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

Optimal. Leaf size=105 \[ \frac{a^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{d x^5 (2 b c-a d)}{5 b^2}+\frac{x^3 (b c-a d)^2}{3 b^3}-\frac{a x (b c-a d)^2}{b^4}+\frac{d^2 x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.069337, antiderivative size = 105, 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} \[ \frac{a^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{d x^5 (2 b c-a d)}{5 b^2}+\frac{x^3 (b c-a d)^2}{3 b^3}-\frac{a x (b c-a d)^2}{b^4}+\frac{d^2 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0947547, size = 105, normalized size = 1. \[ \frac{a^{3/2} (a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{d x^5 (2 b c-a d)}{5 b^2}+\frac{x^3 (b c-a d)^2}{3 b^3}-\frac{a x (a d-b c)^2}{b^4}+\frac{d^2 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69026, size = 649, normalized size = 6.18 \begin{align*} \left [\frac{30 \, b^{3} d^{2} x^{7} + 42 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 70 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 105 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{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^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{210 \, b^{4}}, \frac{15 \, b^{3} d^{2} x^{7} + 21 \,{\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 35 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 105 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 105 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{105 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.645278, size = 240, normalized size = 2.29 \begin{align*} - \frac{\sqrt{- \frac{a^{3}}{b^{9}}} \left (a d - b c\right )^{2} \log{\left (- \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{9}}} \left (a d - b c\right )^{2}}{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{a^{3}}{b^{9}}} \left (a d - b c\right )^{2} \log{\left (\frac{b^{4} \sqrt{- \frac{a^{3}}{b^{9}}} \left (a d - b c\right )^{2}}{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}} + x \right )}}{2} + \frac{d^{2} x^{7}}{7 b} - \frac{x^{5} \left (a d^{2} - 2 b c d\right )}{5 b^{2}} + \frac{x^{3} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 b^{3}} - \frac{x \left (a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15823, size = 207, normalized size = 1.97 \begin{align*} \frac{{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} d^{2} x^{7} + 42 \, b^{6} c d x^{5} - 21 \, a b^{5} d^{2} x^{5} + 35 \, b^{6} c^{2} x^{3} - 70 \, a b^{5} c d x^{3} + 35 \, a^{2} b^{4} d^{2} x^{3} - 105 \, a b^{5} c^{2} x + 210 \, a^{2} b^{4} c d x - 105 \, a^{3} b^{3} d^{2} x}{105 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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