3.218 \(\int \frac{x^5 \left (c+d x^2\right )^3}{a+b x^2} \, dx\)

Optimal. Leaf size=138 \[ \frac{a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^6}+\frac{d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{6 b^3}-\frac{a x^2 (b c-a d)^3}{2 b^5}+\frac{x^4 (b c-a d)^3}{4 b^4}+\frac{d^2 x^8 (3 b c-a d)}{8 b^2}+\frac{d^3 x^{10}}{10 b} \]

[Out]

-(a*(b*c - a*d)^3*x^2)/(2*b^5) + ((b*c - a*d)^3*x^4)/(4*b^4) + (d*(3*b^2*c^2 - 3
*a*b*c*d + a^2*d^2)*x^6)/(6*b^3) + (d^2*(3*b*c - a*d)*x^8)/(8*b^2) + (d^3*x^10)/
(10*b) + (a^2*(b*c - a*d)^3*Log[a + b*x^2])/(2*b^6)

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Rubi [A]  time = 0.377444, antiderivative size = 138, 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 \[ \frac{a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^6}+\frac{d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{6 b^3}-\frac{a x^2 (b c-a d)^3}{2 b^5}+\frac{x^4 (b c-a d)^3}{4 b^4}+\frac{d^2 x^8 (3 b c-a d)}{8 b^2}+\frac{d^3 x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

-(a*(b*c - a*d)^3*x^2)/(2*b^5) + ((b*c - a*d)^3*x^4)/(4*b^4) + (d*(3*b^2*c^2 - 3
*a*b*c*d + a^2*d^2)*x^6)/(6*b^3) + (d^2*(3*b*c - a*d)*x^8)/(8*b^2) + (d^3*x^10)/
(10*b) + (a^2*(b*c - a*d)^3*Log[a + b*x^2])/(2*b^6)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} \left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{d^{3} x^{10}}{10 b} - \frac{d^{2} x^{8} \left (a d - 3 b c\right )}{8 b^{2}} + \frac{d x^{6} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{6 b^{3}} - \frac{\left (a d - b c\right )^{3} \int ^{x^{2}} x\, dx}{2 b^{4}} + \frac{\left (a d - b c\right )^{3} \int ^{x^{2}} a\, dx}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

-a**2*(a*d - b*c)**3*log(a + b*x**2)/(2*b**6) + d**3*x**10/(10*b) - d**2*x**8*(a
*d - 3*b*c)/(8*b**2) + d*x**6*(a**2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/(6*b**3) - (
a*d - b*c)**3*Integral(x, (x, x**2))/(2*b**4) + (a*d - b*c)**3*Integral(a, (x, x
**2))/(2*b**5)

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Mathematica [A]  time = 0.139149, size = 128, normalized size = 0.93 \[ \frac{20 b^3 d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+60 a^2 (b c-a d)^3 \log \left (a+b x^2\right )+15 b^4 d^2 x^8 (3 b c-a d)+30 b^2 x^4 (b c-a d)^3+60 a b x^2 (a d-b c)^3+12 b^5 d^3 x^{10}}{120 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(60*a*b*(-(b*c) + a*d)^3*x^2 + 30*b^2*(b*c - a*d)^3*x^4 + 20*b^3*d*(3*b^2*c^2 -
3*a*b*c*d + a^2*d^2)*x^6 + 15*b^4*d^2*(3*b*c - a*d)*x^8 + 12*b^5*d^3*x^10 + 60*a
^2*(b*c - a*d)^3*Log[a + b*x^2])/(120*b^6)

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Maple [B]  time = 0.006, size = 263, normalized size = 1.9 \[{\frac{{d}^{3}{x}^{10}}{10\,b}}-{\frac{{x}^{8}a{d}^{3}}{8\,{b}^{2}}}+{\frac{3\,{x}^{8}c{d}^{2}}{8\,b}}+{\frac{{x}^{6}{a}^{2}{d}^{3}}{6\,{b}^{3}}}-{\frac{{x}^{6}ac{d}^{2}}{2\,{b}^{2}}}+{\frac{{x}^{6}{c}^{2}d}{2\,b}}-{\frac{{x}^{4}{a}^{3}{d}^{3}}{4\,{b}^{4}}}+{\frac{3\,{x}^{4}{a}^{2}c{d}^{2}}{4\,{b}^{3}}}-{\frac{3\,{x}^{4}a{c}^{2}d}{4\,{b}^{2}}}+{\frac{{x}^{4}{c}^{3}}{4\,b}}+{\frac{{a}^{4}{d}^{3}{x}^{2}}{2\,{b}^{5}}}-{\frac{3\,{a}^{3}c{d}^{2}{x}^{2}}{2\,{b}^{4}}}+{\frac{3\,{x}^{2}{a}^{2}{c}^{2}d}{2\,{b}^{3}}}-{\frac{a{c}^{3}{x}^{2}}{2\,{b}^{2}}}-{\frac{{a}^{5}\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{6}}}+{\frac{3\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) c{d}^{2}}{2\,{b}^{5}}}-{\frac{3\,{a}^{3}\ln \left ( b{x}^{2}+a \right ){c}^{2}d}{2\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

