3.18 \(\int \frac {(a+b x^2)^3}{(c+d x^2)^2} \, dx\)

Optimal. Leaf size=107 \[ -\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{7/2}}-\frac {x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac {b^3 x^3}{3 d^2} \]

[Out]

-b^2*(-3*a*d+2*b*c)*x/d^3+1/3*b^3*x^3/d^2-1/2*(-a*d+b*c)^3*x/c/d^3/(d*x^2+c)+1/2*(-a*d+b*c)^2*(a*d+5*b*c)*arct
an(x*d^(1/2)/c^(1/2))/c^(3/2)/d^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 107, 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} \[ -\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{7/2}}-\frac {x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac {b^3 x^3}{3 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/(c + d*x^2)^2,x]

[Out]

-((b^2*(2*b*c - 3*a*d)*x)/d^3) + (b^3*x^3)/(3*d^2) - ((b*c - a*d)^3*x)/(2*c*d^3*(c + d*x^2)) + ((b*c - a*d)^2*
(5*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(7/2))

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 )^3}{\left (c+d x^2\right )^2} \, dx &=\int \left (-\frac {b^2 (2 b c-3 a d)}{d^3}+\frac {b^3 x^2}{d^2}+\frac {(b c-a d)^2 (2 b c+a d)+3 b d (b c-a d)^2 x^2}{d^3 \left (c+d x^2\right )^2}\right ) \, dx\\ &=-\frac {b^2 (2 b c-3 a d) x}{d^3}+\frac {b^3 x^3}{3 d^2}+\frac {\int \frac {(b c-a d)^2 (2 b c+a d)+3 b d (b c-a d)^2 x^2}{\left (c+d x^2\right )^2} \, dx}{d^3}\\ &=-\frac {b^2 (2 b c-3 a d) x}{d^3}+\frac {b^3 x^3}{3 d^2}-\frac {(b c-a d)^3 x}{2 c d^3 \left (c+d x^2\right )}+\frac {\left ((b c-a d)^2 (5 b c+a d)\right ) \int \frac {1}{c+d x^2} \, dx}{2 c d^3}\\ &=-\frac {b^2 (2 b c-3 a d) x}{d^3}+\frac {b^3 x^3}{3 d^2}-\frac {(b c-a d)^3 x}{2 c d^3 \left (c+d x^2\right )}+\frac {(b c-a d)^2 (5 b c+a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{7/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 107, normalized size = 1.00 \[ -\frac {b^2 x (2 b c-3 a d)}{d^3}+\frac {(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} d^{7/2}}-\frac {x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac {b^3 x^3}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*(2*b*c - 3*a*d)*x)/d^3) + (b^3*x^3)/(3*d^2) - ((b*c - a*d)^3*x)/(2*c*d^3*(c + d*x^2)) + ((b*c - a*d)^2*
(5*b*c + a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(2*c^(3/2)*d^(7/2))

________________________________________________________________________________________

fricas [B]  time = 0.62, size = 444, normalized size = 4.15 \[ \left [\frac {4 \, b^{3} c^{2} d^{3} x^{5} - 4 \, {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3}\right )} x^{3} - 3 \, {\left (5 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (5 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 6 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x}{12 \, {\left (c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}, \frac {2 \, b^{3} c^{2} d^{3} x^{5} - 2 \, {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3}\right )} x^{3} + 3 \, {\left (5 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (5 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - 3 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} x}{6 \, {\left (c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

[1/12*(4*b^3*c^2*d^3*x^5 - 4*(5*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3)*x^3 - 3*(5*b^3*c^4 - 9*a*b^2*c^3*d + 3*a^2*b*c^
2*d^2 + a^3*c*d^3 + (5*b^3*c^3*d - 9*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + a^3*d^4)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*s
qrt(-c*d)*x - c)/(d*x^2 + c)) - 6*(5*b^3*c^4*d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*x)/(c^2*d^5*x^
2 + c^3*d^4), 1/6*(2*b^3*c^2*d^3*x^5 - 2*(5*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3)*x^3 + 3*(5*b^3*c^4 - 9*a*b^2*c^3*d
+ 3*a^2*b*c^2*d^2 + a^3*c*d^3 + (5*b^3*c^3*d - 9*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + a^3*d^4)*x^2)*sqrt(c*d)*arcta
n(sqrt(c*d)*x/c) - 3*(5*b^3*c^4*d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*x)/(c^2*d^5*x^2 + c^3*d^4)]

________________________________________________________________________________________

giac [A]  time = 0.58, size = 152, normalized size = 1.42 \[ \frac {{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c d^{3}} - \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \, {\left (d x^{2} + c\right )} c d^{3}} + \frac {b^{3} d^{4} x^{3} - 6 \, b^{3} c d^{3} x + 9 \, a b^{2} d^{4} x}{3 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/2*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c*d^3) - 1/2*(b^3*c
^3*x - 3*a*b^2*c^2*d*x + 3*a^2*b*c*d^2*x - a^3*d^3*x)/((d*x^2 + c)*c*d^3) + 1/3*(b^3*d^4*x^3 - 6*b^3*c*d^3*x +
9*a*b^2*d^4*x)/d^6

