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

Optimal. Leaf size=75 \[ \frac{a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{x^4 (b c-a d)}{4 b^2}+\frac{d x^6}{6 b} \]

[Out]

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

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Rubi [A]  time = 0.200813, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2 (b c-a d)}{2 b^3}+\frac{x^4 (b c-a d)}{4 b^2}+\frac{d x^6}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.05094, size = 71, normalized size = 0.95 \[ \frac{b x^2 \left (6 a^2 d-3 a b \left (2 c+d x^2\right )+b^2 x^2 \left (3 c+2 d x^2\right )\right )+6 a^2 (b c-a d) \log \left (a+b x^2\right )}{12 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^2*(6*a^2*d - 3*a*b*(2*c + d*x^2) + b^2*x^2*(3*c + 2*d*x^2)) + 6*a^2*(b*c -
a*d)*Log[a + b*x^2])/(12*b^4)

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Maple [A]  time = 0.004, size = 86, normalized size = 1.2 \[{\frac{d{x}^{6}}{6\,b}}-{\frac{{x}^{4}ad}{4\,{b}^{2}}}+{\frac{c{x}^{4}}{4\,b}}+{\frac{{x}^{2}{a}^{2}d}{2\,{b}^{3}}}-{\frac{a{x}^{2}c}{2\,{b}^{2}}}-{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) d}{2\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) c}{2\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(d*x^2+c)/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.33729, size = 100, normalized size = 1.33 \[ \frac{2 \, b^{2} d x^{6} + 3 \,{\left (b^{2} c - a b d\right )} x^{4} - 6 \,{\left (a b c - a^{2} d\right )} x^{2}}{12 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^2*d*x^6 + 3*(b^2*c - a*b*d)*x^4 - 6*(a*b*c - a^2*d)*x^2)/b^3 + 1/2*(a^
2*b*c - a^3*d)*log(b*x^2 + a)/b^4

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Fricas [A]  time = 0.225877, size = 101, normalized size = 1.35 \[ \frac{2 \, b^{3} d x^{6} + 3 \,{\left (b^{3} c - a b^{2} d\right )} x^{4} - 6 \,{\left (a b^{2} c - a^{2} b d\right )} x^{2} + 6 \,{\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^3*d*x^6 + 3*(b^3*c - a*b^2*d)*x^4 - 6*(a*b^2*c - a^2*b*d)*x^2 + 6*(a^2
*b*c - a^3*d)*log(b*x^2 + a))/b^4

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Sympy [A]  time = 1.76957, size = 65, normalized size = 0.87 \[ - \frac{a^{2} \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{d x^{6}}{6 b} - \frac{x^{4} \left (a d - b c\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d - a b c\right )}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229953, size = 104, normalized size = 1.39 \[ \frac{2 \, b^{2} d x^{6} + 3 \, b^{2} c x^{4} - 3 \, a b d x^{4} - 6 \, a b c x^{2} + 6 \, a^{2} d x^{2}}{12 \, b^{3}} + \frac{{\left (a^{2} b c - a^{3} d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(2*b^2*d*x^6 + 3*b^2*c*x^4 - 3*a*b*d*x^4 - 6*a*b*c*x^2 + 6*a^2*d*x^2)/b^3 +
1/2*(a^2*b*c - a^3*d)*ln(abs(b*x^2 + a))/b^4