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

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

[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 = 0.205817, 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 \[ \frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}}+\frac{d^3 x^5 (4 b c-a d)}{5 b^2}+\frac{d^4 x^7}{7 b} \]

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))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{4} x^{7}}{7 b} - \frac{d^{3} x^{5} \left (a d - 4 b c\right )}{5 b^{2}} + \frac{d^{2} x^{3} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{3 b^{3}} - \frac{\left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right ) \int d\, dx}{b^{4}} + \frac{\left (a d - b c\right )^{4} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**4/(b*x**2+a),x)

[Out]

d**4*x**7/(7*b) - d**3*x**5*(a*d - 4*b*c)/(5*b**2) + d**2*x**3*(a**2*d**2 - 4*a*
b*c*d + 6*b**2*c**2)/(3*b**3) - (a*d - 2*b*c)*(a**2*d**2 - 2*a*b*c*d + 2*b**2*c*
*2)*Integral(d, x)/b**4 + (a*d - b*c)**4*atan(sqrt(b)*x/sqrt(a))/(sqrt(a)*b**(9/
2))

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Mathematica [A]  time = 0.146117, size = 136, normalized size = 0.96 \[ \frac{d x \left (-105 a^3 d^3+35 a^2 b d^2 \left (12 c+d x^2\right )-7 a b^2 d \left (90 c^2+20 c d x^2+3 d^2 x^4\right )+3 b^3 \left (140 c^3+70 c^2 d x^2+28 c d^2 x^4+5 d^3 x^6\right )\right )}{105 b^4}+\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{9/2}} \]

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)))/(10
5*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 = 0.007, size = 246, normalized size = 1.7 \[{\frac{{d}^{4}{x}^{7}}{7\,b}}-{\frac{{d}^{4}{x}^{5}a}{5\,{b}^{2}}}+{\frac{4\,{d}^{3}{x}^{5}c}{5\,b}}+{\frac{{d}^{4}{x}^{3}{a}^{2}}{3\,{b}^{3}}}-{\frac{4\,{d}^{3}{x}^{3}ac}{3\,{b}^{2}}}+2\,{\frac{{d}^{2}{x}^{3}{c}^{2}}{b}}-{\frac{{d}^{4}{a}^{3}x}{{b}^{4}}}+4\,{\frac{{a}^{2}{d}^{3}cx}{{b}^{3}}}-6\,{\frac{a{c}^{2}{d}^{2}x}{{b}^{2}}}+4\,{\frac{d{c}^{3}x}{b}}+{\frac{{a}^{4}{d}^{4}}{{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-4\,{\frac{{a}^{3}c{d}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+6\,{\frac{{a}^{2}{c}^{2}{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-4\,{\frac{a{c}^{3}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{4}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.218328, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (15 \, b^{3} d^{4} x^{7} + 21 \,{\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{5} + 35 \,{\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + 105 \,{\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt{-a b}}{210 \, \sqrt{-a b} b^{4}}, \frac{105 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (15 \, b^{3} d^{4} x^{7} + 21 \,{\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{5} + 35 \,{\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + 105 \,{\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \sqrt{a b}}{105 \, \sqrt{a b} b^{4}}\right ] \]

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

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Sympy [A]  time = 3.38733, size = 323, normalized size = 2.27 \[ - \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4} \log{\left (- \frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4} \log{\left (\frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac{d^{4} x^{7}}{7 b} - \frac{x^{5} \left (a d^{4} - 4 b c d^{3}\right )}{5 b^{2}} + \frac{x^{3} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{3 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \]

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/XCAS [A]  time = 0.234794, size = 267, normalized size = 1.88 \[ \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{15 \, b^{6} d^{4} x^{7} + 84 \, b^{6} c d^{3} x^{5} - 21 \, a b^{5} d^{4} x^{5} + 210 \, b^{6} c^{2} d^{2} x^{3} - 140 \, a b^{5} c d^{3} x^{3} + 35 \, a^{2} b^{4} d^{4} x^{3} + 420 \, b^{6} c^{3} d x - 630 \, a b^{5} c^{2} d^{2} x + 420 \, a^{2} b^{4} c d^{3} x - 105 \, a^{3} b^{3} d^{4} x}{105 \, b^{7}} \]

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 + 4
20*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