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

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

[Out]

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

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Rubi [A]  time = 0.0637283, 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} \[ \frac{b x \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^3}-\frac{b^2 x^3 (b c-3 a d)}{3 d^2}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{7/2}}+\frac{b^3 x^5}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/(c + d*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0599627, size = 93, normalized size = 0.95 \[ \frac{b x \left (45 a^2 d^2+15 a b d \left (d x^2-3 c\right )+b^2 \left (15 c^2-5 c d x^2+3 d^2 x^4\right )\right )}{15 d^3}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(45*a^2*d^2 + 15*a*b*d*(-3*c + d*x^2) + b^2*(15*c^2 - 5*c*d*x^2 + 3*d^2*x^4)))/(15*d^3) - ((b*c - a*d)^3*
ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(7/2))

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Maple [A]  time = 0.003, size = 161, normalized size = 1.6 \begin{align*}{\frac{{b}^{3}{x}^{5}}{5\,d}}+{\frac{a{b}^{2}{x}^{3}}{d}}-{\frac{{b}^{3}{x}^{3}c}{3\,{d}^{2}}}+3\,{\frac{{a}^{2}bx}{d}}-3\,{\frac{a{b}^{2}cx}{{d}^{2}}}+{\frac{{b}^{3}{c}^{2}x}{{d}^{3}}}+{{a}^{3}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-3\,{\frac{{a}^{2}bc}{d\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+3\,{\frac{a{b}^{2}{c}^{2}}{{d}^{2}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }-{\frac{{b}^{3}{c}^{3}}{{d}^{3}}\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((b*x^2+a)^3/(d*x^2+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75112, size = 613, normalized size = 6.26 \begin{align*} \left [\frac{6 \, b^{3} c d^{3} x^{5} - 10 \,{\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c 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{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) + 30 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{30 \, c d^{4}}, \frac{3 \, b^{3} c d^{3} x^{5} - 5 \,{\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c 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{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) + 15 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{15 \, c d^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.671182, size = 240, normalized size = 2.45 \begin{align*} \frac{b^{3} x^{5}}{5 d} - \frac{\sqrt{- \frac{1}{c d^{7}}} \left (a d - b c\right )^{3} \log{\left (- \frac{c d^{3} \sqrt{- \frac{1}{c d^{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}{c d^{7}}} \left (a d - b c\right )^{3} \log{\left (\frac{c d^{3} \sqrt{- \frac{1}{c d^{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{x^{3} \left (3 a b^{2} d - b^{3} c\right )}{3 d^{2}} + \frac{x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.28352, size = 176, normalized size = 1.8 \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{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{3}} + \frac{3 \, b^{3} d^{4} x^{5} - 5 \, b^{3} c d^{3} x^{3} + 15 \, a b^{2} d^{4} x^{3} + 15 \, b^{3} c^{2} d^{2} x - 45 \, a b^{2} c d^{3} x + 45 \, a^{2} b d^{4} x}{15 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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