3.713 \(\int \frac{x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b (b c-a d)^{3/2}}-\frac{c x}{d \sqrt{c+d x^2} (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.310914, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b (b c-a d)^{3/2}}-\frac{c x}{d \sqrt{c+d x^2} (b c-a d)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x^2)*(c + d*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 42.8231, size = 92, normalized size = 0.84 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b \left (a d - b c\right )^{\frac{3}{2}}} + \frac{c x}{d \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{b d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.2704, size = 111, normalized size = 1.02 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b (b c-a d)^{3/2}}+\frac{c x}{d \sqrt{c+d x^2} (a d-b c)}+\frac{\log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{b d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 720, normalized size = 6.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)/(d*x^2+c)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.466243, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="fricas")

[Out]

[-1/4*(4*sqrt(d*x^2 + c)*b*c*sqrt(d)*x + (a*d^2*x^2 + a*c*d)*sqrt(d)*sqrt(-a/(b*
c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 -
4*a^2*c*d)*x^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*
x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(b*c^2
- a*c*d + (b*c*d - a*d^2)*x^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(
d)))/((b^2*c^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x^2)*sqrt(d)), -1/4*(4*sqrt
(d*x^2 + c)*b*c*sqrt(-d)*x + (a*d^2*x^2 + a*c*d)*sqrt(-d)*sqrt(-a/(b*c - a*d))*l
og(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*
x^2 - 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x
^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(b*c^2 - a*c*d +
(b*c*d - a*d^2)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((b^2*c^2*d - a*b*c*d^2
+ (b^2*c*d^2 - a*b*d^3)*x^2)*sqrt(-d)), -1/2*(2*sqrt(d*x^2 + c)*b*c*sqrt(d)*x -
(a*d^2*x^2 + a*c*d)*sqrt(d)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*c - 2*a*d)*x^2 -
a*c)/(sqrt(d*x^2 + c)*(b*c - a*d)*x*sqrt(a/(b*c - a*d)))) - (b*c^2 - a*c*d + (b
*c*d - a*d^2)*x^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/((b^2*c^
2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x^2)*sqrt(d)), -1/2*(2*sqrt(d*x^2 + c)*b
*c*sqrt(-d)*x - (a*d^2*x^2 + a*c*d)*sqrt(-d)*sqrt(a/(b*c - a*d))*arctan(1/2*((b*
c - 2*a*d)*x^2 - a*c)/(sqrt(d*x^2 + c)*(b*c - a*d)*x*sqrt(a/(b*c - a*d)))) - 2*(
b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/((b^2*c
^2*d - a*b*c*d^2 + (b^2*c*d^2 - a*b*d^3)*x^2)*sqrt(-d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**2+a)/(d*x**2+c)**(3/2),x)

[Out]

Integral(x**4/((a + b*x**2)*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.256757, size = 201, normalized size = 1.84 \[ -\frac{b^{2} c x}{{\left (b^{3} c d - a b^{2} d^{2}\right )} \sqrt{d x^{2} + c}} - \frac{a^{2} \sqrt{d} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}}{\left (b^{2} c - a b d\right )}} - \frac{{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{2 \, b d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^(3/2)),x, algorithm="giac")

[Out]

-b^2*c*x/((b^3*c*d - a*b^2*d^2)*sqrt(d*x^2 + c)) - a^2*sqrt(d)*arctan(1/2*((sqrt
(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c
*d - a^2*d^2)*(b^2*c - a*b*d)) - 1/2*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b*d^(3
/2))