1/10*d^3*x^10/b-1/8/b^2*x^8*a*d^3+3/8/b*x^8*c*d^2+1/6/b^3*x^6*a^2*d^3-1/2/b^2*x^
6*a*c*d^2+1/2/b*x^6*c^2*d-1/4/b^4*x^4*a^3*d^3+3/4/b^3*x^4*a^2*c*d^2-3/4/b^2*x^4*
a*c^2*d+1/4/b*x^4*c^3+1/2/b^5*a^4*d^3*x^2-3/2/b^4*a^3*c*d^2*x^2+3/2/b^3*a^2*c^2*
d*x^2-1/2/b^2*a*c^3*x^2-1/2*a^5/b^6*ln(b*x^2+a)*d^3+3/2*a^4/b^5*ln(b*x^2+a)*c*d^
2-3/2*a^3/b^4*ln(b*x^2+a)*c^2*d+1/2*a^2/b^3*ln(b*x^2+a)*c^3

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Maxima [A]  time = 1.34387, size = 296, normalized size = 2.14 \[ \frac{12 \, b^{4} d^{3} x^{10} + 15 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{8} + 20 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{6} + 30 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} - 60 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2}}{120 \, b^{5}} + \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^5/(b*x^2 + a),x, algorithm="maxima")

[Out]

1/120*(12*b^4*d^3*x^10 + 15*(3*b^4*c*d^2 - a*b^3*d^3)*x^8 + 20*(3*b^4*c^2*d - 3*
a*b^3*c*d^2 + a^2*b^2*d^3)*x^6 + 30*(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^4 - 60*(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^2
)/b^5 + 1/2*(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)*log(b*x^2
+ a)/b^6

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Fricas [A]  time = 0.212227, size = 297, normalized size = 2.15 \[ \frac{12 \, b^{5} d^{3} x^{10} + 15 \,{\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{8} + 20 \,{\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{6} + 30 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{4} - 60 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^5/(b*x^2 + a),x, algorithm="fricas")

[Out]

1/120*(12*b^5*d^3*x^10 + 15*(3*b^5*c*d^2 - a*b^4*d^3)*x^8 + 20*(3*b^5*c^2*d - 3*
a*b^4*c*d^2 + a^2*b^3*d^3)*x^6 + 30*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 -
a^3*b^2*d^3)*x^4 - 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^
3)*x^2 + 60*(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)*log(b*x^2
+ a))/b^6

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Sympy [A]  time = 2.85442, size = 187, normalized size = 1.36 \[ - \frac{a^{2} \left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{d^{3} x^{10}}{10 b} - \frac{x^{8} \left (a d^{3} - 3 b c d^{2}\right )}{8 b^{2}} + \frac{x^{6} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{6 b^{3}} - \frac{x^{4} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{4 b^{4}} + \frac{x^{2} \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{2 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

-a**2*(a*d - b*c)**3*log(a + b*x**2)/(2*b**6) + d**3*x**10/(10*b) - x**8*(a*d**3
- 3*b*c*d**2)/(8*b**2) + x**6*(a**2*d**3 - 3*a*b*c*d**2 + 3*b**2*c**2*d)/(6*b**
3) - x**4*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - b**3*c**3)/(4*b**4) +
x**2*(a**4*d**3 - 3*a**3*b*c*d**2 + 3*a**2*b**2*c**2*d - a*b**3*c**3)/(2*b**5)

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GIAC/XCAS [A]  time = 0.227672, size = 321, normalized size = 2.33 \[ \frac{12 \, b^{4} d^{3} x^{10} + 45 \, b^{4} c d^{2} x^{8} - 15 \, a b^{3} d^{3} x^{8} + 60 \, b^{4} c^{2} d x^{6} - 60 \, a b^{3} c d^{2} x^{6} + 20 \, a^{2} b^{2} d^{3} x^{6} + 30 \, b^{4} c^{3} x^{4} - 90 \, a b^{3} c^{2} d x^{4} + 90 \, a^{2} b^{2} c d^{2} x^{4} - 30 \, a^{3} b d^{3} x^{4} - 60 \, a b^{3} c^{3} x^{2} + 180 \, a^{2} b^{2} c^{2} d x^{2} - 180 \, a^{3} b c d^{2} x^{2} + 60 \, a^{4} d^{3} x^{2}}{120 \, b^{5}} + \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^5/(b*x^2 + a),x, algorithm="giac")

[Out]

1/120*(12*b^4*d^3*x^10 + 45*b^4*c*d^2*x^8 - 15*a*b^3*d^3*x^8 + 60*b^4*c^2*d*x^6
- 60*a*b^3*c*d^2*x^6 + 20*a^2*b^2*d^3*x^6 + 30*b^4*c^3*x^4 - 90*a*b^3*c^2*d*x^4
+ 90*a^2*b^2*c*d^2*x^4 - 30*a^3*b*d^3*x^4 - 60*a*b^3*c^3*x^2 + 180*a^2*b^2*c^2*d
*x^2 - 180*a^3*b*c*d^2*x^2 + 60*a^4*d^3*x^2)/b^5 + 1/2*(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)*ln(abs(b*x^2 + a))/b^6