3.219 \(\int \frac{x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx\)

Optimal. Leaf size=140 \[ \frac{a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}+\frac{d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}-\frac{a x (b c-a d)^3}{b^5}+\frac{x^3 (b c-a d)^3}{3 b^4}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{d^3 x^9}{9 b} \]

[Out]

-((a*(b*c - a*d)^3*x)/b^5) + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b
*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b)
+ (a^(3/2)*(b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)

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Rubi [A]  time = 0.217926, antiderivative size = 140, 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 \[ \frac{a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}+\frac{d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}-\frac{a x (b c-a d)^3}{b^5}+\frac{x^3 (b c-a d)^3}{3 b^4}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{d^3 x^9}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

-((a*(b*c - a*d)^3*x)/b^5) + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b
*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b)
+ (a^(3/2)*(b*c - a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{\frac{3}{2}} \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{11}{2}}} + \frac{d^{3} x^{9}}{9 b} - \frac{d^{2} x^{7} \left (a d - 3 b c\right )}{7 b^{2}} + \frac{d x^{5} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{5 b^{3}} - \frac{x^{3} \left (a d - b c\right )^{3}}{3 b^{4}} + \frac{\left (a d - b c\right )^{3} \int a\, dx}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

-a**(3/2)*(a*d - b*c)**3*atan(sqrt(b)*x/sqrt(a))/b**(11/2) + d**3*x**9/(9*b) - d
**2*x**7*(a*d - 3*b*c)/(7*b**2) + d*x**5*(a**2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/(
5*b**3) - x**3*(a*d - b*c)**3/(3*b**4) + (a*d - b*c)**3*Integral(a, x)/b**5

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Mathematica [A]  time = 0.078615, size = 140, normalized size = 1. \[ -\frac{a^{3/2} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}+\frac{d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{a x (a d-b c)^3}{b^5}+\frac{x^3 (b c-a d)^3}{3 b^4}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{d^3 x^9}{9 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(a*(-(b*c) + a*d)^3*x)/b^5 + ((b*c - a*d)^3*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b
*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^2*(3*b*c - a*d)*x^7)/(7*b^2) + (d^3*x^9)/(9*b)
- (a^(3/2)*(-(b*c) + a*d)^3*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)

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Maple [B]  time = 0.006, size = 276, normalized size = 2. \[{\frac{{d}^{3}{x}^{9}}{9\,b}}-{\frac{{x}^{7}a{d}^{3}}{7\,{b}^{2}}}+{\frac{3\,{x}^{7}c{d}^{2}}{7\,b}}+{\frac{{x}^{5}{a}^{2}{d}^{3}}{5\,{b}^{3}}}-{\frac{3\,{x}^{5}ac{d}^{2}}{5\,{b}^{2}}}+{\frac{3\,{x}^{5}{c}^{2}d}{5\,b}}-{\frac{{x}^{3}{a}^{3}{d}^{3}}{3\,{b}^{4}}}+{\frac{{x}^{3}{a}^{2}c{d}^{2}}{{b}^{3}}}-{\frac{{x}^{3}a{c}^{2}d}{{b}^{2}}}+{\frac{{x}^{3}{c}^{3}}{3\,b}}+{\frac{{a}^{4}{d}^{3}x}{{b}^{5}}}-3\,{\frac{{a}^{3}c{d}^{2}x}{{b}^{4}}}+3\,{\frac{x{a}^{2}{c}^{2}d}{{b}^{3}}}-{\frac{a{c}^{3}x}{{b}^{2}}}-{\frac{{a}^{5}{d}^{3}}{{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{{a}^{4}c{d}^{2}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{{a}^{3}{c}^{2}d}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{a}^{2}{c}^{3}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

1/9*d^3*x^9/b-1/7/b^2*x^7*a*d^3+3/7/b*x^7*c*d^2+1/5/b^3*x^5*a^2*d^3-3/5/b^2*x^5*
a*c*d^2+3/5/b*x^5*c^2*d-1/3/b^4*x^3*a^3*d^3+1/b^3*x^3*a^2*c*d^2-1/b^2*x^3*a*c^2*
d+1/3/b*x^3*c^3+1/b^5*a^4*d^3*x-3/b^4*a^3*c*d^2*x+3/b^3*a^2*c^2*d*x-1/b^2*a*c^3*
x-a^5/b^5/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d^3+3*a^4/b^4/(a*b)^(1/2)*arctan(x
*b/(a*b)^(1/2))*c*d^2-3*a^3/b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^2*d+a^2/b^
2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^3

