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

Optimal. Leaf size=130 \[ \frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.28252, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{x (a d+2 b c)}{2 b \sqrt{c+d x^2} (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 (b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 52.0721, size = 110, normalized size = 0.85 \[ - \frac{3 \sqrt{a} c \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 \left (a d - b c\right )^{\frac{5}{2}}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} + \frac{x \left (a d + 2 b c\right )}{2 b \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.435764, size = 106, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{3 a c x+a d x^3+2 b c x^3}{\left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)^2}-\frac{3 \sqrt{a} c \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.032, size = 1498, normalized size = 11.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/b^2*x/c/(d*x^2+c)^(1/2)-1/4*a/b^2/(a*d-b*c)/(x-1/b*(-a*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)+3/4*a/b^
2*d*(-a*b)^(1/2)/(a*d-b*c)^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)+3/4*a^2/b^2*d^2/(a*d-b*c)^2/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-3/4*a/b^2
*d*(-a*b)^(1/2)/(a*d-b*c)^2/(-(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)))-5
/4*a/b^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-1/4*a/b^2/(a*d-b*c)/(x+1/b*(-a*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)-3/4*
a/b^2*d*(-a*b)^(1/2)/(a*d-b*c)^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)+3/4*a^2/b^2*d^2/(a*d-b*c)^2/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+3/4*a
/b^2*d*(-a*b)^(1/2)/(a*d-b*c)^2/(-(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)
))-5/4*a/b^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+3/4/b*a/(-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)-3/4/b*a/(-
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)))-3/4/b*a/
(-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)+3/4/b*a/(-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. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.50435, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b c d x^{4} + a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{d x^{2} + c}}{8 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (b c d x^{4} + a c^{2} +{\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} x \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \,{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt{d x^{2} + c}}{4 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  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)**2/(d*x**2+c)**(3/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 3.34081, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x