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

Optimal. Leaf size=109 \[ \frac{a^4 \log (a+b x)}{b^4 (b c-a d)}+\frac{x \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}-\frac{x^2 (a d+b c)}{2 b^2 d^2}-\frac{c^4 \log (c+d x)}{d^4 (b c-a d)}+\frac{x^3}{3 b d} \]

[Out]

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

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Rubi [A]  time = 0.180058, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a^4 \log (a+b x)}{b^4 (b c-a d)}+\frac{x \left (a^2 d^2+a b c d+b^2 c^2\right )}{b^3 d^3}-\frac{x^2 (a d+b c)}{2 b^2 d^2}-\frac{c^4 \log (c+d x)}{d^4 (b c-a d)}+\frac{x^3}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4} \log{\left (a + b x \right )}}{b^{4} \left (a d - b c\right )} + \frac{c^{4} \log{\left (c + d x \right )}}{d^{4} \left (a d - b c\right )} + \frac{\left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right ) \int \frac{1}{b^{3}}\, dx}{d^{3}} + \frac{x^{3}}{3 b d} - \frac{\left (a d + b c\right ) \int x\, dx}{b^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0864412, size = 105, normalized size = 0.96 \[ \frac{6 a^4 d^4 \log (a+b x)+b d x (b c-a d) \left (6 a^2 d^2-3 a b d (d x-2 c)+b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )-6 b^4 c^4 \log (c+d x)}{6 b^4 d^4 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 116, normalized size = 1.1 \[{\frac{{x}^{3}}{3\,bd}}-{\frac{{x}^{2}a}{2\,{b}^{2}d}}-{\frac{c{x}^{2}}{2\,b{d}^{2}}}+{\frac{{a}^{2}x}{{b}^{3}d}}+{\frac{acx}{{b}^{2}{d}^{2}}}+{\frac{{c}^{2}x}{b{d}^{3}}}+{\frac{{c}^{4}\ln \left ( dx+c \right ) }{{d}^{4} \left ( ad-bc \right ) }}-{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{4} \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)/(d*x+c),x)

[Out]

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

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Maxima [A]  time = 1.35448, size = 153, normalized size = 1.4 \[ \frac{a^{4} \log \left (b x + a\right )}{b^{5} c - a b^{4} d} - \frac{c^{4} \log \left (d x + c\right )}{b c d^{4} - a d^{5}} + \frac{2 \, b^{2} d^{2} x^{3} - 3 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 6 \,{\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215863, size = 165, normalized size = 1.51 \[ \frac{6 \, a^{4} d^{4} \log \left (b x + a\right ) - 6 \, b^{4} c^{4} \log \left (d x + c\right ) + 2 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 3 \,{\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} + 6 \,{\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x}{6 \,{\left (b^{5} c d^{4} - a b^{4} d^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.23302, size = 252, normalized size = 2.31 \[ - \frac{a^{4} \log{\left (x + \frac{\frac{a^{6} d^{5}}{b \left (a d - b c\right )} - \frac{2 a^{5} c d^{4}}{a d - b c} + \frac{a^{4} b c^{2} d^{3}}{a d - b c} + a^{4} c d^{3} + a b^{3} c^{4}}{a^{4} d^{4} + b^{4} c^{4}} \right )}}{b^{4} \left (a d - b c\right )} + \frac{c^{4} \log{\left (x + \frac{a^{4} c d^{3} - \frac{a^{2} b^{3} c^{4} d}{a d - b c} + \frac{2 a b^{4} c^{5}}{a d - b c} + a b^{3} c^{4} - \frac{b^{5} c^{6}}{d \left (a d - b c\right )}}{a^{4} d^{4} + b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )} + \frac{x^{3}}{3 b d} - \frac{x^{2} \left (a d + b c\right )}{2 b^{2} d^{2}} + \frac{x \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{b^{3} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError