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

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

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Rubi [A]  time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {778, 205} \begin {gather*} \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 )} \end {gather*}

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)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^
(3/2)*c^(3/2))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 778

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*(e*f + d*g) -
(c*d*f - a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {a (B d+A e)-(A c d-a B e) x}{2 a c \left (a+c x^2\right )}+\frac {(A c d+a B e) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {a (B d+A e)-(A c d-a B e) x}{2 a c \left (a+c x^2\right )}+\frac {(A c d+a B e) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 78, normalized size = 0.99 \begin {gather*} \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 )} \end {gather*}

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)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*
a^(3/2)*c^(3/2))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)}{\left (a+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 225, normalized size = 2.85 \begin {gather*} \left [-\frac {2 \, B a^{2} c d + 2 \, A a^{2} c e + {\left (A a c d + B a^{2} e + {\left (A c^{2} d + B a c e\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (A a c^{2} d - B a^{2} c e\right )} x}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac {B a^{2} c d + A a^{2} c e - {\left (A a c d + B a^{2} e + {\left (A c^{2} d + B a c e\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (A a c^{2} d - B a^{2} c e\right )} x}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \end {gather*}

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

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giac [A]  time = 0.15, size = 74, normalized size = 0.94 \begin {gather*} \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} \end {gather*}

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)

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maple [A]  time = 0.06, size = 86, normalized size = 1.09 \begin {gather*} \frac {A d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}+\frac {B e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {\frac {\left (A c d -a B e \right ) x}{2 a c}-\frac {A e +B d}{2 c}}{c \,x^{2}+a} \end {gather*}

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

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maxima [A]  time = 1.20, size = 72, normalized size = 0.91 \begin {gather*} -\frac {B a d + A a e - {\left (A c d - B a e\right )} x}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {{\left (A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \end {gather*}

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]

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

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mupad [B]  time = 1.77, size = 70, normalized size = 0.89 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (A\,c\,d+B\,a\,e\right )}{2\,a^{3/2}\,c^{3/2}}-\frac {\frac {A\,e+B\,d}{2\,c}-\frac {x\,\left (A\,c\,d-B\,a\,e\right )}{2\,a\,c}}{c\,x^2+a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((c^(1/2)*x)/a^(1/2))*(A*c*d + B*a*e))/(2*a^(3/2)*c^(3/2)) - ((A*e + B*d)/(2*c) - (x*(A*c*d - B*a*e))/(2*
a*c))/(a + c*x^2)

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sympy [A]  time = 0.92, size = 133, normalized size = 1.68 \begin {gather*} - \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}} \end {gather*}

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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