3.184 \(\int \frac {(a+b x^2)^2}{(c+d x^2)^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac {(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{5/2}}+\frac {x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {b^2 x}{d^2} \]

[Out]

b^2*x/d^2+1/2*(-a*d+b*c)^2*x/c/d^2/(d*x^2+c)-1/2*(-a*d+b*c)*(a*d+3*b*c)*arctan(x*d^(1/2)/c^(1/2))/c^(3/2)/d^(5
/2)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {390, 385, 205} \[ -\frac {(b c-a d) (a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{5/2}}+\frac {x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {b^2 x}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(c + d*x^2)^2,x]

[Out]

(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(2*c*d^2*(c + d*x^2)) - ((b*c - a*d)*(3*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]
)/(2*c^(3/2)*d^(5/2))

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 89, normalized size = 1.09 \[ -\frac {\left (-a^2 d^2-2 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{5/2}}+\frac {x (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {b^2 x}{d^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/(c + d*x^2)^2,x]

[Out]

(b^2*x)/d^2 + ((b*c - a*d)^2*x)/(2*c*d^2*(c + d*x^2)) - ((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[d]*x)/
Sqrt[c]])/(2*c^(3/2)*d^(5/2))

________________________________________________________________________________________

fricas [B]  time = 0.51, size = 302, normalized size = 3.68 \[ \left [\frac {4 \, b^{2} c^{2} d^{2} x^{3} + {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, {\left (3 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x}{4 \, {\left (c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}}, \frac {2 \, b^{2} c^{2} d^{2} x^{3} - {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2} + {\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (3 \, b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x}{2 \, {\left (c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

[1/4*(4*b^2*c^2*d^2*x^3 + (3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*x^2)*sq
rt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + 2*(3*b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x)/(c^2*d
^4*x^2 + c^3*d^3), 1/2*(2*b^2*c^2*d^2*x^3 - (3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b*c*d^2
- a^2*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) + (3*b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*x)/(c^2*d^4*x^2 +
c^3*d^3)]

________________________________________________________________________________________

giac [A]  time = 0.36, size = 95, normalized size = 1.16 \[ \frac {b^{2} x}{d^{2}} - \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c d^{2}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (d x^{2} + c\right )} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/(d*x^2+c)^2,x, algorithm="giac")

[Out]

b^2*x/d^2 - 1/2*(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c*d^2) + 1/2*(b^2*c^2*x - 2
*a*b*c*d*x + a^2*d^2*x)/((d*x^2 + c)*c*d^2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 129, normalized size = 1.57 \[ \frac {a^{2} x}{2 \left (d \,x^{2}+c \right ) c}+\frac {a^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, c}-\frac {a b x}{\left (d \,x^{2}+c \right ) d}+\frac {a b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d}+\frac {b^{2} c x}{2 \left (d \,x^{2}+c \right ) d^{2}}-\frac {3 b^{2} c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, d^{2}}+\frac {b^{2} x}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

b^2*x/d^2+1/2/c*x/(d*x^2+c)*a^2-1/d*x/(d*x^2+c)*a*b+1/2/d^2*c*x/(d*x^2+c)*b^2+1/2/c/(c*d)^(1/2)*arctan(1/(c*d)
^(1/2)*d*x)*a^2+1/d/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*a*b-3/2/d^2*c/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*
b^2

________________________________________________________________________________________

maxima [A]  time = 2.38, size = 96, normalized size = 1.17 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {b^{2} x}{d^{2}} - \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x/(c*d^3*x^2 + c^2*d^2) + b^2*x/d^2 - 1/2*(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2
)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c*d^2)

________________________________________________________________________________________

mupad [B]  time = 0.16, size = 124, normalized size = 1.51 \[ \frac {b^2\,x}{d^2}+\frac {x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c\,\left (d^3\,x^2+c\,d^2\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,\left (a\,d-b\,c\right )\,\left (a\,d+3\,b\,c\right )}{\sqrt {c}\,\left (a^2\,d^2+2\,a\,b\,c\,d-3\,b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a\,d+3\,b\,c\right )}{2\,c^{3/2}\,d^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/(c + d*x^2)^2,x)

[Out]

(b^2*x)/d^2 + (x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(2*c*(c*d^2 + d^3*x^2)) + (atan((d^(1/2)*x*(a*d - b*c)*(a*d
+ 3*b*c))/(c^(1/2)*(a^2*d^2 - 3*b^2*c^2 + 2*a*b*c*d)))*(a*d - b*c)*(a*d + 3*b*c))/(2*c^(3/2)*d^(5/2))

________________________________________________________________________________________

sympy [B]  time = 0.73, size = 236, normalized size = 2.88 \[ \frac {b^{2} x}{d^{2}} + \frac {x \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac {\sqrt {- \frac {1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log {\left (- \frac {c^{2} d^{2} \sqrt {- \frac {1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right ) \log {\left (\frac {c^{2} d^{2} \sqrt {- \frac {1}{c^{3} d^{5}}} \left (a d - b c\right ) \left (a d + 3 b c\right )}{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

b**2*x/d**2 + x*(a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(2*c**2*d**2 + 2*c*d**3*x**2) - sqrt(-1/(c**3*d**5))*(a*d
- b*c)*(a*d + 3*b*c)*log(-c**2*d**2*sqrt(-1/(c**3*d**5))*(a*d - b*c)*(a*d + 3*b*c)/(a**2*d**2 + 2*a*b*c*d - 3*
b**2*c**2) + x)/4 + sqrt(-1/(c**3*d**5))*(a*d - b*c)*(a*d + 3*b*c)*log(c**2*d**2*sqrt(-1/(c**3*d**5))*(a*d - b
*c)*(a*d + 3*b*c)/(a**2*d**2 + 2*a*b*c*d - 3*b**2*c**2) + x)/4

________________________________________________________________________________________