3.1351 \(\int \frac{A+B x}{\left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{3 A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{3 A x}{8 a^2 \left (a+c x^2\right )}-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0526052, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 A \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{3 A x}{8 a^2 \left (a+c x^2\right )}-\frac{a B-A c x}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(a + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.97193, size = 66, normalized size = 0.88 \[ \frac{3 A x}{8 a^{2} \left (a + c x^{2}\right )} + \frac{3 A \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{c}} - \frac{- A c x + B a}{4 a c \left (a + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(c*x**2+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0880168, size = 71, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 B+5 a A c x+3 A c^2 x^3\right )}{\left (a+c x^2\right )^2}+3 A \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(-2*a^2*B + 5*a*A*c*x + 3*A*c^2*x^3))/(a + c*x^2)^2 + 3*A*Sqrt[c]*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c)

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Maple [A]  time = 0.005, size = 65, normalized size = 0.9 \[{\frac{2\,Acx-2\,Ba}{8\,ac \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{3\,Ax}{8\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{3\,A}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+a)^3,x)

[Out]

1/8*(2*A*c*x-2*B*a)/a/c/(c*x^2+a)^2+3/8*A*x/a^2/(c*x^2+a)+3/8*A/a^2/(a*c)^(1/2)*
arctan(c*x/(a*c)^(1/2))

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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)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272397, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (A c^{3} x^{4} + 2 \, A a c^{2} x^{2} + A a^{2} c\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (3 \, A c^{2} x^{3} + 5 \, A a c x - 2 \, B a^{2}\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{-a c}}, \frac{3 \,{\left (A c^{3} x^{4} + 2 \, A a c^{2} x^{2} + A a^{2} c\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (3 \, A c^{2} x^{3} + 5 \, A a c x - 2 \, B a^{2}\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.2985, size = 124, normalized size = 1.65 \[ A \left (- \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16}\right ) + \frac{5 A a c x + 3 A c^{2} x^{3} - 2 B a^{2}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(c*x**2+a)**3,x)

[Out]

A*(-3*sqrt(-1/(a**5*c))*log(-a**3*sqrt(-1/(a**5*c)) + x)/16 + 3*sqrt(-1/(a**5*c)
)*log(a**3*sqrt(-1/(a**5*c)) + x)/16) + (5*A*a*c*x + 3*A*c**2*x**3 - 2*B*a**2)/(
8*a**4*c + 16*a**3*c**2*x**2 + 8*a**2*c**3*x**4)

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GIAC/XCAS [A]  time = 0.303442, size = 81, normalized size = 1.08 \[ \frac{3 \, A \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2}} + \frac{3 \, A c^{2} x^{3} + 5 \, A a c x - 2 \, B a^{2}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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