________________________________________________________________________________________

maple [B]  time = 0.01, size = 205, normalized size = 1.92 \[ \frac {b^{3} x^{3}}{3 d^{2}}+\frac {a^{3} x}{2 \left (d \,x^{2}+c \right ) c}+\frac {a^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, c}-\frac {3 a^{2} b x}{2 \left (d \,x^{2}+c \right ) d}+\frac {3 a^{2} b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, d}+\frac {3 a \,b^{2} c x}{2 \left (d \,x^{2}+c \right ) d^{2}}-\frac {9 a \,b^{2} c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, d^{2}}-\frac {b^{3} c^{2} x}{2 \left (d \,x^{2}+c \right ) d^{3}}+\frac {5 b^{3} c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}\, d^{3}}+\frac {3 a \,b^{2} x}{d^{2}}-\frac {2 b^{3} c x}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/(d*x^2+c)^2,x)

[Out]

1/3*b^3*x^3/d^2+3*b^2/d^2*a*x-2*b^3/d^3*x*c+1/2/c*x/(d*x^2+c)*a^3-3/2/d*x/(d*x^2+c)*a^2*b+3/2/d^2*c*x/(d*x^2+c
)*a*b^2-1/2/d^3*c^2*x/(d*x^2+c)*b^3+1/2/c/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*a^3+3/2/d/(c*d)^(1/2)*arctan(1
/(c*d)^(1/2)*d*x)*a^2*b-9/2/d^2*c/(c*d)^(1/2)*arctan(1/(c*d)^(1/2)*d*x)*a*b^2+5/2/d^3*c^2/(c*d)^(1/2)*arctan(1
/(c*d)^(1/2)*d*x)*b^3

________________________________________________________________________________________

maxima [A]  time = 2.99, size = 147, normalized size = 1.37 \[ -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{2 \, {\left (c d^{4} x^{2} + c^{2} d^{3}\right )}} + \frac {b^{3} d x^{3} - 3 \, {\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x}{3 \, d^{3}} + \frac {{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x/(c*d^4*x^2 + c^2*d^3) + 1/3*(b^3*d*x^3 - 3*(2*b^3*c
- 3*a*b^2*d)*x)/d^3 + 1/2*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*arctan(d*x/sqrt(c*d))/(sqrt(c
*d)*c*d^3)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 181, normalized size = 1.69 \[ x\,\left (\frac {3\,a\,b^2}{d^2}-\frac {2\,b^3\,c}{d^3}\right )+\frac {b^3\,x^3}{3\,d^2}+\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,c\,\left (d^4\,x^2+c\,d^3\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+5\,b\,c\right )}{\sqrt {c}\,\left (a^3\,d^3+3\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+5\,b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d+5\,b\,c\right )}{2\,c^{3/2}\,d^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/(c + d*x^2)^2,x)

[Out]

x*((3*a*b^2)/d^2 - (2*b^3*c)/d^3) + (b^3*x^3)/(3*d^2) + (x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)
)/(2*c*(c*d^3 + d^4*x^2)) + (atan((d^(1/2)*x*(a*d - b*c)^2*(a*d + 5*b*c))/(c^(1/2)*(a^3*d^3 + 5*b^3*c^3 - 9*a*
b^2*c^2*d + 3*a^2*b*c*d^2)))*(a*d - b*c)^2*(a*d + 5*b*c))/(2*c^(3/2)*d^(7/2))

________________________________________________________________________________________

sympy [B]  time = 1.05, size = 314, normalized size = 2.93 \[ \frac {b^{3} x^{3}}{3 d^{2}} + x \left (\frac {3 a b^{2}}{d^{2}} - \frac {2 b^{3} c}{d^{3}}\right ) + \frac {x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 c^{2} d^{3} + 2 c d^{4} x^{2}} - \frac {\sqrt {- \frac {1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log {\left (- \frac {c^{2} d^{3} \sqrt {- \frac {1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log {\left (\frac {c^{2} d^{3} \sqrt {- \frac {1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/(d*x**2+c)**2,x)

[Out]

b**3*x**3/(3*d**2) + x*(3*a*b**2/d**2 - 2*b**3*c/d**3) + x*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b*
*3*c**3)/(2*c**2*d**3 + 2*c*d**4*x**2) - sqrt(-1/(c**3*d**7))*(a*d - b*c)**2*(a*d + 5*b*c)*log(-c**2*d**3*sqrt
(-1/(c**3*d**7))*(a*d - b*c)**2*(a*d + 5*b*c)/(a**3*d**3 + 3*a**2*b*c*d**2 - 9*a*b**2*c**2*d + 5*b**3*c**3) +
x)/4 + sqrt(-1/(c**3*d**7))*(a*d - b*c)**2*(a*d + 5*b*c)*log(c**2*d**3*sqrt(-1/(c**3*d**7))*(a*d - b*c)**2*(a*
d + 5*b*c)/(a**3*d**3 + 3*a**2*b*c*d**2 - 9*a*b**2*c**2*d + 5*b**3*c**3) + x)/4

________________________________________________________________________________________