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

Optimal. Leaf size=169 \[ \frac{d x^3 \left (3 a^2 d^2-7 a b c d+5 b^2 c^2\right )}{2 b^4}+\frac{3 d^2 x^5 (7 b c-3 a d)}{10 b^3}+\frac{3 x (b c-3 a d) (b c-a d)^2}{2 b^5}-\frac{3 \sqrt{a} (b c-3 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}-\frac{x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{9 d^3 x^7}{14 b^2} \]

[Out]

(3*(b*c - 3*a*d)*(b*c - a*d)^2*x)/(2*b^5) + (d*(5*b^2*c^2 - 7*a*b*c*d + 3*a^2*d^2)*x^3)/(2*b^4) + (3*d^2*(7*b*
c - 3*a*d)*x^5)/(10*b^3) + (9*d^3*x^7)/(14*b^2) - (x^3*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*Sqrt[a]*(b*c - 3*
a*d)*(b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

________________________________________________________________________________________

Rubi [A]  time = 0.156172, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {467, 570, 205} \[ \frac{d x^3 \left (3 a^2 d^2-7 a b c d+5 b^2 c^2\right )}{2 b^4}+\frac{3 d^2 x^5 (7 b c-3 a d)}{10 b^3}+\frac{3 x (b c-3 a d) (b c-a d)^2}{2 b^5}-\frac{3 \sqrt{a} (b c-3 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}-\frac{x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{9 d^3 x^7}{14 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(b*c - 3*a*d)*(b*c - a*d)^2*x)/(2*b^5) + (d*(5*b^2*c^2 - 7*a*b*c*d + 3*a^2*d^2)*x^3)/(2*b^4) + (3*d^2*(7*b*
c - 3*a*d)*x^5)/(10*b^3) + (9*d^3*x^7)/(14*b^2) - (x^3*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*Sqrt[a]*(b*c - 3*
a*d)*(b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

Rule 467

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

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 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{x^4 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \left (c+d x^2\right )^2 \left (3 c+9 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=-\frac{x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\int \left (\frac{3 (b c-3 a d) (b c-a d)^2}{b^4}+\frac{3 d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^2}{b^3}+\frac{3 d^2 (7 b c-3 a d) x^4}{b^2}+\frac{9 d^3 x^6}{b}+\frac{3 \left (-a b^3 c^3+5 a^2 b^2 c^2 d-7 a^3 b c d^2+3 a^4 d^3\right )}{b^4 \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=\frac{3 (b c-3 a d) (b c-a d)^2 x}{2 b^5}+\frac{d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^3}{2 b^4}+\frac{3 d^2 (7 b c-3 a d) x^5}{10 b^3}+\frac{9 d^3 x^7}{14 b^2}-\frac{x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{\left (3 a (b c-3 a d) (b c-a d)^2\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^5}\\ &=\frac{3 (b c-3 a d) (b c-a d)^2 x}{2 b^5}+\frac{d \left (5 b^2 c^2-7 a b c d+3 a^2 d^2\right ) x^3}{2 b^4}+\frac{3 d^2 (7 b c-3 a d) x^5}{10 b^3}+\frac{9 d^3 x^7}{14 b^2}-\frac{x^3 \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{3 \sqrt{a} (b c-3 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0843146, size = 151, normalized size = 0.89 \[ \frac{d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac{d x^3 (b c-a d)^2}{b^4}+\frac{a x (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac{x (b c-4 a d) (b c-a d)^2}{b^5}+\frac{3 \sqrt{a} (b c-a d)^2 (3 a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}+\frac{d^3 x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - 4*a*d)*(b*c - a*d)^2*x)/b^5 + (d*(b*c - a*d)^2*x^3)/b^4 + (d^2*(3*b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^7
)/(7*b^2) + (a*(b*c - a*d)^3*x)/(2*b^5*(a + b*x^2)) + (3*Sqrt[a]*(b*c - a*d)^2*(-(b*c) + 3*a*d)*ArcTan[(Sqrt[b
]*x)/Sqrt[a]])/(2*b^(11/2))

