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

Optimal. Leaf size=98 \[ \frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}+\frac{d^3 x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0555838, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {390, 205} \[ \frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^3 (3 b c-a d)}{3 b^2}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}}+\frac{d^3 x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(a + b*x^2),x]

[Out]

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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

Mathematica [A]  time = 0.066009, size = 92, normalized size = 0.94 \[ \frac{d x \left (15 a^2 d^2-5 a b d \left (9 c+d x^2\right )+3 b^2 \left (15 c^2+5 c d x^2+d^2 x^4\right )\right )}{15 b^3}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(15*a^2*d^2 - 5*a*b*d*(9*c + d*x^2) + 3*b^2*(15*c^2 + 5*c*d*x^2 + d^2*x^4)))/(15*b^3) + ((b*c - a*d)^3*Ar
cTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))

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Maple [A]  time = 0., size = 161, normalized size = 1.6 \begin{align*}{\frac{{d}^{3}{x}^{5}}{5\,b}}-{\frac{{d}^{3}{x}^{3}a}{3\,{b}^{2}}}+{\frac{{d}^{2}{x}^{3}c}{b}}+{\frac{{a}^{2}{d}^{3}x}{{b}^{3}}}-3\,{\frac{a{d}^{2}cx}{{b}^{2}}}+3\,{\frac{d{c}^{2}x}{b}}-{\frac{{a}^{3}{d}^{3}}{{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{{a}^{2}c{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{a{c}^{2}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{3}\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((d*x^2+c)^3/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5064, size = 613, normalized size = 6.26 \begin{align*} \left [\frac{6 \, a b^{3} d^{3} x^{5} + 10 \,{\left (3 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + 15 \,{\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{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 30 \,{\left (3 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x}{30 \, a b^{4}}, \frac{3 \, a b^{3} d^{3} x^{5} + 5 \,{\left (3 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + 15 \,{\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{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 15 \,{\left (3 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x}{15 \, a b^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(6*a*b^3*d^3*x^5 + 10*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^3 + 15*(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*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 30*(3*a*b^3*c^2*d - 3*a^2*b^2*c*d^2 + a^3*
b*d^3)*x)/(a*b^4), 1/15*(3*a*b^3*d^3*x^5 + 5*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^3 + 15*(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(sqrt(a*b)*x/a) + 15*(3*a*b^3*c^2*d - 3*a^2*b^2*c*d^2 + a^3*b*d^3)*x
)/(a*b^4)]

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Sympy [B]  time = 0.719599, size = 240, normalized size = 2.45 \begin{align*} \frac{\sqrt{- \frac{1}{a b^{7}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a b^{3} \sqrt{- \frac{1}{a b^{7}}} \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{1}{a b^{7}}} \left (a d - b c\right )^{3} \log{\left (\frac{a b^{3} \sqrt{- \frac{1}{a b^{7}}} \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^{5}}{5 b} - \frac{x^{3} \left (a d^{3} - 3 b c d^{2}\right )}{3 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-1/(a*b**7))*(a*d - b*c)**3*log(-a*b**3*sqrt(-1/(a*b**7))*(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(-1/(a*b**7))*(a*d - b*c)**3*log(a*b**3*sqrt(-1/(a*b**7))*(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**5/(5*b) - x**3*(a*d**3 - 3*b*
c*d**2)/(3*b**2) + x*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d)/b**3

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Giac [A]  time = 1.17654, size = 174, normalized size = 1.78 \begin{align*} \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{3 \, b^{4} d^{3} x^{5} + 15 \, b^{4} c d^{2} x^{3} - 5 \, a b^{3} d^{3} x^{3} + 45 \, b^{4} c^{2} d x - 45 \, a b^{3} c d^{2} x + 15 \, a^{2} b^{2} d^{3} x}{15 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*b^4*d^3*x^
5 + 15*b^4*c*d^2*x^3 - 5*a*b^3*d^3*x^3 + 45*b^4*c^2*d*x - 45*a*b^3*c*d^2*x + 15*a^2*b^2*d^3*x)/b^5