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*(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))

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Rubi [A]  time = 0.0961711, antiderivative size = 107, normalized size of antiderivative = 1., 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 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]

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 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]

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 = 0.0582583, size = 107, normalized size = 1. \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]

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))

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Maple [B]  time = 0.01, size = 205, normalized size = 1.9 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3\,{d}^{2}}}+3\,{\frac{a{b}^{2}x}{{d}^{2}}}-2\,{\frac{{b}^{3}xc}{{d}^{3}}}+{\frac{x{a}^{3}}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{3\,{a}^{2}bx}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{3\,acx{b}^{2}}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}x{b}^{3}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{3}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{a}^{2}b}{2\,d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{9\,a{b}^{2}c}{2\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{5\,{b}^{3}{c}^{2}}{2\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.79082, size = 896, normalized size = 8.37 \begin{align*} \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 ] \end{align*}

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)]

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Sympy [B]  time = 1.21979, size = 313, normalized size = 2.93 \begin{align*} \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^{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} + \frac{x \left (3 a b^{2} d - 2 b^{3} c\right )}{d^{3}} \end{align*}

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*(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 + x*(3*a*b**2*d - 2*b**3*c)/d**3

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Giac [A]  time = 1.63629, size = 205, normalized size = 1.92 \begin{align*} \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}} \end{align*}

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