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

Optimal. Leaf size=142 \[ \frac{d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}}+\frac{d^4 x^7}{7 b} \]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^3)/(3*b^3
) + (d^3*(4*b*c - a*d)*x^5)/(5*b^2) + (d^4*x^7)/(7*b) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b
^(9/2))

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Rubi [A]  time = 0.093036, antiderivative size = 142, 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} \[ \frac{d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}}+\frac{d^4 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4/(a + b*x^2),x]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^3)/(3*b^3
) + (d^3*(4*b*c - a*d)*x^5)/(5*b^2) + (d^4*x^7)/(7*b) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b
^(9/2))

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

Mathematica [A]  time = 0.0855929, size = 136, normalized size = 0.96 \[ \frac{d x \left (35 a^2 b d^2 \left (12 c+d x^2\right )-105 a^3 d^3-7 a b^2 d \left (90 c^2+20 c d x^2+3 d^2 x^4\right )+3 b^3 \left (70 c^2 d x^2+140 c^3+28 c d^2 x^4+5 d^3 x^6\right )\right )}{105 b^4}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(-105*a^3*d^3 + 35*a^2*b*d^2*(12*c + d*x^2) - 7*a*b^2*d*(90*c^2 + 20*c*d*x^2 + 3*d^2*x^4) + 3*b^3*(140*c^
3 + 70*c^2*d*x^2 + 28*c*d^2*x^4 + 5*d^3*x^6)))/(105*b^4) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a
]*b^(9/2))

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Maple [A]  time = 0.004, size = 246, normalized size = 1.7 \begin{align*}{\frac{{d}^{4}{x}^{7}}{7\,b}}-{\frac{{d}^{4}{x}^{5}a}{5\,{b}^{2}}}+{\frac{4\,{d}^{3}{x}^{5}c}{5\,b}}+{\frac{{d}^{4}{x}^{3}{a}^{2}}{3\,{b}^{3}}}-{\frac{4\,{d}^{3}{x}^{3}ac}{3\,{b}^{2}}}+2\,{\frac{{d}^{2}{x}^{3}{c}^{2}}{b}}-{\frac{{d}^{4}{a}^{3}x}{{b}^{4}}}+4\,{\frac{{a}^{2}{d}^{3}cx}{{b}^{3}}}-6\,{\frac{a{c}^{2}{d}^{2}x}{{b}^{2}}}+4\,{\frac{d{c}^{3}x}{b}}+{\frac{{a}^{4}{d}^{4}}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-4\,{\frac{{a}^{3}c{d}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+6\,{\frac{{a}^{2}{c}^{2}{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-4\,{\frac{a{c}^{3}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{4}\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)^4/(b*x^2+a),x)

[Out]

1/7*d^4*x^7/b-1/5*d^4/b^2*x^5*a+4/5*d^3/b*x^5*c+1/3*d^4/b^3*x^3*a^2-4/3*d^3/b^2*x^3*a*c+2*d^2/b*x^3*c^2-d^4/b^
4*a^3*x+4*d^3/b^3*a^2*c*x-6*d^2/b^2*a*c^2*x+4*d/b*c^3*x+1/b^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*a^4*d^4-4/b^
3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*a^3*c*d^3+6/b^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*a^2*c^2*d^2-4/b/(a*b
)^(1/2)*arctan(b*x/(a*b)^(1/2))*a*c^3*d+1/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^4

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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)^4/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.85122, size = 892, normalized size = 6.28 \begin{align*} \left [\frac{30 \, a b^{4} d^{4} x^{7} + 42 \,{\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 70 \,{\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} - 105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 210 \,{\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{210 \, a b^{5}}, \frac{15 \, a b^{4} d^{4} x^{7} + 21 \,{\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 35 \,{\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} + 105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 105 \,{\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{105 \, a b^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(30*a*b^4*d^4*x^7 + 42*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^5 + 70*(6*a*b^4*c^2*d^2 - 4*a^2*b^3*c*d^3 + a^3*
b^2*d^4)*x^3 - 105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a*b)*log((b*x
^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 210*(4*a*b^4*c^3*d - 6*a^2*b^3*c^2*d^2 + 4*a^3*b^2*c*d^3 - a^4*b*d^4)*
x)/(a*b^5), 1/105*(15*a*b^4*d^4*x^7 + 21*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^5 + 35*(6*a*b^4*c^2*d^2 - 4*a^2*b^3*c
*d^3 + a^3*b^2*d^4)*x^3 + 105*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a*b
)*arctan(sqrt(a*b)*x/a) + 105*(4*a*b^4*c^3*d - 6*a^2*b^3*c^2*d^2 + 4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)/(a*b^5)]

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Sympy [B]  time = 0.881547, size = 323, normalized size = 2.27 \begin{align*} - \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4} \log{\left (- \frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4} \log{\left (\frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac{d^{4} x^{7}}{7 b} - \frac{x^{5} \left (a d^{4} - 4 b c d^{3}\right )}{5 b^{2}} + \frac{x^{3} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{3 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(a*b**9))*(a*d - b*c)**4*log(-a*b**4*sqrt(-1/(a*b**9))*(a*d - b*c)**4/(a**4*d**4 - 4*a**3*b*c*d**3 +
6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4) + x)/2 + sqrt(-1/(a*b**9))*(a*d - b*c)**4*log(a*b**4*sqrt
(-1/(a*b**9))*(a*d - b*c)**4/(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**
4) + x)/2 + d**4*x**7/(7*b) - x**5*(a*d**4 - 4*b*c*d**3)/(5*b**2) + x**3*(a**2*d**4 - 4*a*b*c*d**3 + 6*b**2*c*
*2*d**2)/(3*b**3) - x*(a**3*d**4 - 4*a**2*b*c*d**3 + 6*a*b**2*c**2*d**2 - 4*b**3*c**3*d)/b**4

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Giac [A]  time = 1.11394, size = 267, normalized size = 1.88 \begin{align*} \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} d^{4} x^{7} + 84 \, b^{6} c d^{3} x^{5} - 21 \, a b^{5} d^{4} x^{5} + 210 \, b^{6} c^{2} d^{2} x^{3} - 140 \, a b^{5} c d^{3} x^{3} + 35 \, a^{2} b^{4} d^{4} x^{3} + 420 \, b^{6} c^{3} d x - 630 \, a b^{5} c^{2} d^{2} x + 420 \, a^{2} b^{4} c d^{3} x - 105 \, a^{3} b^{3} d^{4} x}{105 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4)
+ 1/105*(15*b^6*d^4*x^7 + 84*b^6*c*d^3*x^5 - 21*a*b^5*d^4*x^5 + 210*b^6*c^2*d^2*x^3 - 140*a*b^5*c*d^3*x^3 + 35
*a^2*b^4*d^4*x^3 + 420*b^6*c^3*d*x - 630*a*b^5*c^2*d^2*x + 420*a^2*b^4*c*d^3*x - 105*a^3*b^3*d^4*x)/b^7