3.189 \(\int \frac{x^4 (a+b x^2)^2}{(c+d x^2)^3} \, dx\)

Optimal. Leaf size=163 \[ -\frac{x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}+\frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac{b^2 x^3}{3 d^3} \]

[Out]

-((13*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*x)/(4*c*d^4) + (b^2*x^3)/(3*d^3) + ((b*c - a*d)^2*x^5)/(4*c*d^2*(c + d*x
^2)^2) - ((b*c - a*d)*(9*b*c - a*d)*x)/(8*d^4*(c + d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sq
rt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*d^(9/2))

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Rubi [A]  time = 0.157701, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {463, 455, 1153, 205} \[ -\frac{x \left (a^2 d^2-10 a b c d+13 b^2 c^2\right )}{4 c d^4}+\frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}+\frac{x^5 (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac{x (b c-a d) (9 b c-a d)}{8 d^4 \left (c+d x^2\right )}+\frac{b^2 x^3}{3 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

-((13*b^2*c^2 - 10*a*b*c*d + a^2*d^2)*x)/(4*c*d^4) + (b^2*x^3)/(3*d^3) + ((b*c - a*d)^2*x^5)/(4*c*d^2*(c + d*x
^2)^2) - ((b*c - a*d)*(9*b*c - a*d)*x)/(8*d^4*(c + d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sq
rt[d]*x)/Sqrt[c]])/(8*Sqrt[c]*d^(9/2))

Rule 463

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> -Simp[((b*c - a*
d)^2*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b^2*e*n*(p + 1)), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a + b
*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 455

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -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]

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac{(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac{\int \frac{x^4 \left (-4 a^2 d^2+5 (b c-a d)^2-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac{(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac{\int \frac{c d (b c-a d) (9 b c-a d)-2 d^2 (b c-a d) (9 b c-a d) x^2+8 b^2 c d^3 x^4}{c+d x^2} \, dx}{8 c d^5}\\ &=\frac{(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac{\int \left (-2 d \left (13 b^2 c^2-10 a b c d+a^2 d^2\right )+8 b^2 c d^2 x^2+\frac{35 b^2 c^3 d-30 a b c^2 d^2+3 a^2 c d^3}{c+d x^2}\right ) \, dx}{8 c d^5}\\ &=-\frac{\left (13 b^2 c^2-10 a b c d+a^2 d^2\right ) x}{4 c d^4}+\frac{b^2 x^3}{3 d^3}+\frac{(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \int \frac{1}{c+d x^2} \, dx}{8 d^4}\\ &=-\frac{\left (13 b^2 c^2-10 a b c d+a^2 d^2\right ) x}{4 c d^4}+\frac{b^2 x^3}{3 d^3}+\frac{(b c-a d)^2 x^5}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c-a d) x}{8 d^4 \left (c+d x^2\right )}+\frac{\left (35 b^2 c^2-30 a b c d+3 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0890152, size = 148, normalized size = 0.91 \[ -\frac{x \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right )}{8 d^4 \left (c+d x^2\right )}+\frac{\left (3 a^2 d^2-30 a b c d+35 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{9/2}}+\frac{c x (b c-a d)^2}{4 d^4 \left (c+d x^2\right )^2}-\frac{b x (3 b c-2 a d)}{d^4}+\frac{b^2 x^3}{3 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2)^3,x]

[Out]

-((b*(3*b*c - 2*a*d)*x)/d^4) + (b^2*x^3)/(3*d^3) + (c*(b*c - a*d)^2*x)/(4*d^4*(c + d*x^2)^2) - ((13*b^2*c^2 -
18*a*b*c*d + 5*a^2*d^2)*x)/(8*d^4*(c + d*x^2)) + ((35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqr
t[c]])/(8*Sqrt[c]*d^(9/2))

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Maple [A]  time = 0.011, size = 223, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{3}}{3\,{d}^{3}}}+2\,{\frac{abx}{{d}^{3}}}-3\,{\frac{{b}^{2}cx}{{d}^{4}}}-{\frac{5\,{x}^{3}{a}^{2}}{8\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{9\,{x}^{3}abc}{4\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{13\,{x}^{3}{b}^{2}{c}^{2}}{8\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{a}^{2}cx}{8\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{7\,ab{c}^{2}x}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{3}x}{8\,{d}^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}}{8\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,abc}{4\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{35\,{b}^{2}{c}^{2}}{8\,{d}^{4}}\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(x^4*(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

1/3*b^2*x^3/d^3+2*b/d^3*a*x-3*b^2/d^4*x*c-5/8/d/(d*x^2+c)^2*x^3*a^2+9/4/d^2/(d*x^2+c)^2*x^3*a*b*c-13/8/d^3/(d*
x^2+c)^2*x^3*b^2*c^2-3/8/d^2/(d*x^2+c)^2*a^2*c*x+7/4/d^3/(d*x^2+c)^2*a*b*c^2*x-11/8/d^4/(d*x^2+c)^2*b^2*c^3*x+
3/8/d^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-15/4/d^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*c*a*b+35/8/d^4/(c
*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2*c^2

