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

Optimal. Leaf size=106 \[ \frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0932229, antiderivative size = 106, 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} \[ \frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(a + b*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0604665, size = 106, normalized size = 1. \[ \frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.008, size = 205, normalized size = 1.9 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{a{d}^{3}x}{{b}^{3}}}+3\,{\frac{{d}^{2}xc}{{b}^{2}}}-{\frac{{a}^{2}x{d}^{3}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,acx{d}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,x{c}^{2}d}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{3}}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}{d}^{3}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{9\,ac{d}^{2}}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{2}d}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*d^3*x^3/b^2-2*d^3/b^3*a*x+3*d^2/b^2*x*c-1/2/b^3*a^2*x/(b*x^2+a)*d^3+3/2/b^2*a*x/(b*x^2+a)*c*d^2-3/2/b*x/(b
*x^2+a)*c^2*d+1/2/a*x/(b*x^2+a)*c^3+5/2/b^3*a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^3-9/2/b^2*a/(a*b)^(1/2)*
arctan(b*x/(a*b)^(1/2))*c*d^2+3/2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^2*d+1/2/a/(a*b)^(1/2)*arctan(b*x/(a*
b)^(1/2))*c^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((d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.81831, size = 896, normalized size = 8.45 \begin{align*} \left [\frac{4 \, a^{2} b^{3} d^{3} x^{5} + 4 \,{\left (9 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{3} - 3 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 6 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{12 \,{\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, \frac{2 \, a^{2} b^{3} d^{3} x^{5} + 2 \,{\left (9 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{6 \,{\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.23759, size = 313, normalized size = 2.95 \begin{align*} - \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 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right ) \log{\left (- \frac{a^{2} b^{3} \sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right ) \log{\left (\frac{a^{2} b^{3} \sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac{d^{3} x^{3}}{3 b^{2}} - \frac{x \left (2 a d^{3} - 3 b c d^{2}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15905, size = 205, normalized size = 1.93 \begin{align*} \frac{{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{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 (b x^{2} + a\right )} a b^{3}} + \frac{b^{4} d^{3} x^{3} + 9 \, b^{4} c d^{2} x - 6 \, a b^{3} d^{3} x}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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