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

Optimal. Leaf size=160 \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 b^{5/2}}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}-\frac{d \sqrt{c+d x^2} (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.627101, antiderivative size = 160, 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 \[ \frac{(b c-a d)^{3/2} (3 a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 a^2 b^{5/2}}-\frac{c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a^2}-\frac{d \sqrt{c+d x^2} (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 68.9213, size = 138, normalized size = 0.86 \[ - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b \left (a + b x^{2}\right )} + \frac{d \sqrt{c + d x^{2}} \left (3 a d - b c\right )}{2 a b^{2}} - \frac{c^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a^{2}} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (3 a d + 2 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 a^{2} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.803669, size = 344, normalized size = 2.15 \[ \frac{1}{4} \left (\frac{(b c-a d)^{3/2} (3 a d+2 b c) \log \left (-\frac{4 a^2 b^{5/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{5/2} (3 a d+2 b c)}\right )}{a^2 b^{5/2}}+\frac{(b c-a d)^{3/2} (3 a d+2 b c) \log \left (-\frac{4 a^2 b^{5/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{5/2} (3 a d+2 b c)}\right )}{a^2 b^{5/2}}-\frac{4 c^{5/2} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{a^2}+\frac{4 c^{5/2} \log (x)}{a^2}+\frac{2 \sqrt{c+d x^2} \left (\frac{(b c-a d)^2}{a \left (a+b x^2\right )}+2 d^2\right )}{b^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 7477, normalized size = 46.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(5/2)/x/(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. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.54751, 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),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.233599, size = 290, normalized size = 1.81 \[ \frac{1}{2} \, d^{2}{\left (\frac{2 \, c^{3} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} + \frac{2 \, \sqrt{d x^{2} + c}}{b^{2}} + \frac{\sqrt{d x^{2} + c} b^{2} c^{2} - 2 \, \sqrt{d x^{2} + c} a b c d + \sqrt{d x^{2} + c} a^{2} d^{2}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} a b^{2} d} - \frac{{\left (2 \, b^{3} c^{3} - a b^{2} c^{2} d - 4 \, a^{2} b c d^{2} + 3 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b^{2} d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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