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

Optimal. Leaf size=170 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^2} (2 b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac{c \sqrt{c+d x^2}}{2 a x^2 \left (a+b x^2\right )} \]

[Out]

-((2*b*c - a*d)*Sqrt[c + d*x^2])/(2*a^2*(a + b*x^2)) - (c*Sqrt[c + d*x^2])/(2*a*
x^2*(a + b*x^2)) + (Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2
*a^3) - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*
c - a*d]])/(2*a^3*Sqrt[b])

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Rubi [A]  time = 0.695402, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{c} (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3}-\frac{\sqrt{b c-a d} (4 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 \sqrt{b}}-\frac{\sqrt{c+d x^2} (2 b c-a d)}{2 a^2 \left (a+b x^2\right )}-\frac{c \sqrt{c+d x^2}}{2 a x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((2*b*c - a*d)*Sqrt[c + d*x^2])/(2*a^2*(a + b*x^2)) - (c*Sqrt[c + d*x^2])/(2*a*
x^2*(a + b*x^2)) + (Sqrt[c]*(4*b*c - 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2
*a^3) - (Sqrt[b*c - a*d]*(4*b*c - a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*
c - a*d]])/(2*a^3*Sqrt[b])

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Rubi in Sympy [A]  time = 83.3755, size = 153, normalized size = 0.9 \[ - \frac{\sqrt{c + d x^{2}} \left (a d - b c\right )}{2 a b x^{2} \left (a + b x^{2}\right )} + \frac{\sqrt{c + d x^{2}} \left (a d - 2 b c\right )}{2 a^{2} b x^{2}} - \frac{\sqrt{c} \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{3}} + \frac{\left (a d - 4 b c\right ) \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{3} \sqrt{b}} \]

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)**2,x)

[Out]

-sqrt(c + d*x**2)*(a*d - b*c)/(2*a*b*x**2*(a + b*x**2)) + sqrt(c + d*x**2)*(a*d
- 2*b*c)/(2*a**2*b*x**2) - sqrt(c)*(3*a*d - 4*b*c)*atanh(sqrt(c + d*x**2)/sqrt(c
))/(2*a**3) + (a*d - 4*b*c)*sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a
*d - b*c))/(2*a**3*sqrt(b))

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Mathematica [C]  time = 0.838654, size = 405, normalized size = 2.38 \[ -\frac{\frac{\left (a^2 d^2-5 a b c d+4 b^2 c^2\right ) \log \left (\frac{4 a^3 \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) \sqrt{b c-a d} \left (a^2 d^2-5 a b c d+4 b^2 c^2\right )}\right )}{\sqrt{b} \sqrt{b c-a d}}+\frac{\left (a^2 d^2-5 a b c d+4 b^2 c^2\right ) \log \left (\frac{4 i a^3 \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{a}+i \sqrt{b} x\right ) \sqrt{b c-a d} \left (a^2 d^2-5 a b c d+4 b^2 c^2\right )}\right )}{\sqrt{b} \sqrt{b c-a d}}+\frac{2 a \sqrt{c+d x^2} \left (a \left (c-d x^2\right )+2 b c x^2\right )}{x^2 \left (a+b x^2\right )}-2 \sqrt{c} (4 b c-3 a d) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+2 \sqrt{c} \log (x) (4 b c-3 a d)}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a*Sqrt[c + d*x^2]*(2*b*c*x^2 + a*(c - d*x^2)))/(x^2*(a + b*x^2)) + 2*Sqrt[c
]*(4*b*c - 3*a*d)*Log[x] - 2*Sqrt[c]*(4*b*c - 3*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*
x^2]] + ((4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*Log[(4*a^3*Sqrt[b]*(Sqrt[b]*c - I*Sqr
t[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b*c - a*d]*(4*b^2*c^2 - 5*a*b
*c*d + a^2*d^2)*(I*Sqrt[a] + Sqrt[b]*x))])/(Sqrt[b]*Sqrt[b*c - a*d]) + ((4*b^2*c
^2 - 5*a*b*c*d + a^2*d^2)*Log[((4*I)*a^3*Sqrt[b]*(Sqrt[b]*c + I*Sqrt[a]*d*x + Sq
rt[b*c - a*d]*Sqrt[c + d*x^2]))/(Sqrt[b*c - a*d]*(4*b^2*c^2 - 5*a*b*c*d + a^2*d^
2)*(Sqrt[a] + I*Sqrt[b]*x))])/(Sqrt[b]*Sqrt[b*c - a*d]))/(4*a^3)

