3.12.76 \(\int \frac {(A+B x) (d+e x)}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=110 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+3 A c d)}{8 a^{5/2} c^{3/2}}+\frac {x (a B e+3 A c d)}{8 a^2 c \left (a+c x^2\right )}-\frac {a (A e+B d)-x (A c d-a B e)}{4 a c \left (a+c x^2\right )^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {778, 199, 205} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+3 A c d)}{8 a^{5/2} c^{3/2}}+\frac {x (a B e+3 A c d)}{8 a^2 c \left (a+c x^2\right )}-\frac {a (A e+B d)-x (A c d-a B e)}{4 a c \left (a+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x))/(a + c*x^2)^3,x]

[Out]

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 )^3} \, dx &=-\frac {a (B d+A e)-(A c d-a B e) x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 A c d+a B e) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {a (B d+A e)-(A c d-a B e) x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 A c d+a B e) x}{8 a^2 c \left (a+c x^2\right )}+\frac {(3 A c d+a B e) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c}\\ &=-\frac {a (B d+A e)-(A c d-a B e) x}{4 a c \left (a+c x^2\right )^2}+\frac {(3 A c d+a B e) x}{8 a^2 c \left (a+c x^2\right )}+\frac {(3 A c d+a B e) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 101, normalized size = 0.92 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) (a B e+3 A c d)}{8 a^{5/2} c^{3/2}}+\frac {-a^2 (2 A e+2 B d+B e x)+a c x \left (5 A d+B e x^2\right )+3 A c^2 d x^3}{8 a^2 c \left (a+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*A*c^2*d*x^3 - a^2*(2*B*d + 2*A*e + B*e*x) + a*c*x*(5*A*d + B*e*x^2))/(8*a^2*c*(a + c*x^2)^2) + ((3*A*c*d +
a*B*e)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*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 )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x))/(a + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(d + e*x))/(a + c*x^2)^3, x]

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fricas [A]  time = 0.43, size = 355, normalized size = 3.23 \begin {gather*} \left [-\frac {4 \, B a^{3} c d + 4 \, A a^{3} c e - 2 \, {\left (3 \, A a c^{3} d + B a^{2} c^{2} e\right )} x^{3} + {\left (3 \, A a^{2} c d + B a^{3} e + {\left (3 \, A c^{3} d + B a c^{2} e\right )} x^{4} + 2 \, {\left (3 \, A a c^{2} d + B a^{2} 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 (5 \, A a^{2} c^{2} d - B a^{3} c e\right )} x}{16 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac {2 \, B a^{3} c d + 2 \, A a^{3} c e - {\left (3 \, A a c^{3} d + B a^{2} c^{2} e\right )} x^{3} - {\left (3 \, A a^{2} c d + B a^{3} e + {\left (3 \, A c^{3} d + B a c^{2} e\right )} x^{4} + 2 \, {\left (3 \, A a c^{2} d + B a^{2} c e\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, A a^{2} c^{2} d - B a^{3} c e\right )} x}{8 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} 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)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 102, normalized size = 0.93 \begin {gather*} \frac {{\left (3 \, A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} + \frac {3 \, A c^{2} d x^{3} + B a c x^{3} e + 5 \, A a c d x - B a^{2} x e - 2 \, B a^{2} d - 2 \, A a^{2} e}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} 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)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 108, normalized size = 0.98 \begin {gather*} \frac {3 A d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}+\frac {B e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}+\frac {\frac {\left (3 A c d +a B e \right ) x^{3}}{8 a^{2}}+\frac {\left (5 A c d -a B e \right ) x}{8 a c}-\frac {A e +B d}{4 c}}{\left (c \,x^{2}+a \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)/(c*x^2+a)^3,x)

[Out]

(1/8*(3*A*c*d+B*a*e)/a^2*x^3+1/8*(5*A*c*d-B*a*e)/a/c*x-1/4*(A*e+B*d)/c)/(c*x^2+a)^2+3/8/a^2/(a*c)^(1/2)*arctan
(1/(a*c)^(1/2)*c*x)*A*d+1/8/a/c/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*e

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maxima [A]  time = 1.12, size = 114, normalized size = 1.04 \begin {gather*} -\frac {2 \, B a^{2} d + 2 \, A a^{2} e - {\left (3 \, A c^{2} d + B a c e\right )} x^{3} - {\left (5 \, A a c d - B a^{2} e\right )} x}{8 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} + \frac {{\left (3 \, A c d + B a e\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} 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)^3,x, algorithm="maxima")

[Out]

-1/8*(2*B*a^2*d + 2*A*a^2*e - (3*A*c^2*d + B*a*c*e)*x^3 - (5*A*a*c*d - B*a^2*e)*x)/(a^2*c^3*x^4 + 2*a^3*c^2*x^
2 + a^4*c) + 1/8*(3*A*c*d + B*a*e)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c)

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mupad [B]  time = 1.77, size = 100, normalized size = 0.91 \begin {gather*} \frac {\frac {x^3\,\left (3\,A\,c\,d+B\,a\,e\right )}{8\,a^2}-\frac {A\,e+B\,d}{4\,c}+\frac {x\,\left (5\,A\,c\,d-B\,a\,e\right )}{8\,a\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (3\,A\,c\,d+B\,a\,e\right )}{8\,a^{5/2}\,c^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x))/(a + c*x^2)^3,x)

[Out]

((x^3*(3*A*c*d + B*a*e))/(8*a^2) - (A*e + B*d)/(4*c) + (x*(5*A*c*d - B*a*e))/(8*a*c))/(a^2 + c^2*x^4 + 2*a*c*x
^2) + (atan((c^(1/2)*x)/a^(1/2))*(3*A*c*d + B*a*e))/(8*a^(5/2)*c^(3/2))

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sympy [A]  time = 1.93, size = 180, normalized size = 1.64 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} c^{3}}} \left (3 A c d + B a e\right ) \log {\left (- a^{3} c \sqrt {- \frac {1}{a^{5} c^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} c^{3}}} \left (3 A c d + B a e\right ) \log {\left (a^{3} c \sqrt {- \frac {1}{a^{5} c^{3}}} + x \right )}}{16} + \frac {- 2 A a^{2} e - 2 B a^{2} d + x^{3} \left (3 A c^{2} d + B a c e\right ) + x \left (5 A a c d - B a^{2} e\right )}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)/(c*x**2+a)**3,x)

[Out]

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

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