3.28 \(\int \frac{(c+d x^2)^4}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=142 \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac{d^4 x^5}{5 b^2} \]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*
c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(
9/2))

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Rubi [A]  time = 0.120468, antiderivative size = 142, 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} \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac{d^4 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4/(a + b*x^2)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*
c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(
9/2))

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 )^4}{\left (a+b x^2\right )^2} \, dx &=\int \left (\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac{2 d^3 (2 b c-a d) x^2}{b^3}+\frac{d^4 x^4}{b^2}+\frac{(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^2}{b^4 \left (a+b x^2\right )^2}\right ) \, dx\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac{d^4 x^5}{5 b^2}+\frac{\int \frac{(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^2}{\left (a+b x^2\right )^2} \, dx}{b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac{d^4 x^5}{5 b^2}+\frac{(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac{\left ((b c-a d)^3 (b c+7 a d)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac{d^4 x^5}{5 b^2}+\frac{(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac{(b c-a d)^3 (b c+7 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0865084, size = 142, normalized size = 1. \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac{d^4 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^4/(a + b*x^2)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*
c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(
9/2))

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Maple [B]  time = 0.009, size = 296, normalized size = 2.1 \begin{align*}{\frac{{d}^{4}{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,{d}^{4}{x}^{3}a}{3\,{b}^{3}}}+{\frac{4\,{d}^{3}{x}^{3}c}{3\,{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{4}x}{{b}^{4}}}-8\,{\frac{ac{d}^{3}x}{{b}^{3}}}+6\,{\frac{{c}^{2}{d}^{2}x}{{b}^{2}}}+{\frac{{a}^{3}x{d}^{4}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{{a}^{2}cx{d}^{3}}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{a{c}^{2}x{d}^{2}}{{b}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{x{c}^{3}d}{b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{4}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{a}^{3}{d}^{4}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+10\,{\frac{{a}^{2}c{d}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-9\,{\frac{a{c}^{2}{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+2\,{\frac{{c}^{3}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{c}^{4}}{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)^4/(b*x^2+a)^2,x)

[Out]

1/5*d^4*x^5/b^2-2/3*d^4/b^3*x^3*a+4/3*d^3/b^2*x^3*c+3*d^4/b^4*a^2*x-8*d^3/b^3*a*c*x+6*d^2/b^2*c^2*x+1/2/b^4*a^
3*x/(b*x^2+a)*d^4-2/b^3*a^2*x/(b*x^2+a)*c*d^3+3/b^2*a*x/(b*x^2+a)*c^2*d^2-2/b*x/(b*x^2+a)*c^3*d+1/2/a*x/(b*x^2
+a)*c^4-7/2/b^4*a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^4+10/b^3*a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c*d
^3-9/b^2*a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^2*d^2+2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^3*d+1/2/a/(a*
b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^4

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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)^4/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.78316, size = 1261, normalized size = 8.88 \begin{align*} \left [\frac{12 \, a^{2} b^{4} d^{4} x^{7} + 4 \,{\left (20 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 20 \,{\left (18 \, a^{2} b^{4} c^{2} d^{2} - 20 \, a^{3} b^{3} c d^{3} + 7 \, a^{4} b^{2} d^{4}\right )} x^{3} + 15 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 30 \,{\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 20 \, a^{4} b^{2} c d^{3} + 7 \, a^{5} b d^{4}\right )} x}{60 \,{\left (a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}, \frac{6 \, a^{2} b^{4} d^{4} x^{7} + 2 \,{\left (20 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 10 \,{\left (18 \, a^{2} b^{4} c^{2} d^{2} - 20 \, a^{3} b^{3} c d^{3} + 7 \, a^{4} b^{2} d^{4}\right )} x^{3} + 15 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 15 \,{\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 20 \, a^{4} b^{2} c d^{3} + 7 \, a^{5} b d^{4}\right )} x}{30 \,{\left (a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/60*(12*a^2*b^4*d^4*x^7 + 4*(20*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^5 + 20*(18*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d
^3 + 7*a^4*b^2*d^4)*x^3 + 15*(a*b^4*c^4 + 4*a^2*b^3*c^3*d - 18*a^3*b^2*c^2*d^2 + 20*a^4*b*c*d^3 - 7*a^5*d^4 +
(b^5*c^4 + 4*a*b^4*c^3*d - 18*a^2*b^3*c^2*d^2 + 20*a^3*b^2*c*d^3 - 7*a^4*b*d^4)*x^2)*sqrt(-a*b)*log((b*x^2 + 2
*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 30*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 18*a^3*b^3*c^2*d^2 - 20*a^4*b^2*c*d^3 + 7*
a^5*b*d^4)*x)/(a^2*b^6*x^2 + a^3*b^5), 1/30*(6*a^2*b^4*d^4*x^7 + 2*(20*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^5 + 10
*(18*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d^3 + 7*a^4*b^2*d^4)*x^3 + 15*(a*b^4*c^4 + 4*a^2*b^3*c^3*d - 18*a^3*b^2*c^
2*d^2 + 20*a^4*b*c*d^3 - 7*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d - 18*a^2*b^3*c^2*d^2 + 20*a^3*b^2*c*d^3 - 7*a^4*
b*d^4)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + 15*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 18*a^3*b^3*c^2*d^2 - 20*a^4*b^
2*c*d^3 + 7*a^5*b*d^4)*x)/(a^2*b^6*x^2 + a^3*b^5)]

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Sympy [B]  time = 1.70845, size = 398, normalized size = 2.8 \begin{align*} \frac{x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{2 a^{2} b^{4} + 2 a b^{5} x^{2}} + \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \log{\left (- \frac{a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \log{\left (\frac{a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} + \frac{d^{4} x^{5}}{5 b^{2}} - \frac{x^{3} \left (2 a d^{4} - 4 b c d^{3}\right )}{3 b^{3}} + \frac{x \left (3 a^{2} d^{4} - 8 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**4/(b*x**2+a)**2,x)

[Out]

x*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4)/(2*a**2*b**4 + 2*a*b**5*
x**2) + sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)*log(-a**2*b**4*sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(
7*a*d + b*c)/(7*a**4*d**4 - 20*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - b**4*c**4) + x)/4 -
sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)*log(a**2*b**4*sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b
*c)/(7*a**4*d**4 - 20*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - b**4*c**4) + x)/4 + d**4*x**5
/(5*b**2) - x**3*(2*a*d**4 - 4*b*c*d**3)/(3*b**3) + x*(3*a**2*d**4 - 8*a*b*c*d**3 + 6*b**2*c**2*d**2)/b**4

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Giac [A]  time = 1.12159, size = 297, normalized size = 2.09 \begin{align*} \frac{{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{4}} + \frac{b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{2 \,{\left (b x^{2} + a\right )} a b^{4}} + \frac{3 \, b^{8} d^{4} x^{5} + 20 \, b^{8} c d^{3} x^{3} - 10 \, a b^{7} d^{4} x^{3} + 90 \, b^{8} c^{2} d^{2} x - 120 \, a b^{7} c d^{3} x + 45 \, a^{2} b^{6} d^{4} x}{15 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/2*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*
b)*a*b^4) + 1/2*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4*d^4*x)/((b*x^2 + a)
*a*b^4) + 1/15*(3*b^8*d^4*x^5 + 20*b^8*c*d^3*x^3 - 10*a*b^7*d^4*x^3 + 90*b^8*c^2*d^2*x - 120*a*b^7*c*d^3*x + 4
5*a^2*b^6*d^4*x)/b^10