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Maple [B]  time = 0.026, size = 4820, normalized size = 28.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*b^2/a^2/(-a*b)^(1/2)/(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)^(5/2)-3/4*b/a^2*d/(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)*c-3/2/a*d^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))
)*c+1/2/a^3*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*b/a^2*d/(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)))*c^2+1/4*b^2/a^2/(-a*b)^(1/2)*d/(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)*x+3/8*b
^2/a^2/(-a*b)^(1/2)*d^(1/2)/(a*d-b*c)*c^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))-3/8*b/a/(-a*b)^(1/2)*d^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)*x+1/4*b^2/a^2/(-a*
b)^(1/2)/(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)^(5/2)-3/4*b/a^2*d/(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)*c-3/2/a*
d^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)))*c+3/8*b^2/a^2/(
-a*b)^(1/2)*d/(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+3/4/a*d^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/(-a*b)^(1/2)*d^(5/
2)/(a*d-b*c)*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/a^2
/c/x^2*(d*x^2+c)^(5/2)+b/a^3*((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+3/4/a*d^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/(-a*b)^(1/2)*d^
(5/2)/(a*d-b*c)*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))+b/a^3
*((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^2*d/c*(d*x^2+c)^(3/2)-3/8*b^2/a^2/(-a*b)^(1/2)*d/(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-3/2/a^2*d*c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+2*b/a^3*c^(3/2)*ln((2*
c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)-2*b/a^3*(d*x^2+c)^(1/2)*c+1/3*b/a^3*((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/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)*d+3/2/a^2*d*(d*x^2+c)^(1/2)+1/3*b/a^3*((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/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)*d-2/3*b/a^3*(d*x^2+c)^
(3/2)+3/2/a^3*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+3/8*b/a/(-a*b)^(1/2)*d^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)*x+9/8*b/a/(-a*b)^(1/2)*d
^(3/2)/(a*d-b*c)*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/
4*b/a^2*d/(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)))*c^2-1/4*b
^2/a^2/(-a*b)^(1/2)*d/(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)*x-3/8*b^2/a^2/(-a*b)^(1/2)*d^(1/2)/(a*d-b*c
)*c^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))-9/8*b/a/(-a*b)^
(1/2)*d^(3/2)/(a*d-b*c)*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/b/a^2*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/b/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^2+2/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)))*d*c-b/a^3/(-(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/b/a^2*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/b/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^2+2/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
)))*d*c-b/a^3/(-(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*b/a^2*d
/(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)+3/4/b*d^3/(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)))-1/4*b/a^2*d/(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)+3/4/b*d^3/(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/a^3*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-3/2/a^3*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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.523474, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224614, size = 300, normalized size = 1.76 \[ -\frac{1}{2} \, d^{3}{\left (\frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b c - 2 \, \sqrt{d x^{2} + c} b c^{2} -{\left (d x^{2} + c\right )}^{\frac{3}{2}} a d + 2 \, \sqrt{d x^{2} + c} a c d}{{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )} a^{2} d^{2}} - \frac{{\left (4 \, b^{2} c^{2} - 5 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} d^{3}} + \frac{{\left (4 \, b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d^3*((2*(d*x^2 + c)^(3/2)*b*c - 2*sqrt(d*x^2 + c)*b*c^2 - (d*x^2 + c)^(3/2)
*a*d + 2*sqrt(d*x^2 + c)*a*c*d)/(((d*x^2 + c)^2*b - 2*(d*x^2 + c)*b*c + b*c^2 +
(d*x^2 + c)*a*d - a*c*d)*a^2*d^2) - (4*b^2*c^2 - 5*a*b*c*d + a^2*d^2)*arctan(sqr
t(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3*d^3) + (4*b*c^2 -
3*a*c*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*d^3))