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Maple [B]  time = 0.01, size = 302, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{7}}{7\,{b}^{2}}}-{\frac{2\,{x}^{5}a{d}^{3}}{5\,{b}^{3}}}+{\frac{3\,{x}^{5}c{d}^{2}}{5\,{b}^{2}}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{{b}^{4}}}-2\,{\frac{{x}^{3}ac{d}^{2}}{{b}^{3}}}+{\frac{{x}^{3}{c}^{2}d}{{b}^{2}}}-4\,{\frac{{a}^{3}{d}^{3}x}{{b}^{5}}}+9\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{4}}}-6\,{\frac{a{c}^{2}dx}{{b}^{3}}}+{\frac{{c}^{3}x}{{b}^{2}}}-{\frac{{a}^{4}x{d}^{3}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{a}^{3}cx{d}^{2}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}{c}^{2}dx}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{a{c}^{3}x}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{a}^{4}{d}^{3}}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{21\,{a}^{3}c{d}^{2}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{15\,{a}^{2}{c}^{2}d}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,a{c}^{3}}{2\,{b}^{2}}\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(x^4*(d*x^2+c)^3/(b*x^2+a)^2,x)

[Out]

1/7*d^3*x^7/b^2-2/5/b^3*x^5*a*d^3+3/5/b^2*x^5*c*d^2+1/b^4*x^3*a^2*d^3-2/b^3*x^3*a*c*d^2+1/b^2*x^3*c^2*d-4/b^5*
a^3*d^3*x+9/b^4*a^2*c*d^2*x-6/b^3*a*c^2*d*x+1/b^2*c^3*x-1/2*a^4/b^5*x/(b*x^2+a)*d^3+3/2*a^3/b^4*x/(b*x^2+a)*c*
d^2-3/2*a^2/b^3*x/(b*x^2+a)*c^2*d+1/2*a/b^2*x/(b*x^2+a)*c^3+9/2*a^4/b^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^
3-21/2*a^3/b^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c*d^2+15/2*a^2/b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^2*
d-3/2*a/b^2/(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(x^4*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57987, size = 1196, normalized size = 7.08 \begin{align*} \left [\frac{20 \, b^{4} d^{3} x^{9} + 12 \,{\left (7 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{7} + 28 \,{\left (5 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 3 \, a^{2} b^{2} d^{3}\right )} x^{5} + 140 \,{\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{3} - 105 \,{\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 210 \,{\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x}{140 \,{\left (b^{6} x^{2} + a b^{5}\right )}}, \frac{10 \, b^{4} d^{3} x^{9} + 6 \,{\left (7 \, b^{4} c d^{2} - 3 \, a b^{3} d^{3}\right )} x^{7} + 14 \,{\left (5 \, b^{4} c^{2} d - 7 \, a b^{3} c d^{2} + 3 \, a^{2} b^{2} d^{3}\right )} x^{5} + 70 \,{\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{3} - 105 \,{\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 5 \, a b^{3} c^{2} d + 7 \, a^{2} b^{2} c d^{2} - 3 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 105 \,{\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x}{70 \,{\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/140*(20*b^4*d^3*x^9 + 12*(7*b^4*c*d^2 - 3*a*b^3*d^3)*x^7 + 28*(5*b^4*c^2*d - 7*a*b^3*c*d^2 + 3*a^2*b^2*d^3)
*x^5 + 140*(b^4*c^3 - 5*a*b^3*c^2*d + 7*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^3 - 105*(a*b^3*c^3 - 5*a^2*b^2*c^2*d +
7*a^3*b*c*d^2 - 3*a^4*d^3 + (b^4*c^3 - 5*a*b^3*c^2*d + 7*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^2)*sqrt(-a/b)*log((b*x
^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + 210*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*x)/(b^
6*x^2 + a*b^5), 1/70*(10*b^4*d^3*x^9 + 6*(7*b^4*c*d^2 - 3*a*b^3*d^3)*x^7 + 14*(5*b^4*c^2*d - 7*a*b^3*c*d^2 + 3
*a^2*b^2*d^3)*x^5 + 70*(b^4*c^3 - 5*a*b^3*c^2*d + 7*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^3 - 105*(a*b^3*c^3 - 5*a^2*
b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3 + (b^4*c^3 - 5*a*b^3*c^2*d + 7*a^2*b^2*c*d^2 - 3*a^3*b*d^3)*x^2)*sqrt(a/
b)*arctan(b*x*sqrt(a/b)/a) + 105*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*x)/(b^6*x^2 + a*b^5
)]

