3.241 \(\int \frac{x (c+d x)^3}{(a+b x)^2} \, dx\)

Optimal. Leaf size=103 \[ \frac{a (b c-a d)^3}{b^5 (a+b x)}+\frac{(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}+\frac{3 d x (b c-a d)^2}{b^4}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.199916, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a (b c-a d)^3}{b^5 (a+b x)}+\frac{(b c-4 a d) (b c-a d)^2 \log (a+b x)}{b^5}+\frac{3 d x (b c-a d)^2}{b^4}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a*d - b*c)**3/(b**5*(a + b*x)) + d**3*x**3/(3*b**2) - d**2*(2*a*d - 3*b*c)*I
ntegral(x, x)/b**3 + 3*d*x*(a*d - b*c)**2/b**4 - (a*d - b*c)**2*(4*a*d - b*c)*lo
g(a + b*x)/b**5

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Mathematica [A]  time = 0.121971, size = 100, normalized size = 0.97 \[ \frac{3 b^2 d^2 x^2 (3 b c-2 a d)-\frac{6 a (a d-b c)^3}{a+b x}+18 b d x (b c-a d)^2+6 (b c-4 a d) (b c-a d)^2 \log (a+b x)+2 b^3 d^3 x^3}{6 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 205, normalized size = 2. \[{\frac{{d}^{3}{x}^{3}}{3\,{b}^{2}}}-{\frac{{d}^{3}{x}^{2}a}{{b}^{3}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{3}x}{{b}^{4}}}-6\,{\frac{ac{d}^{2}x}{{b}^{3}}}+3\,{\frac{{c}^{2}dx}{{b}^{2}}}-4\,{\frac{\ln \left ( bx+a \right ){a}^{3}{d}^{3}}{{b}^{5}}}+9\,{\frac{\ln \left ( bx+a \right ){a}^{2}c{d}^{2}}{{b}^{4}}}-6\,{\frac{\ln \left ( bx+a \right ) a{c}^{2}d}{{b}^{3}}}+{\frac{\ln \left ( bx+a \right ){c}^{3}}{{b}^{2}}}-{\frac{{a}^{4}{d}^{3}}{{b}^{5} \left ( bx+a \right ) }}+3\,{\frac{{a}^{3}c{d}^{2}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}{c}^{2}d}{{b}^{3} \left ( bx+a \right ) }}+{\frac{a{c}^{3}}{{b}^{2} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(d*x+c)^3/(b*x+a)^2,x)

[Out]

1/3*d^3*x^3/b^2-d^3/b^3*x^2*a+3/2*d^2/b^2*x^2*c+3*d^3/b^4*a^2*x-6*d^2/b^3*a*c*x+
3*d/b^2*c^2*x-4/b^5*ln(b*x+a)*a^3*d^3+9/b^4*ln(b*x+a)*a^2*c*d^2-6/b^3*ln(b*x+a)*
a*c^2*d+1/b^2*ln(b*x+a)*c^3-a^4/b^5/(b*x+a)*d^3+3*a^3/b^4/(b*x+a)*c*d^2-3*a^2/b^
3/(b*x+a)*c^2*d+a/b^2/(b*x+a)*c^3

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Maxima [A]  time = 1.33623, size = 224, normalized size = 2.17 \[ \frac{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{b^{6} x + a b^{5}} + \frac{2 \, b^{2} d^{3} x^{3} + 3 \,{\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{2} + 18 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{4}} + \frac{{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)/(b^6*x + a*b^5) + 1/6*(2
*b^2*d^3*x^3 + 3*(3*b^2*c*d^2 - 2*a*b*d^3)*x^2 + 18*(b^2*c^2*d - 2*a*b*c*d^2 + a
^2*d^3)*x)/b^4 + (b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*log(b*x +
a)/b^5

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Fricas [A]  time = 0.210492, size = 332, normalized size = 3.22 \[ \frac{2 \, b^{4} d^{3} x^{4} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} +{\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 3 \,{\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{2} + 18 \,{\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x + 6 \,{\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{6} x + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b^4*d^3*x^4 + 6*a*b^3*c^3 - 18*a^2*b^2*c^2*d + 18*a^3*b*c*d^2 - 6*a^4*d^3
+ (9*b^4*c*d^2 - 4*a*b^3*d^3)*x^3 + 3*(6*b^4*c^2*d - 9*a*b^3*c*d^2 + 4*a^2*b^2*
d^3)*x^2 + 18*(a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + a^3*b*d^3)*x + 6*(a*b^3*c^3 - 6*a
^2*b^2*c^2*d + 9*a^3*b*c*d^2 - 4*a^4*d^3 + (b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2*b^2*
c*d^2 - 4*a^3*b*d^3)*x)*log(b*x + a))/(b^6*x + a*b^5)

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Sympy [A]  time = 5.39421, size = 146, normalized size = 1.42 \[ - \frac{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}}{a b^{5} + b^{6} x} + \frac{d^{3} x^{3}}{3 b^{2}} - \frac{x^{2} \left (2 a d^{3} - 3 b c d^{2}\right )}{2 b^{3}} + \frac{x \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{4}} - \frac{\left (a d - b c\right )^{2} \left (4 a d - b c\right ) \log{\left (a + b x \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.319496, size = 312, normalized size = 3.03 \[ \frac{\frac{{\left (2 \, d^{3} + \frac{3 \,{\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}{\left (b x + a\right )}^{3}}{b^{4}} - \frac{6 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} + \frac{6 \,{\left (\frac{a b^{6} c^{3}}{b x + a} - \frac{3 \, a^{2} b^{5} c^{2} d}{b x + a} + \frac{3 \, a^{3} b^{4} c d^{2}}{b x + a} - \frac{a^{4} b^{3} d^{3}}{b x + a}\right )}}{b^{7}}}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*((2*d^3 + 3*(3*b^2*c*d^2 - 4*a*b*d^3)/((b*x + a)*b) + 18*(b^4*c^2*d - 3*a*b^
3*c*d^2 + 2*a^2*b^2*d^3)/((b*x + a)^2*b^2))*(b*x + a)^3/b^4 - 6*(b^3*c^3 - 6*a*b
^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^4
+ 6*(a*b^6*c^3/(b*x + a) - 3*a^2*b^5*c^2*d/(b*x + a) + 3*a^3*b^4*c*d^2/(b*x + a)
- a^4*b^3*d^3/(b*x + a))/b^7)/b