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

Optimal. Leaf size=102 \[ -\frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} b}-\frac{c \sqrt{c+d x^2}}{a x}+\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.260762, antiderivative size = 102, 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 \[ -\frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} b}-\frac{c \sqrt{c+d x^2}}{a x}+\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.1378, size = 85, normalized size = 0.83 \[ \frac{d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{b} - \frac{c \sqrt{c + d x^{2}}}{a x} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{a^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.213369, size = 105, normalized size = 1.03 \[ -\frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2} b}-\frac{c \sqrt{c+d x^2}}{a x}+\frac{d^{3/2} \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 1956, normalized size = 19.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/a/c/x*(d*x^2+c)^(5/2)+1/a*d/c*x*(d*x^2+c)^(3/2)+3/2/a*d*x*(d*x^2+c)^(1/2)+3/2
/a*d^(1/2)*c*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/6*b/a/(-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)^(3/2)-1/4/a*d*((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*d^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(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))*c+1
/2/(-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)*d-1/2*b/a/(-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)*c+1/2/b*d^(3/2)*ln((d*(-a*b)^(1
/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(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))+1/2/b*a/(-a*b)^(1/2)/(-(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)))*d^2-1/(-a*b)^(1/2)/(-(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)))*d*c+1/2*b/a/(-a*b)^(1/2)/(-(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)))*c^2+1/6*b/a/(-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)^(3/2)-1/4/a*d*((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*d^(1/2)
*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(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))*c-1/2/(-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)*d+1/2*b/a/(-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)*c+1/2/b*d^(3/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-
a*b)^(1/2))*d)/d^(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))-1/2/b*a/(-a*b)^(1/2)/(-(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)))*d^2+1/(-a*b)^(1/2)/(-(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)))*d*c-1/2*b/a/(-a*b)^(1/2)/(-(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)))*c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.380138, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a d^{\frac{3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) -{\left (b c - a d\right )} x \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt{d x^{2} + c} b c}{4 \, a b x}, \frac{4 \, a \sqrt{-d} d x \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) -{\left (b c - a d\right )} x \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, \sqrt{d x^{2} + c} b c}{4 \, a b x}, \frac{a d^{\frac{3}{2}} x \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) +{\left (b c - a d\right )} x \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \, \sqrt{d x^{2} + c} b c}{2 \, a b x}, \frac{2 \, a \sqrt{-d} d x \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) +{\left (b c - a d\right )} x \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right ) - 2 \, \sqrt{d x^{2} + c} b c}{2 \, a b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.241393, size = 220, normalized size = 2.16 \[ -\frac{d^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{2 \, b} + \frac{2 \, c^{2} \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )} a} + \frac{{\left (b^{2} c^{2} \sqrt{d} - 2 \, a b c d^{\frac{3}{2}} + a^{2} d^{\frac{5}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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