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

Optimal. Leaf size=141 \[ -\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d} \]

[Out]

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

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Rubi [A]  time = 0.420212, antiderivative size = 141, 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 \[ -\frac{\left (4 a b c d-3 (a d+b c)^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{8 b^{5/2} d^{5/2}}-\frac{3 \sqrt{a+b x^2} \sqrt{c+d x^2} (a d+b c)}{8 b^2 d^2}+\frac{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.4774, size = 128, normalized size = 0.91 \[ \frac{x^{2} \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{4 b d} - \frac{3 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d + b c\right )}{8 b^{2} d^{2}} - \frac{\left (a b c d - \frac{3 \left (a d + b c\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{2 b^{\frac{5}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.183064, size = 135, normalized size = 0.96 \[ \frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{16 b^{5/2} d^{5/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (-3 a d-3 b c+2 b d x^2\right )}{8 b^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-3*b*c - 3*a*d + 2*b*d*x^2))/(8*b^2*d^2) + ((3
*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*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]])/(16*b^(5/2)*d^(5/2))

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Maple [B]  time = 0.046, size = 340, normalized size = 2.4 \[{\frac{1}{16\,{b}^{2}{d}^{2}} \left ( 4\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}db\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{d}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) cadb+3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}ad\sqrt{bd}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}cb\sqrt{bd} \right ) \sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277308, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b d x^{2} - 3 \, b c - 3 \, a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} +{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{32 \, \sqrt{b d} b^{2} d^{2}}, \frac{2 \,{\left (2 \, b d x^{2} - 3 \, b c - 3 \, a d\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} +{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{16 \, \sqrt{-b d} b^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.245114, size = 212, normalized size = 1.5 \[ \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (\frac{2 \,{\left (b x^{2} + a\right )}}{b d} - \frac{3 \, b^{2} c d + 5 \, a b d^{2}}{b^{2} d^{3}}\right )} - \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{2}}}{8 \, b{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a)/(b*d)
- (3*b^2*c*d + 5*a*b*d^2)/(b^2*d^3)) - (3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ln(ab
s(-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^2))/(b*abs(b))