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

Optimal. Leaf size=180 \[ -\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3}-\frac{\sqrt{c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left (a+b x^2\right )}-\frac{c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )} \]

[Out]

-((b*c - a*d)*(2*b*c - a*d)*Sqrt[c + d*x^2])/(2*a^2*b*(a + b*x^2)) - (c*(c + d*x
^2)^(3/2))/(2*a*x^2*(a + b*x^2)) + (c^(3/2)*(4*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x
^2]/Sqrt[c]])/(2*a^3) - ((b*c - a*d)^(3/2)*(4*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c
+ d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*b^(3/2))

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Rubi [A]  time = 0.719622, antiderivative size = 180, 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 \[ -\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^3 b^{3/2}}+\frac{c^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 a^3}-\frac{\sqrt{c+d x^2} (b c-a d) (2 b c-a d)}{2 a^2 b \left (a+b x^2\right )}-\frac{c \left (c+d x^2\right )^{3/2}}{2 a x^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*(2*b*c - a*d)*Sqrt[c + d*x^2])/(2*a^2*b*(a + b*x^2)) - (c*(c + d*x
^2)^(3/2))/(2*a*x^2*(a + b*x^2)) + (c^(3/2)*(4*b*c - 5*a*d)*ArcTanh[Sqrt[c + d*x
^2]/Sqrt[c]])/(2*a^3) - ((b*c - a*d)^(3/2)*(4*b*c + a*d)*ArcTanh[(Sqrt[b]*Sqrt[c
+ d*x^2])/Sqrt[b*c - a*d]])/(2*a^3*b^(3/2))

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Rubi in Sympy [A]  time = 72.7551, size = 155, normalized size = 0.86 \[ - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b x^{2} \left (a + b x^{2}\right )} + \frac{c \sqrt{c + d x^{2}} \left (a d - 2 b c\right )}{2 a^{2} b x^{2}} - \frac{c^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 a^{3}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (a d + 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{3} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(c + d*x**2)**(3/2)*(a*d - b*c)/(2*a*b*x**2*(a + b*x**2)) + c*sqrt(c + d*x**2)*
(a*d - 2*b*c)/(2*a**2*b*x**2) - c**(3/2)*(5*a*d - 4*b*c)*atanh(sqrt(c + d*x**2)/
sqrt(c))/(2*a**3) + (a*d - b*c)**(3/2)*(a*d + 4*b*c)*atan(sqrt(b)*sqrt(c + d*x**
2)/sqrt(a*d - b*c))/(2*a**3*b**(3/2))

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Mathematica [C]  time = 1.18527, size = 349, normalized size = 1.94 \[ -\frac{\frac{(b c-a d)^{3/2} (a d+4 b c) \log \left (\frac{4 a^3 b^{3/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{5/2} (a d+4 b c)}\right )}{b^{3/2}}+\frac{(b c-a d)^{3/2} (a d+4 b c) \log \left (\frac{4 a^3 b^{3/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{5/2} (a d+4 b c)}\right )}{b^{3/2}}-2 c^{3/2} (4 b c-5 a d) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+2 c^{3/2} \log (x) (4 b c-5 a d)+2 a \sqrt{c+d x^2} \left (\frac{(b c-a d)^2}{b \left (a+b x^2\right )}+\frac{c^2}{x^2}\right )}{4 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.031, size = 7590, normalized size = 42.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.64854, 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)^(5/2)/((b*x^2 + a)^2*x^3),x, algorithm="fricas")

[Out]

[-1/8*(((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 - 3*a^2*b*c*d -
a^3*d^2)*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^3*c^2 - 5*a*
b^2*c*d)*x^4 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x
^2 + c)*sqrt(c) + 2*c)/x^2) + 4*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^
2)*x^2)*sqrt(d*x^2 + c))/(a^3*b^2*x^4 + a^4*b*x^2), 1/8*(4*((4*b^3*c^2 - 5*a*b^2
*c*d)*x^4 + (4*a*b^2*c^2 - 5*a^2*b*c*d)*x^2)*sqrt(-c)*arctan(c/(sqrt(d*x^2 + c)*
sqrt(-c))) - ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 - 3*a^2*b
*c*d - a^3*d^2)*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)*sq
rt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(a^2*b*c^2 +
(2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x^2)*sqrt(d*x^2 + c))/(a^3*b^2*x^4 + a^4*
b*x^2), -1/4*(((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 - 3*a^2*
b*c*d - a^3*d^2)*x^2)*sqrt(-(b*c - a*d)/b)*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)/(s
qrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b))) + ((4*b^3*c^2 - 5*a*b^2*c*d)*x^4 + (4*a*
b^2*c^2 - 5*a^2*b*c*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*
c)/x^2) + 2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x^2)*sqrt(d*x^2 +
c))/(a^3*b^2*x^4 + a^4*b*x^2), 1/4*(2*((4*b^3*c^2 - 5*a*b^2*c*d)*x^4 + (4*a*b^2
*c^2 - 5*a^2*b*c*d)*x^2)*sqrt(-c)*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) - ((4*b^3
*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^4 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*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))) - 2*(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x^2)*s
qrt(d*x^2 + c))/(a^3*b^2*x^4 + a^4*b*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)**(5/2)/x**3/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.238189, size = 390, normalized size = 2.17 \[ -\frac{1}{2} \, d^{3}{\left (\frac{{\left (4 \, b c^{3} - 5 \, a c^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{3} \sqrt{-c} d^{3}} + \frac{2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{2} - 2 \, \sqrt{d x^{2} + c} b^{2} c^{3} - 2 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d + 3 \, \sqrt{d x^{2} + c} a b c^{2} d +{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} - \sqrt{d x^{2} + c} a^{2} c d^{2}}{{\left ({\left (d x^{2} + c\right )}^{2} b - 2 \,{\left (d x^{2} + c\right )} b c + b c^{2} +{\left (d x^{2} + c\right )} a d - a c d\right )} a^{2} b d^{2}} - \frac{{\left (4 \, b^{3} c^{3} - 7 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{3} b d^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d^3*((4*b*c^3 - 5*a*c^2*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*d
^3) + (2*(d*x^2 + c)^(3/2)*b^2*c^2 - 2*sqrt(d*x^2 + c)*b^2*c^3 - 2*(d*x^2 + c)^(
3/2)*a*b*c*d + 3*sqrt(d*x^2 + c)*a*b*c^2*d + (d*x^2 + c)^(3/2)*a^2*d^2 - sqrt(d*
x^2 + c)*a^2*c*d^2)/(((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*b*d^2) - (4*b^3*c^3 - 7*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)*
arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a^3*b*d^3))