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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)^3*x^4/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239898, size = 1, normalized size = 0.01 \[ \left [\frac{70 \, b^{4} d^{3} x^{9} + 90 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 126 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 210 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} - 315 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 630 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{630 \, b^{5}}, \frac{35 \, b^{4} d^{3} x^{9} + 45 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 63 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 105 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} + 315 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 315 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{315 \, b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^4/(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/630*(70*b^4*d^3*x^9 + 90*(3*b^4*c*d^2 - a*b^3*d^3)*x^7 + 126*(3*b^4*c^2*d - 3
*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 + 210*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2
- a^3*b*d^3)*x^3 - 315*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*
sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 630*(a*b^3*c^3 - 3*
a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5, 1/315*(35*b^4*d^3*x^9 + 45*(3*b
^4*c*d^2 - a*b^3*d^3)*x^7 + 63*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^5 +
105*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^3 + 315*(a*b^3*c^
3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*sqrt(a/b)*arctan(x/sqrt(a/b)) - 3
15*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*x)/b^5]

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Sympy [A]  time = 3.18588, size = 338, normalized size = 2.41 \[ \frac{\sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log{\left (- \frac{b^{5} \sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log{\left (\frac{b^{5} \sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{9}}{9 b} - \frac{x^{7} \left (a d^{3} - 3 b c d^{2}\right )}{7 b^{2}} + \frac{x^{5} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{5 b^{3}} - \frac{x^{3} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{3 b^{4}} + \frac{x \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

sqrt(-a**3/b**11)*(a*d - b*c)**3*log(-b**5*sqrt(-a**3/b**11)*(a*d - b*c)**3/(a**
4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3) + x)/2 - sqrt(-a**3
/b**11)*(a*d - b*c)**3*log(b**5*sqrt(-a**3/b**11)*(a*d - b*c)**3/(a**4*d**3 - 3*
a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3) + x)/2 + d**3*x**9/(9*b) - x**
7*(a*d**3 - 3*b*c*d**2)/(7*b**2) + x**5*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*
d)/(5*b**3) - x**3*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/(
3*b**4) + x*(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3)/b**
5

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GIAC/XCAS [A]  time = 0.223493, size = 325, normalized size = 2.32 \[ \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{35 \, b^{8} d^{3} x^{9} + 135 \, b^{8} c d^{2} x^{7} - 45 \, a b^{7} d^{3} x^{7} + 189 \, b^{8} c^{2} d x^{5} - 189 \, a b^{7} c d^{2} x^{5} + 63 \, a^{2} b^{6} d^{3} x^{5} + 105 \, b^{8} c^{3} x^{3} - 315 \, a b^{7} c^{2} d x^{3} + 315 \, a^{2} b^{6} c d^{2} x^{3} - 105 \, a^{3} b^{5} d^{3} x^{3} - 315 \, a b^{7} c^{3} x + 945 \, a^{2} b^{6} c^{2} d x - 945 \, a^{3} b^{5} c d^{2} x + 315 \, a^{4} b^{4} d^{3} x}{315 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^4/(b*x^2 + a),x, algorithm="giac")

[Out]

(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*arctan(b*x/sqrt(a*b))/
(sqrt(a*b)*b^5) + 1/315*(35*b^8*d^3*x^9 + 135*b^8*c*d^2*x^7 - 45*a*b^7*d^3*x^7 +
189*b^8*c^2*d*x^5 - 189*a*b^7*c*d^2*x^5 + 63*a^2*b^6*d^3*x^5 + 105*b^8*c^3*x^3
- 315*a*b^7*c^2*d*x^3 + 315*a^2*b^6*c*d^2*x^3 - 105*a^3*b^5*d^3*x^3 - 315*a*b^7*
c^3*x + 945*a^2*b^6*c^2*d*x - 945*a^3*b^5*c*d^2*x + 315*a^4*b^4*d^3*x)/b^9