3.1342 \(\int \frac{(A+B x) (d+e x)}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=79 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}-\frac{a (A e+B d)-x (A c d-a B e)}{2 a c \left (a+c x^2\right )} \]

[Out]

-(a*(B*d + A*e) - (A*c*d - a*B*e)*x)/(2*a*c*(a + c*x^2)) + ((A*c*d + a*B*e)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))

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Rubi [A]  time = 0.0948209, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}-\frac{a (A e+B d)-x (A c d-a B e)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x))/(a + c*x^2)^2,x]

[Out]

-(a*(B*d + A*e) - (A*c*d - a*B*e)*x)/(2*a*c*(a + c*x^2)) + ((A*c*d + a*B*e)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))

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Rubi in Sympy [A]  time = 10.6076, size = 60, normalized size = 0.76 \[ - \frac{\left (d + e x\right ) \left (- A c x + B a\right )}{2 a c \left (a + c x^{2}\right )} + \frac{\left (A c d + B a e\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)/(c*x**2+a)**2,x)

[Out]

-(d + e*x)*(-A*c*x + B*a)/(2*a*c*(a + c*x**2)) + (A*c*d + B*a*e)*atan(sqrt(c)*x/
sqrt(a))/(2*a**(3/2)*c**(3/2))

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Mathematica [A]  time = 0.122524, size = 78, normalized size = 0.99 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) (a B e+A c d)}{2 a^{3/2} c^{3/2}}+\frac{-a A e-a B d-a B e x+A c d x}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x))/(a + c*x^2)^2,x]

[Out]

(-(a*B*d) - a*A*e + A*c*d*x - a*B*e*x)/(2*a*c*(a + c*x^2)) + ((A*c*d + a*B*e)*Ar
cTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(3/2))

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Maple [A]  time = 0.009, size = 86, normalized size = 1.1 \[{\frac{1}{c{x}^{2}+a} \left ({\frac{ \left ( Acd-aBe \right ) x}{2\,ac}}-{\frac{Ae+Bd}{2\,c}} \right ) }+{\frac{Ad}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Be}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)/(c*x^2+a)^2,x)

[Out]

(1/2*(A*c*d-B*a*e)/a/c*x-1/2*(A*e+B*d)/c)/(c*x^2+a)+1/2/a/(a*c)^(1/2)*arctan(c*x
/(a*c)^(1/2))*A*d+1/2/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*B*e

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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 + A)*(e*x + d)/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278179, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (A a c d + B a^{2} e +{\left (A c^{2} d + B a c e\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (B a d + A a e -{\left (A c d - B a e\right )} x\right )} \sqrt{-a c}}{4 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-a c}}, \frac{{\left (A a c d + B a^{2} e +{\left (A c^{2} d + B a c e\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (B a d + A a e -{\left (A c d - B a e\right )} x\right )} \sqrt{a c}}{2 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((A*a*c*d + B*a^2*e + (A*c^2*d + B*a*c*e)*x^2)*log((2*a*c*x + (c*x^2 - a)*s
qrt(-a*c))/(c*x^2 + a)) - 2*(B*a*d + A*a*e - (A*c*d - B*a*e)*x)*sqrt(-a*c))/((a*
c^2*x^2 + a^2*c)*sqrt(-a*c)), 1/2*((A*a*c*d + B*a^2*e + (A*c^2*d + B*a*c*e)*x^2)
*arctan(sqrt(a*c)*x/a) - (B*a*d + A*a*e - (A*c*d - B*a*e)*x)*sqrt(a*c))/((a*c^2*
x^2 + a^2*c)*sqrt(a*c))]

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Sympy [A]  time = 4.46201, size = 133, normalized size = 1.68 \[ - \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (A c d + B a e\right ) \log{\left (- a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (A c d + B a e\right ) \log{\left (a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} - \frac{A a e + B a d + x \left (- A c d + B a e\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)/(c*x**2+a)**2,x)

[Out]

-sqrt(-1/(a**3*c**3))*(A*c*d + B*a*e)*log(-a**2*c*sqrt(-1/(a**3*c**3)) + x)/4 +
sqrt(-1/(a**3*c**3))*(A*c*d + B*a*e)*log(a**2*c*sqrt(-1/(a**3*c**3)) + x)/4 - (A
*a*e + B*a*d + x*(-A*c*d + B*a*e))/(2*a**2*c + 2*a*c**2*x**2)

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GIAC/XCAS [A]  time = 0.297646, size = 100, normalized size = 1.27 \[ \frac{{\left (A c d + B a e\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} + \frac{A c d x - B a x e - B a d - A a e}{2 \,{\left (c x^{2} + a\right )} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(A*c*d + B*a*e)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c) + 1/2*(A*c*d*x - B*a*x
*e - B*a*d - A*a*e)/((c*x^2 + a)*a*c)