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

Optimal. Leaf size=175 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 d^3}+\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^3}+\frac{b x \sqrt{a+b x^2} (2 b c-a d)}{2 c d^2}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.526759, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 d^3}+\frac{(b c-a d)^{3/2} (a d+4 b c) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{2 c^{3/2} d^3}+\frac{b x \sqrt{a+b x^2} (2 b c-a d)}{2 c d^2}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 75.1903, size = 155, normalized size = 0.89 \[ \frac{b^{\frac{3}{2}} \left (5 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 d^{3}} - \frac{b x \sqrt{a + b x^{2}} \left (a d - 2 b c\right )}{2 c d^{2}} + \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 c d \left (c + d x^{2}\right )} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (a d + 4 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{2 c^{\frac{3}{2}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.279217, size = 144, normalized size = 0.82 \[ \frac{b^{3/2} (-(4 b c-5 a d)) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+d x \sqrt{a+b x^2} \left (\frac{(b c-a d)^2}{c \left (c+d x^2\right )}+b^2\right )+\frac{(a d-b c)^{3/2} (a d+4 b c) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{3/2}}}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.037, size = 7345, normalized size = 42. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.00061, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.793526, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x