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

Optimal. Leaf size=88 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{3/2} d^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.236678, antiderivative size = 88, 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 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{2 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.9862, size = 75, normalized size = 0.85 \[ \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{2 b d} - \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{2 b^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(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*b*d) - (a*d + b*c)*atanh(sqrt(d)*sqrt(a + b
*x**2)/(sqrt(b)*sqrt(c + d*x**2)))/(2*b**(3/2)*d**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0875192, size = 103, normalized size = 1.17 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{2 b d}-\frac{(a d+b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{4 b^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

_______________________________________________________________________________________

Maple [B]  time = 0.024, size = 200, normalized size = 2.3 \[ -{\frac{1}{4\,bd} \left ( a\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) d+b\ln \left ({\frac{1}{2} \left ( 2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) c-2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd} \right ) \sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(a*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*d+b*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b
*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c-2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1
/2))*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d/(b*d)^(1/2)/b/(b*d*x^4+a*d*x^2+b*c*x^2+a*
c)^(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.263777, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right ) + 4 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d}}{8 \, \sqrt{b d} b d}, -\frac{{\left (b c + a d\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right ) - 2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d}}{4 \, \sqrt{-b d} b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.241798, size = 140, normalized size = 1.59 \[ \frac{\frac{{\left (b c + a d\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d} + \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}}{b d}}{2 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((b*c + a*d)*ln(abs(-sqrt(b*x^2 + a)*sqrt(b*d) + sqrt(b^2*c + (b*x^2 + a)*b*
d - a*b*d)))/(sqrt(b*d)*d) + sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 +
a)/(b*d))/abs(b)