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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(x^4*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.50765, size = 1107, normalized size = 6.79 \begin{align*} \left [\frac{16 \, b^{2} c d^{4} x^{7} - 16 \,{\left (7 \, b^{2} c^{2} d^{3} - 6 \, a b c d^{4}\right )} x^{5} - 10 \,{\left (35 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 3 \, a^{2} c d^{4}\right )} x^{3} - 3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c 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 ) - 6 \,{\left (35 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3}\right )} x}{48 \,{\left (c d^{7} x^{4} + 2 \, c^{2} d^{6} x^{2} + c^{3} d^{5}\right )}}, \frac{8 \, b^{2} c d^{4} x^{7} - 8 \,{\left (7 \, b^{2} c^{2} d^{3} - 6 \, a b c d^{4}\right )} x^{5} - 5 \,{\left (35 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 3 \, a^{2} c d^{4}\right )} x^{3} + 3 \,{\left (35 \, b^{2} c^{4} - 30 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2} +{\left (35 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x^{4} + 2 \,{\left (35 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} x^{2}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) - 3 \,{\left (35 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 3 \, a^{2} c^{2} d^{3}\right )} x}{24 \,{\left (c d^{7} x^{4} + 2 \, c^{2} d^{6} x^{2} + c^{3} d^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

[1/48*(16*b^2*c*d^4*x^7 - 16*(7*b^2*c^2*d^3 - 6*a*b*c*d^4)*x^5 - 10*(35*b^2*c^3*d^2 - 30*a*b*c^2*d^3 + 3*a^2*c
*d^4)*x^3 - 3*(35*b^2*c^4 - 30*a*b*c^3*d + 3*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 30*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2
*(35*b^2*c^3*d - 30*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) -
6*(35*b^2*c^4*d - 30*a*b*c^3*d^2 + 3*a^2*c^2*d^3)*x)/(c*d^7*x^4 + 2*c^2*d^6*x^2 + c^3*d^5), 1/24*(8*b^2*c*d^4
*x^7 - 8*(7*b^2*c^2*d^3 - 6*a*b*c*d^4)*x^5 - 5*(35*b^2*c^3*d^2 - 30*a*b*c^2*d^3 + 3*a^2*c*d^4)*x^3 + 3*(35*b^2
*c^4 - 30*a*b*c^3*d + 3*a^2*c^2*d^2 + (35*b^2*c^2*d^2 - 30*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(35*b^2*c^3*d - 30*a
*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) - 3*(35*b^2*c^4*d - 30*a*b*c^3*d^2 + 3*a^2*c^2*
d^3)*x)/(c*d^7*x^4 + 2*c^2*d^6*x^2 + c^3*d^5)]

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Sympy [A]  time = 2.25218, size = 238, normalized size = 1.46 \begin{align*} \frac{b^{2} x^{3}}{3 d^{3}} - \frac{\sqrt{- \frac{1}{c d^{9}}} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log{\left (- c d^{4} \sqrt{- \frac{1}{c d^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c d^{9}}} \left (3 a^{2} d^{2} - 30 a b c d + 35 b^{2} c^{2}\right ) \log{\left (c d^{4} \sqrt{- \frac{1}{c d^{9}}} + x \right )}}{16} - \frac{x^{3} \left (5 a^{2} d^{3} - 18 a b c d^{2} + 13 b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} - 14 a b c^{2} d + 11 b^{2} c^{3}\right )}{8 c^{2} d^{4} + 16 c d^{5} x^{2} + 8 d^{6} x^{4}} + \frac{x \left (2 a b d - 3 b^{2} c\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

b**2*x**3/(3*d**3) - sqrt(-1/(c*d**9))*(3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)*log(-c*d**4*sqrt(-1/(c*d**9))
+ x)/16 + sqrt(-1/(c*d**9))*(3*a**2*d**2 - 30*a*b*c*d + 35*b**2*c**2)*log(c*d**4*sqrt(-1/(c*d**9)) + x)/16 -
(x**3*(5*a**2*d**3 - 18*a*b*c*d**2 + 13*b**2*c**2*d) + x*(3*a**2*c*d**2 - 14*a*b*c**2*d + 11*b**2*c**3))/(8*c*
*2*d**4 + 16*c*d**5*x**2 + 8*d**6*x**4) + x*(2*a*b*d - 3*b**2*c)/d**4

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Giac [A]  time = 1.13401, size = 208, normalized size = 1.28 \begin{align*} \frac{{\left (35 \, b^{2} c^{2} - 30 \, a b c d + 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} d^{4}} - \frac{13 \, b^{2} c^{2} d x^{3} - 18 \, a b c d^{2} x^{3} + 5 \, a^{2} d^{3} x^{3} + 11 \, b^{2} c^{3} x - 14 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} d^{4}} + \frac{b^{2} d^{6} x^{3} - 9 \, b^{2} c d^{5} x + 6 \, a b d^{6} x}{3 \, d^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/8*(35*b^2*c^2 - 30*a*b*c*d + 3*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d^4) - 1/8*(13*b^2*c^2*d*x^3 - 18*a
*b*c*d^2*x^3 + 5*a^2*d^3*x^3 + 11*b^2*c^3*x - 14*a*b*c^2*d*x + 3*a^2*c*d^2*x)/((d*x^2 + c)^2*d^4) + 1/3*(b^2*d
^6*x^3 - 9*b^2*c*d^5*x + 6*a*b*d^6*x)/d^9