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

Optimal. Leaf size=114 \[ -\frac{(b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2}-\frac{c \sqrt{c+d x^2}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.411838, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{(b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a^2 \sqrt{b}}+\frac{\sqrt{c} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^2}-\frac{c \sqrt{c+d x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.4144, size = 100, normalized size = 0.88 \[ - \frac{c \sqrt{c + d x^{2}}}{2 a x^{2}} - \frac{\sqrt{c} \left (3 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{2}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a^{2} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.33947, size = 284, normalized size = 2.49 \[ -\frac{\frac{(b c-a d)^{3/2} \log \left (\frac{2 a^2 \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{5/2}}\right )}{\sqrt{b}}+\frac{(b c-a d)^{3/2} \log \left (\frac{2 a^2 \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{5/2}}\right )}{\sqrt{b}}-\sqrt{c} (2 b c-3 a d) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\sqrt{c} \log (x) (2 b c-3 a d)+\frac{a c \sqrt{c+d x^2}}{x^2}}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a*c*Sqrt[c + d*x^2])/x^2 + Sqrt[c]*(2*b*c - 3*a*d)*Log[x] - Sqrt[c]*(2*b*c -
3*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]] + ((b*c - a*d)^(3/2)*Log[(2*a^2*Sqrt[b]*
(Sqrt[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/((b*c - a*d)^(5/2
)*(I*Sqrt[a] + Sqrt[b]*x))])/Sqrt[b] + ((b*c - a*d)^(3/2)*Log[(2*a^2*Sqrt[b]*(Sq
rt[b]*c + I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/((b*c - a*d)^(5/2)*(
(-I)*Sqrt[a] + Sqrt[b]*x))])/Sqrt[b])/(2*a^2)

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Maple [B]  time = 0.02, size = 2003, normalized size = 17.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.581458, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} x^{2} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) +{\left (2 \, b c - 3 \, a d\right )} \sqrt{c} x^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt{d x^{2} + c} a c}{4 \, a^{2} x^{2}}, \frac{2 \,{\left (2 \, b c - 3 \, a d\right )} \sqrt{-c} x^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (b c - a d\right )} x^{2} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 2 \, \sqrt{d x^{2} + c} a c}{4 \, a^{2} x^{2}}, -\frac{2 \,{\left (b c - a d\right )} x^{2} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (2 \, b c - 3 \, a d\right )} \sqrt{c} x^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt{d x^{2} + c} a c}{4 \, a^{2} x^{2}}, \frac{{\left (2 \, b c - 3 \, a d\right )} \sqrt{-c} x^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (b c - a d\right )} x^{2} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) - \sqrt{d x^{2} + c} a c}{2 \, a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.238803, size = 182, normalized size = 1.6 \[ \frac{1}{2} \, d^{2}{\left (\frac{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} - \frac{\sqrt{d x^{2} + c} c}{a d^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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