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

Optimal. Leaf size=92 \[ \frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}+\frac{x (3 a d+b c)}{8 c^2 d \left (c+d x^2\right )}-\frac{x (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]

[Out]

-((b*c - a*d)*x)/(4*c*d*(c + d*x^2)^2) + ((b*c + 3*a*d)*x)/(8*c^2*d*(c + d*x^2))
+ ((b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(3/2))

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Rubi [A]  time = 0.086278, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}+\frac{x (3 a d+b c)}{8 c^2 d \left (c+d x^2\right )}-\frac{x (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x)/(4*c*d*(c + d*x^2)^2) + ((b*c + 3*a*d)*x)/(8*c^2*d*(c + d*x^2))
+ ((b*c + 3*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(3/2))

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Rubi in Sympy [A]  time = 12.9023, size = 78, normalized size = 0.85 \[ \frac{x \left (a d - b c\right )}{4 c d \left (c + d x^{2}\right )^{2}} + \frac{x \left (3 a d + b c\right )}{8 c^{2} d \left (c + d x^{2}\right )} + \frac{\left (3 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{5}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*d - b*c)/(4*c*d*(c + d*x**2)**2) + x*(3*a*d + b*c)/(8*c**2*d*(c + d*x**2))
+ (3*a*d + b*c)*atan(sqrt(d)*x/sqrt(c))/(8*c**(5/2)*d**(3/2))

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Mathematica [A]  time = 0.101606, size = 82, normalized size = 0.89 \[ \frac{(3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} d^{3/2}}+\frac{x \left (a d \left (5 c+3 d x^2\right )+b c \left (d x^2-c\right )\right )}{8 c^2 d \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*c*(-c + d*x^2) + a*d*(5*c + 3*d*x^2)))/(8*c^2*d*(c + d*x^2)^2) + ((b*c + 3
*a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(8*c^(5/2)*d^(3/2))

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Maple [A]  time = 0.011, size = 90, normalized size = 1. \[{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ({\frac{ \left ( 3\,ad+bc \right ){x}^{3}}{8\,{c}^{2}}}+{\frac{ \left ( 5\,ad-bc \right ) x}{8\,cd}} \right ) }+{\frac{3\,a}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{b}{8\,cd}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(d*x^2+c)^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.211059, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b c d^{2} + 3 \, a d^{3}\right )} x^{4} + b c^{3} + 3 \, a c^{2} d + 2 \,{\left (b c^{2} d + 3 \, a c d^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left ({\left (b c d + 3 \, a d^{2}\right )} x^{3} -{\left (b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{-c d}}{16 \,{\left (c^{2} d^{3} x^{4} + 2 \, c^{3} d^{2} x^{2} + c^{4} d\right )} \sqrt{-c d}}, \frac{{\left ({\left (b c d^{2} + 3 \, a d^{3}\right )} x^{4} + b c^{3} + 3 \, a c^{2} d + 2 \,{\left (b c^{2} d + 3 \, a c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left ({\left (b c d + 3 \, a d^{2}\right )} x^{3} -{\left (b c^{2} - 5 \, a c d\right )} x\right )} \sqrt{c d}}{8 \,{\left (c^{2} d^{3} x^{4} + 2 \, c^{3} d^{2} x^{2} + c^{4} d\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(((b*c*d^2 + 3*a*d^3)*x^4 + b*c^3 + 3*a*c^2*d + 2*(b*c^2*d + 3*a*c*d^2)*x^
2)*log((2*c*d*x + (d*x^2 - c)*sqrt(-c*d))/(d*x^2 + c)) + 2*((b*c*d + 3*a*d^2)*x^
3 - (b*c^2 - 5*a*c*d)*x)*sqrt(-c*d))/((c^2*d^3*x^4 + 2*c^3*d^2*x^2 + c^4*d)*sqrt
(-c*d)), 1/8*(((b*c*d^2 + 3*a*d^3)*x^4 + b*c^3 + 3*a*c^2*d + 2*(b*c^2*d + 3*a*c*
d^2)*x^2)*arctan(sqrt(c*d)*x/c) + ((b*c*d + 3*a*d^2)*x^3 - (b*c^2 - 5*a*c*d)*x)*
sqrt(c*d))/((c^2*d^3*x^4 + 2*c^3*d^2*x^2 + c^4*d)*sqrt(c*d))]

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Sympy [A]  time = 2.7462, size = 150, normalized size = 1.63 \[ - \frac{\sqrt{- \frac{1}{c^{5} d^{3}}} \left (3 a d + b c\right ) \log{\left (- c^{3} d \sqrt{- \frac{1}{c^{5} d^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{5} d^{3}}} \left (3 a d + b c\right ) \log{\left (c^{3} d \sqrt{- \frac{1}{c^{5} d^{3}}} + x \right )}}{16} + \frac{x^{3} \left (3 a d^{2} + b c d\right ) + x \left (5 a c d - b c^{2}\right )}{8 c^{4} d + 16 c^{3} d^{2} x^{2} + 8 c^{2} d^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(c**5*d**3))*(3*a*d + b*c)*log(-c**3*d*sqrt(-1/(c**5*d**3)) + x)/16 + s
qrt(-1/(c**5*d**3))*(3*a*d + b*c)*log(c**3*d*sqrt(-1/(c**5*d**3)) + x)/16 + (x**
3*(3*a*d**2 + b*c*d) + x*(5*a*c*d - b*c**2))/(8*c**4*d + 16*c**3*d**2*x**2 + 8*c
**2*d**3*x**4)

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GIAC/XCAS [A]  time = 0.232243, size = 105, normalized size = 1.14 \[ \frac{{\left (b c + 3 \, a d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{2} d} + \frac{b c d x^{3} + 3 \, a d^{2} x^{3} - b c^{2} x + 5 \, a c d x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b*c + 3*a*d)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^2*d) + 1/8*(b*c*d*x^3 + 3*a
*d^2*x^3 - b*c^2*x + 5*a*c*d*x)/((d*x^2 + c)^2*c^2*d)