3.723 \(\int \frac{x^3}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=103 \[ -\frac{a}{\sqrt{c+d x^2} (b c-a d)^2}-\frac{c}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.263603, antiderivative size = 103, 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 \[ -\frac{a}{\sqrt{c+d x^2} (b c-a d)^2}-\frac{c}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac{a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^2)*(c + d*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 31.6656, size = 85, normalized size = 0.83 \[ - \frac{a \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{\left (a d - b c\right )^{\frac{5}{2}}} - \frac{a}{\sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{c}{3 d \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**2+a)/(d*x**2+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.316848, size = 99, normalized size = 0.96 \[ \frac{-a d \left (2 c+3 d x^2\right )-b c^2}{3 d \left (c+d x^2\right )^{3/2} (b c-a d)^2}+\frac{a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/((a + b*x^2)*(c + d*x^2)^(5/2)),x]

[Out]

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

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Maple [B]  time = 0.02, size = 1123, normalized size = 10.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a)/(d*x^2+c)^(5/2),x)

[Out]

-1/3/b/d/(d*x^2+c)^(3/2)+1/6*a/b/(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)^(3/2)-1/6*a/b^2*d*(-a*b)^(1/2)/(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)^(3/2)*x-1/3*a/b^2*d*(-a*b)^(1/2)/(a*d-b*c)/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)*x-1/2*a/(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)
+1/2*a/b/(a*d-b*c)^2*(-a*b)^(1/2)/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*d+1/2*a/(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)))+1/6*a/b/(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)^(3/2)+1/6*a/b^2*d*(-a*b)^(1/2
)/(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)^(3/2)*x+1/3*a/b^2*d*(-a*b)^(1/2)/(a*d-b*c)/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)*x-1/2*a/(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)-1/2*a/b/(a*d-b*c)^2*(-a*b)^(1/2)/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*d+1/2*a/(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)))

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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^3/((b*x^2 + a)*(d*x^2 + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279152, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a d^{3} x^{4} + 2 \, a c d^{2} x^{2} + a c^{2} d\right )} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left (3 \, a d^{2} x^{2} + b c^{2} + 2 \, a c d\right )} \sqrt{d x^{2} + c}}{12 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}, -\frac{3 \,{\left (a d^{3} x^{4} + 2 \, a c d^{2} x^{2} + a c^{2} d\right )} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}\right ) + 2 \,{\left (3 \, a d^{2} x^{2} + b c^{2} + 2 \, a c d\right )} \sqrt{d x^{2} + c}}{6 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/12*(3*(a*d^3*x^4 + 2*a*c*d^2*x^2 + a*c^2*d)*sqrt(b/(b*c - a*d))*log((b^2*d^2*
x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(2*b^2
*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*
c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(3*a*d^2*x^2 + b*c^2 + 2*a*c*d)*sqrt
(d*x^2 + c))/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d
^4 + a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2), -1/6*(3*(a
*d^3*x^4 + 2*a*c*d^2*x^2 + a*c^2*d)*sqrt(-b/(b*c - a*d))*arctan(-1/2*(b*d*x^2 +
2*b*c - a*d)/(sqrt(d*x^2 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d)))) + 2*(3*a*d^2*x^
2 + b*c^2 + 2*a*c*d)*sqrt(d*x^2 + c))/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 +
(b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^
2*c*d^4)*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**2+a)/(d*x**2+c)**(5/2),x)

[Out]

Integral(x**3/((a + b*x**2)*(c + d*x**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.244476, size = 171, normalized size = 1.66 \[ -\frac{\frac{3 \, a b d \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{b c^{2} + 3 \,{\left (d x^{2} + c\right )} a d - a c d}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^(5/2)),x, algorithm="giac")

[Out]

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