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

Optimal. Leaf size=104 \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{c x (b c-a d)^2}{d^4}+\frac{b^2 x^7}{7 d} \]

[Out]

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

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Rubi [A]  time = 0.0737845, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {461, 205} \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{c x (b c-a d)^2}{d^4}+\frac{b^2 x^7}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 461

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

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

Mathematica [A]  time = 0.0930724, size = 104, normalized size = 1. \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{x^3 (a d-b c)^2}{3 d^3}-\frac{c x (b c-a d)^2}{d^4}+\frac{b^2 x^7}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 176, normalized size = 1.7 \begin{align*}{\frac{{b}^{2}{x}^{7}}{7\,d}}+{\frac{2\,{x}^{5}ab}{5\,d}}-{\frac{{x}^{5}{b}^{2}c}{5\,{d}^{2}}}+{\frac{{x}^{3}{a}^{2}}{3\,d}}-{\frac{2\,{x}^{3}abc}{3\,{d}^{2}}}+{\frac{{x}^{3}{b}^{2}{c}^{2}}{3\,{d}^{3}}}-{\frac{{a}^{2}cx}{{d}^{2}}}+2\,{\frac{ab{c}^{2}x}{{d}^{3}}}-{\frac{{b}^{2}{c}^{3}x}{{d}^{4}}}+{\frac{{a}^{2}{c}^{2}}{{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{ab{c}^{3}}{{d}^{3}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}{c}^{4}}{{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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.30505, size = 649, normalized size = 6.24 \begin{align*} \left [\frac{30 \, b^{2} d^{3} x^{7} - 42 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 70 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} + 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 210 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{210 \, d^{4}}, \frac{15 \, b^{2} d^{3} x^{7} - 21 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) - 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}}\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),x, algorithm="fricas")

[Out]

[1/210*(30*b^2*d^3*x^7 - 42*(b^2*c*d^2 - 2*a*b*d^3)*x^5 + 70*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^3 + 105*(b^
2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) - 210*(b^2*c^3 - 2
*a*b*c^2*d + a^2*c*d^2)*x)/d^4, 1/105*(15*b^2*d^3*x^7 - 21*(b^2*c*d^2 - 2*a*b*d^3)*x^5 + 35*(b^2*c^2*d - 2*a*b
*c*d^2 + a^2*d^3)*x^3 + 105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - 105*(b^2*c
^3 - 2*a*b*c^2*d + a^2*c*d^2)*x)/d^4]

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Sympy [B]  time = 0.65842, size = 240, normalized size = 2.31 \begin{align*} \frac{b^{2} x^{7}}{7 d} - \frac{\sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log{\left (- \frac{d^{4} \sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log{\left (\frac{d^{4} \sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac{x^{5} \left (2 a b d - b^{2} c\right )}{5 d^{2}} + \frac{x^{3} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 d^{3}} - \frac{x \left (a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}\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),x)

[Out]

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

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Giac [A]  time = 1.20064, size = 207, normalized size = 1.99 \begin{align*} \frac{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{4}} + \frac{15 \, b^{2} d^{6} x^{7} - 21 \, b^{2} c d^{5} x^{5} + 42 \, a b d^{6} x^{5} + 35 \, b^{2} c^{2} d^{4} x^{3} - 70 \, a b c d^{5} x^{3} + 35 \, a^{2} d^{6} x^{3} - 105 \, b^{2} c^{3} d^{3} x + 210 \, a b c^{2} d^{4} x - 105 \, a^{2} c d^{5} x}{105 \, d^{7}} \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),x, algorithm="giac")

[Out]

(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d^4) + 1/105*(15*b^2*d^6*x^7 - 21*b^2*c
*d^5*x^5 + 42*a*b*d^6*x^5 + 35*b^2*c^2*d^4*x^3 - 70*a*b*c*d^5*x^3 + 35*a^2*d^6*x^3 - 105*b^2*c^3*d^3*x + 210*a
*b*c^2*d^4*x - 105*a^2*c*d^5*x)/d^7