3.1790 \(\int \frac{(a+b x)^4}{a c+(b c+a d) x+b d x^2} \, dx\)

Optimal. Leaf size=74 \[ -\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}+\frac{(a+b x)^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.11041, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{(b c-a d)^3 \log (c+d x)}{d^4}+\frac{b x (b c-a d)^2}{d^3}-\frac{(a+b x)^2 (b c-a d)}{2 d^2}+\frac{(a+b x)^3}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0527716, size = 74, normalized size = 1. \[ \frac{b d x \left (18 a^2 d^2+9 a b d (d x-2 c)+b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )-6 (b c-a d)^3 \log (c+d x)}{6 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*d*x*(18*a^2*d^2 + 9*a*b*d*(-2*c + d*x) + b^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2)) -
6*(b*c - a*d)^3*Log[c + d*x])/(6*d^4)

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Maple [A]  time = 0.004, size = 133, normalized size = 1.8 \[{\frac{{b}^{3}{x}^{3}}{3\,d}}+{\frac{3\,{b}^{2}{x}^{2}a}{2\,d}}-{\frac{{b}^{3}{x}^{2}c}{2\,{d}^{2}}}+3\,{\frac{{a}^{2}bx}{d}}-3\,{\frac{ac{b}^{2}x}{{d}^{2}}}+{\frac{{b}^{3}{c}^{2}x}{{d}^{3}}}+{\frac{\ln \left ( dx+c \right ){a}^{3}}{d}}-3\,{\frac{\ln \left ( dx+c \right ){a}^{2}cb}{{d}^{2}}}+3\,{\frac{\ln \left ( dx+c \right ) a{c}^{2}{b}^{2}}{{d}^{3}}}-{\frac{\ln \left ( dx+c \right ){c}^{3}{b}^{3}}{{d}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.741577, size = 154, normalized size = 2.08 \[ \frac{2 \, b^{3} d^{2} x^{3} - 3 \,{\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x}{6 \, d^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203959, size = 155, normalized size = 2.09 \[ \frac{2 \, b^{3} d^{3} x^{3} - 3 \,{\left (b^{3} c d^{2} - 3 \, a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.86333, size = 82, normalized size = 1.11 \[ \frac{b^{3} x^{3}}{3 d} + \frac{x^{2} \left (3 a b^{2} d - b^{3} c\right )}{2 d^{2}} + \frac{x \left (3 a^{2} b d^{2} - 3 a b^{2} c d + b^{3} c^{2}\right )}{d^{3}} + \frac{\left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212946, size = 157, normalized size = 2.12 \[ \frac{2 \, b^{3} d^{2} x^{3} - 3 \, b^{3} c d x^{2} + 9 \, a b^{2} d^{2} x^{2} + 6 \, b^{3} c^{2} x - 18 \, a b^{2} c d x + 18 \, a^{2} b d^{2} x}{6 \, d^{3}} - \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b^3*d^2*x^3 - 3*b^3*c*d*x^2 + 9*a*b^2*d^2*x^2 + 6*b^3*c^2*x - 18*a*b^2*c*
d*x + 18*a^2*b*d^2*x)/d^3 - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
ln(abs(d*x + c))/d^4