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

Optimal. Leaf size=77 \[ \frac{a^{3/2} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^3 (b c-a d)}{3 b^2}-\frac{a x (b c-a d)}{b^3}+\frac{d x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0516394, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {459, 302, 205} \[ \frac{a^{3/2} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^3 (b c-a d)}{3 b^2}-\frac{a x (b c-a d)}{b^3}+\frac{d x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^2))/(a + b*x^2),x]

[Out]

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

Rule 459

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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

Mathematica [A]  time = 0.050827, size = 77, normalized size = 1. \[ -\frac{a^{3/2} (a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{7/2}}+\frac{x^3 (b c-a d)}{3 b^2}+\frac{a x (a d-b c)}{b^3}+\frac{d x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 92, normalized size = 1.2 \begin{align*}{\frac{d{x}^{5}}{5\,b}}-{\frac{{x}^{3}ad}{3\,{b}^{2}}}+{\frac{c{x}^{3}}{3\,b}}+{\frac{{a}^{2}dx}{{b}^{3}}}-{\frac{acx}{{b}^{2}}}-{\frac{{a}^{3}d}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{a}^{2}c}{{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)/(b*x^2+a),x)

[Out]

1/5*d*x^5/b-1/3/b^2*x^3*a*d+1/3/b*x^3*c+1/b^3*a^2*d*x-1/b^2*a*c*x-a^3/b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*
d+a^2/b^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c

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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)/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48366, size = 381, normalized size = 4.95 \begin{align*} \left [\frac{6 \, b^{2} d x^{5} + 10 \,{\left (b^{2} c - a b d\right )} x^{3} - 15 \,{\left (a b c - a^{2} d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 30 \,{\left (a b c - a^{2} d\right )} x}{30 \, b^{3}}, \frac{3 \, b^{2} d x^{5} + 5 \,{\left (b^{2} c - a b d\right )} x^{3} + 15 \,{\left (a b c - a^{2} d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 15 \,{\left (a b c - a^{2} d\right )} x}{15 \, b^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.472932, size = 150, normalized size = 1.95 \begin{align*} \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right ) \log{\left (- \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right )}{a^{2} d - a b c} + x \right )}}{2} - \frac{\sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right ) \log{\left (\frac{b^{3} \sqrt{- \frac{a^{3}}{b^{7}}} \left (a d - b c\right )}{a^{2} d - a b c} + x \right )}}{2} + \frac{d x^{5}}{5 b} - \frac{x^{3} \left (a d - b c\right )}{3 b^{2}} + \frac{x \left (a^{2} d - a b c\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17764, size = 113, normalized size = 1.47 \begin{align*} \frac{{\left (a^{2} b c - a^{3} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, b^{4} d x^{5} + 5 \, b^{4} c x^{3} - 5 \, a b^{3} d x^{3} - 15 \, a b^{3} c x + 15 \, a^{2} b^{2} d x}{15 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*b*c - a^3*d)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*b^4*d*x^5 + 5*b^4*c*x^3 - 5*a*b^3*d*x^3 - 15
*a*b^3*c*x + 15*a^2*b^2*d*x)/b^5