{b}^{2}de+{b}^{4}ce}{{b}^{4}}}}}}{\frac{1}{\sqrt{df{x}^{4}+c{x}^{2}f+{x}^{2}de+ce}}}} \right ) \sqrt{-{\frac{d}{c}}}a \right ) \sqrt{f{x}^{2}+e}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{{\frac{{a}^{4}df+{a}^{2}{b}^{2}cf+{a}^{2}{b}^{2}de+{b}^{4}ce}{{b}^{4}}}}}}{\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-2*((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*((a^4*d*f+a^2*b^2*c*f+a^2*b^2*d
*e+b^4*c*e)/b^4)^(1/2)*EllipticPi(x*(-1/c*d)^(1/2),-b^2*c/a^2/d,(-f/e)^(1/2)/(-1
/c*d)^(1/2))*b+(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*arctanh(1/2*(2*a^2*d*f*x^2+b^
2*c*f*x^2+b^2*d*e*x^2+a^2*c*f+a^2*d*e+2*b^2*c*e)/b^2/((a^4*d*f+a^2*b^2*c*f+a^2*b
^2*d*e+b^4*c*e)/b^4)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2))*(-1/c*d)^(1/2)*a
)*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/b/((a^4*d*f+a^2*b^2*c*f+a^2*b^2*d*e+b^4*c*e)/b
^4)^(1/2)/a/(-1/c*d)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}{\left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \sqrt{c + d x^{2}} \sqrt{e + f x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x^{2} + c} \sqrt{f x^{2} + e}{\left (b x + a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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