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

Optimal. Leaf size=119 \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{x (b c-a d)^3}{b^4}-\frac{\sqrt{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{d^3 x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.0847938, antiderivative size = 119, 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} \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{x (b c-a d)^3}{b^4}-\frac{\sqrt{a} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{d^3 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0390418, size = 118, normalized size = 0.99 \[ \frac{d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{d^2 x^5 (3 b c-a d)}{5 b^2}+\frac{x (b c-a d)^3}{b^4}+\frac{\sqrt{a} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{9/2}}+\frac{d^3 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 218, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{7}}{7\,b}}-{\frac{{x}^{5}a{d}^{3}}{5\,{b}^{2}}}+{\frac{3\,{x}^{5}c{d}^{2}}{5\,b}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{3\,{b}^{3}}}-{\frac{{x}^{3}ac{d}^{2}}{{b}^{2}}}+{\frac{{x}^{3}{c}^{2}d}{b}}-{\frac{{a}^{3}{d}^{3}x}{{b}^{4}}}+3\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{3}}}-3\,{\frac{a{c}^{2}dx}{{b}^{2}}}+{\frac{{c}^{3}x}{b}}+{\frac{{a}^{4}{d}^{3}}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{{a}^{3}c{d}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+3\,{\frac{{a}^{2}{c}^{2}d}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{a{c}^{3}}{b}\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^2*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55906, size = 757, normalized size = 6.36 \begin{align*} \left [\frac{30 \, b^{3} d^{3} x^{7} + 42 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 70 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\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 (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{210 \, b^{4}}, \frac{15 \, b^{3} d^{3} x^{7} + 21 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{5} + 35 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} - 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 105 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{105 \, b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.746042, size = 275, normalized size = 2.31 \begin{align*} - \frac{\sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3} \log{\left (- \frac{b^{4} \sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3} \log{\left (\frac{b^{4} \sqrt{- \frac{a}{b^{9}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{7}}{7 b} - \frac{x^{5} \left (a d^{3} - 3 b c d^{2}\right )}{5 b^{2}} + \frac{x^{3} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{3 b^{3}} - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22829, size = 248, normalized size = 2.08 \begin{align*} -\frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} d^{3} x^{7} + 63 \, b^{6} c d^{2} x^{5} - 21 \, a b^{5} d^{3} x^{5} + 105 \, b^{6} c^{2} d x^{3} - 105 \, a b^{5} c d^{2} x^{3} + 35 \, a^{2} b^{4} d^{3} x^{3} + 105 \, b^{6} c^{3} x - 315 \, a b^{5} c^{2} d x + 315 \, a^{2} b^{4} c d^{2} x - 105 \, a^{3} b^{3} d^{3} x}{105 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/105*(15*b^6
*d^3*x^7 + 63*b^6*c*d^2*x^5 - 21*a*b^5*d^3*x^5 + 105*b^6*c^2*d*x^3 - 105*a*b^5*c*d^2*x^3 + 35*a^2*b^4*d^3*x^3
+ 105*b^6*c^3*x - 315*a*b^5*c^2*d*x + 315*a^2*b^4*c*d^2*x - 105*a^3*b^3*d^3*x)/b^7