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Sympy [B]  time = 1.59973, size = 382, normalized size = 2.26 \begin{align*} - \frac{x \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{3 \sqrt{- \frac{a}{b^{11}}} \left (a d - b c\right )^{2} \left (3 a d - b c\right ) \log{\left (- \frac{3 b^{5} \sqrt{- \frac{a}{b^{11}}} \left (a d - b c\right )^{2} \left (3 a d - b c\right )}{9 a^{3} d^{3} - 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{4} + \frac{3 \sqrt{- \frac{a}{b^{11}}} \left (a d - b c\right )^{2} \left (3 a d - b c\right ) \log{\left (\frac{3 b^{5} \sqrt{- \frac{a}{b^{11}}} \left (a d - b c\right )^{2} \left (3 a d - b c\right )}{9 a^{3} d^{3} - 21 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - 3 b^{3} c^{3}} + x \right )}}{4} + \frac{d^{3} x^{7}}{7 b^{2}} - \frac{x^{5} \left (2 a d^{3} - 3 b c d^{2}\right )}{5 b^{3}} + \frac{x^{3} \left (a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d\right )}{b^{4}} - \frac{x \left (4 a^{3} d^{3} - 9 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3)/(2*a*b**5 + 2*b**6*x**2) - 3*sqrt(-a/b**11
)*(a*d - b*c)**2*(3*a*d - b*c)*log(-3*b**5*sqrt(-a/b**11)*(a*d - b*c)**2*(3*a*d - b*c)/(9*a**3*d**3 - 21*a**2*
b*c*d**2 + 15*a*b**2*c**2*d - 3*b**3*c**3) + x)/4 + 3*sqrt(-a/b**11)*(a*d - b*c)**2*(3*a*d - b*c)*log(3*b**5*s
qrt(-a/b**11)*(a*d - b*c)**2*(3*a*d - b*c)/(9*a**3*d**3 - 21*a**2*b*c*d**2 + 15*a*b**2*c**2*d - 3*b**3*c**3) +
x)/4 + d**3*x**7/(7*b**2) - x**5*(2*a*d**3 - 3*b*c*d**2)/(5*b**3) + x**3*(a**2*d**3 - 2*a*b*c*d**2 + b**2*c**
2*d)/b**4 - x*(4*a**3*d**3 - 9*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3)/b**5

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Giac [A]  time = 1.16649, size = 325, normalized size = 1.92 \begin{align*} -\frac{3 \,{\left (a b^{3} c^{3} - 5 \, a^{2} b^{2} c^{2} d + 7 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{5}} + \frac{a b^{3} c^{3} x - 3 \, a^{2} b^{2} c^{2} d x + 3 \, a^{3} b c d^{2} x - a^{4} d^{3} x}{2 \,{\left (b x^{2} + a\right )} b^{5}} + \frac{5 \, b^{12} d^{3} x^{7} + 21 \, b^{12} c d^{2} x^{5} - 14 \, a b^{11} d^{3} x^{5} + 35 \, b^{12} c^{2} d x^{3} - 70 \, a b^{11} c d^{2} x^{3} + 35 \, a^{2} b^{10} d^{3} x^{3} + 35 \, b^{12} c^{3} x - 210 \, a b^{11} c^{2} d x + 315 \, a^{2} b^{10} c d^{2} x - 140 \, a^{3} b^{9} d^{3} x}{35 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(a*b^3*c^3 - 5*a^2*b^2*c^2*d + 7*a^3*b*c*d^2 - 3*a^4*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/2*(a*
b^3*c^3*x - 3*a^2*b^2*c^2*d*x + 3*a^3*b*c*d^2*x - a^4*d^3*x)/((b*x^2 + a)*b^5) + 1/35*(5*b^12*d^3*x^7 + 21*b^1
2*c*d^2*x^5 - 14*a*b^11*d^3*x^5 + 35*b^12*c^2*d*x^3 - 70*a*b^11*c*d^2*x^3 + 35*a^2*b^10*d^3*x^3 + 35*b^12*c^3*
x - 210*a*b^11*c^2*d*x + 315*a^2*b^10*c*d^2*x - 140*a^3*b^9*d^3*x)/b^14