3.1341 \(\int \frac{(A+B x) (d+e x)^2}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=112 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a e (A e+2 B d)+A c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}+\frac{B e^2 \log \left (a+c x^2\right )}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.178821, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a e (A e+2 B d)+A c d^2\right )}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )}+\frac{B e^2 \log \left (a+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.535, size = 104, normalized size = 0.93 \[ \frac{B e^{2} \log{\left (a + c x^{2} \right )}}{2 c^{2}} - \frac{\left (d + e x\right ) \left (2 a \left (A e + B d\right ) - x \left (2 A c d - 2 B a e\right )\right )}{4 a c \left (a + c x^{2}\right )} + \frac{\left (A c d^{2} + a e \left (A e + 2 B d\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.196523, size = 119, normalized size = 1.06 \[ \frac{\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (a A e^2+2 a B d e+A c d^2\right )}{a^{3/2}}+\frac{a^2 B e^2-a c (A e (2 d+e x)+B d (d+2 e x))+A c^2 d^2 x}{a \left (a+c x^2\right )}+B e^2 \log \left (a+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 154, normalized size = 1.4 \[{\frac{1}{c{x}^{2}+a} \left ( -{\frac{ \left ( Aa{e}^{2}-Ac{d}^{2}+2\,aBde \right ) x}{2\,ac}}-{\frac{2\,Acde-aB{e}^{2}+Bc{d}^{2}}{2\,{c}^{2}}} \right ) }+{\frac{B{e}^{2}\ln \left ( ac \left ( c{x}^{2}+a \right ) \right ) }{2\,{c}^{2}}}+{\frac{A{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{A{d}^{2}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Bde}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-1/2*(A*a*e^2-A*c*d^2+2*B*a*d*e)/a/c*x-1/2*(2*A*c*d*e-B*a*e^2+B*c*d^2)/c^2)/(c*
x^2+a)+1/2*B*e^2/c^2*ln(a*c*(c*x^2+a))+1/2/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))
*A*e^2+1/2/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*A*d^2+1/c/(a*c)^(1/2)*arctan(c*
x/(a*c)^(1/2))*B*d*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)^2/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305528, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2} +{\left (A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2} -{\left (A c^{2} d^{2} - 2 \, B a c d e - A a c e^{2}\right )} x -{\left (B a c e^{2} x^{2} + B a^{2} e^{2}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-a c}}, \frac{{\left (A a c^{2} d^{2} + 2 \, B a^{2} c d e + A a^{2} c e^{2} +{\left (A c^{3} d^{2} + 2 \, B a c^{2} d e + A a c^{2} e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2} -{\left (A c^{2} d^{2} - 2 \, B a c d e - A a c e^{2}\right )} x -{\left (B a c e^{2} x^{2} + B a^{2} e^{2}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((A*a*c^2*d^2 + 2*B*a^2*c*d*e + A*a^2*c*e^2 + (A*c^3*d^2 + 2*B*a*c^2*d*e +
A*a*c^2*e^2)*x^2)*log((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(B*a*c
*d^2 + 2*A*a*c*d*e - B*a^2*e^2 - (A*c^2*d^2 - 2*B*a*c*d*e - A*a*c*e^2)*x - (B*a*
c*e^2*x^2 + B*a^2*e^2)*log(c*x^2 + a))*sqrt(-a*c))/((a*c^3*x^2 + a^2*c^2)*sqrt(-
a*c)), 1/2*((A*a*c^2*d^2 + 2*B*a^2*c*d*e + A*a^2*c*e^2 + (A*c^3*d^2 + 2*B*a*c^2*
d*e + A*a*c^2*e^2)*x^2)*arctan(sqrt(a*c)*x/a) - (B*a*c*d^2 + 2*A*a*c*d*e - B*a^2
*e^2 - (A*c^2*d^2 - 2*B*a*c*d*e - A*a*c*e^2)*x - (B*a*c*e^2*x^2 + B*a^2*e^2)*log
(c*x^2 + a))*sqrt(a*c))/((a*c^3*x^2 + a^2*c^2)*sqrt(a*c))]

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Sympy [A]  time = 12.9849, size = 382, normalized size = 3.41 \[ \left (\frac{B e^{2}}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right ) \log{\left (x + \frac{- 2 B a^{2} e^{2} + 4 a^{2} c^{2} \left (\frac{B e^{2}}{2 c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right )}{A a c e^{2} + A c^{2} d^{2} + 2 B a c d e} \right )} + \left (\frac{B e^{2}}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right ) \log{\left (x + \frac{- 2 B a^{2} e^{2} + 4 a^{2} c^{2} \left (\frac{B e^{2}}{2 c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (A a e^{2} + A c d^{2} + 2 B a d e\right )}{4 a^{3} c^{4}}\right )}{A a c e^{2} + A c^{2} d^{2} + 2 B a c d e} \right )} - \frac{2 A a c d e - B a^{2} e^{2} + B a c d^{2} + x \left (A a c e^{2} - A c^{2} d^{2} + 2 B a c d e\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*e**2/(2*c**2) - sqrt(-a**3*c**5)*(A*a*e**2 + A*c*d**2 + 2*B*a*d*e)/(4*a**3*c*
*4))*log(x + (-2*B*a**2*e**2 + 4*a**2*c**2*(B*e**2/(2*c**2) - sqrt(-a**3*c**5)*(
A*a*e**2 + A*c*d**2 + 2*B*a*d*e)/(4*a**3*c**4)))/(A*a*c*e**2 + A*c**2*d**2 + 2*B
*a*c*d*e)) + (B*e**2/(2*c**2) + sqrt(-a**3*c**5)*(A*a*e**2 + A*c*d**2 + 2*B*a*d*
e)/(4*a**3*c**4))*log(x + (-2*B*a**2*e**2 + 4*a**2*c**2*(B*e**2/(2*c**2) + sqrt(
-a**3*c**5)*(A*a*e**2 + A*c*d**2 + 2*B*a*d*e)/(4*a**3*c**4)))/(A*a*c*e**2 + A*c*
*2*d**2 + 2*B*a*c*d*e)) - (2*A*a*c*d*e - B*a**2*e**2 + B*a*c*d**2 + x*(A*a*c*e**
2 - A*c**2*d**2 + 2*B*a*c*d*e))/(2*a**2*c**2 + 2*a*c**3*x**2)

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GIAC/XCAS [A]  time = 0.287774, size = 171, normalized size = 1.53 \[ \frac{B e^{2}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{{\left (A c d^{2} + 2 \, B a d e + A a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} + \frac{{\left (A c d^{2} - 2 \, B a d e - A a e^{2}\right )} x - \frac{B a c d^{2} + 2 \, A a c d e - B a^{2} e^{2}}{c}}{2 \,{\left (c x^{2} + a\right )} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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