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

Optimal. Leaf size=136 \[ -\frac{a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac{a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac{x (b c-4 a d) (b c-a d)^2}{b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac{d^3 x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.295445, antiderivative size = 136, 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 \[ -\frac{a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac{a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac{x (b c-4 a d) (b c-a d)^2}{b^5}+\frac{3 d x^2 (b c-a d)^2}{2 b^4}+\frac{d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac{d^3 x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} \left (a d - b c\right )^{3}}{b^{6} \left (a + b x\right )} + \frac{a \left (a d - b c\right )^{2} \left (5 a d - 2 b c\right ) \log{\left (a + b x \right )}}{b^{6}} - \left (a d - b c\right )^{2} \left (4 a d - b c\right ) \int \frac{1}{b^{5}}\, dx + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{d^{2} x^{3} \left (2 a d - 3 b c\right )}{3 b^{3}} + \frac{3 d \left (a d - b c\right )^{2} \int x\, dx}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0807202, size = 130, normalized size = 0.96 \[ \frac{\frac{12 a^2 (a d-b c)^3}{a+b x}+4 b^3 d^2 x^3 (3 b c-2 a d)+18 b^2 d x^2 (b c-a d)^2+12 b x (b c-4 a d) (b c-a d)^2+12 a (b c-a d)^2 (5 a d-2 b c) \log (a+b x)+3 b^4 d^3 x^4}{12 b^6} \]

Antiderivative was successfully verified.

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

[Out]

(12*b*(b*c - 4*a*d)*(b*c - a*d)^2*x + 18*b^2*d*(b*c - a*d)^2*x^2 + 4*b^3*d^2*(3*
b*c - 2*a*d)*x^3 + 3*b^4*d^3*x^4 + (12*a^2*(-(b*c) + a*d)^3)/(a + b*x) + 12*a*(b
*c - a*d)^2*(-2*b*c + 5*a*d)*Log[a + b*x])/(12*b^6)

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Maple [A]  time = 0.013, size = 260, normalized size = 1.9 \[{\frac{{d}^{3}{x}^{4}}{4\,{b}^{2}}}-{\frac{2\,{x}^{3}a{d}^{3}}{3\,{b}^{3}}}+{\frac{c{x}^{3}{d}^{2}}{{b}^{2}}}+{\frac{3\,{a}^{2}{x}^{2}{d}^{3}}{2\,{b}^{4}}}-3\,{\frac{{x}^{2}ac{d}^{2}}{{b}^{3}}}+{\frac{3\,{x}^{2}{c}^{2}d}{2\,{b}^{2}}}-4\,{\frac{{a}^{3}{d}^{3}x}{{b}^{5}}}+9\,{\frac{{a}^{2}c{d}^{2}x}{{b}^{4}}}-6\,{\frac{a{c}^{2}dx}{{b}^{3}}}+{\frac{{c}^{3}x}{{b}^{2}}}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ){d}^{3}}{{b}^{6}}}-12\,{\frac{{a}^{3}\ln \left ( bx+a \right ) c{d}^{2}}{{b}^{5}}}+9\,{\frac{{a}^{2}\ln \left ( bx+a \right ){c}^{2}d}{{b}^{4}}}-2\,{\frac{a\ln \left ( bx+a \right ){c}^{3}}{{b}^{3}}}+{\frac{{a}^{5}{d}^{3}}{{b}^{6} \left ( bx+a \right ) }}-3\,{\frac{{a}^{4}c{d}^{2}}{{b}^{5} \left ( bx+a \right ) }}+3\,{\frac{{a}^{3}{c}^{2}d}{{b}^{4} \left ( bx+a \right ) }}-{\frac{{a}^{2}{c}^{3}}{{b}^{3} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(d*x+c)^3/(b*x+a)^2,x)

[Out]

1/4*d^3*x^4/b^2-2/3/b^3*x^3*a*d^3+1/b^2*x^3*c*d^2+3/2/b^4*x^2*a^2*d^3-3/b^3*x^2*
a*c*d^2+3/2/b^2*x^2*c^2*d-4/b^5*a^3*d^3*x+9/b^4*a^2*c*d^2*x-6/b^3*a*c^2*d*x+1/b^
2*c^3*x+5*a^4/b^6*ln(b*x+a)*d^3-12*a^3/b^5*ln(b*x+a)*c*d^2+9*a^2/b^4*ln(b*x+a)*c
^2*d-2*a/b^3*ln(b*x+a)*c^3+a^5/b^6/(b*x+a)*d^3-3*a^4/b^5/(b*x+a)*c*d^2+3*a^3/b^4
/(b*x+a)*c^2*d-a^2/b^3/(b*x+a)*c^3

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Maxima [A]  time = 1.36073, size = 297, normalized size = 2.18 \[ -\frac{a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}}{b^{7} x + a b^{6}} + \frac{3 \, b^{3} d^{3} x^{4} + 4 \,{\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{3} + 18 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x}{12 \, b^{5}} - \frac{{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(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)/(b^7*x + a*b^6) + 1/1
2*(3*b^3*d^3*x^4 + 4*(3*b^3*c*d^2 - 2*a*b^2*d^3)*x^3 + 18*(b^3*c^2*d - 2*a*b^2*c
*d^2 + a^2*b*d^3)*x^2 + 12*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)
*x)/b^5 - (2*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 - 5*a^4*d^3)*log(b*x +
a)/b^6

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Fricas [A]  time = 0.206712, size = 424, normalized size = 3.12 \[ \frac{3 \, b^{5} d^{3} x^{5} - 12 \, a^{2} b^{3} c^{3} + 36 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 12 \, a^{5} d^{3} +{\left (12 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \,{\left (9 \, b^{5} c^{2} d - 12 \, a b^{4} c d^{2} + 5 \, a^{2} b^{3} d^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 12 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{2} + 12 \,{\left (a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\right )} x - 12 \,{\left (2 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 12 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} +{\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b^5*d^3*x^5 - 12*a^2*b^3*c^3 + 36*a^3*b^2*c^2*d - 36*a^4*b*c*d^2 + 12*a^
5*d^3 + (12*b^5*c*d^2 - 5*a*b^4*d^3)*x^4 + 2*(9*b^5*c^2*d - 12*a*b^4*c*d^2 + 5*a
^2*b^3*d^3)*x^3 + 6*(2*b^5*c^3 - 9*a*b^4*c^2*d + 12*a^2*b^3*c*d^2 - 5*a^3*b^2*d^
3)*x^2 + 12*(a*b^4*c^3 - 6*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 4*a^4*b*d^3)*x - 12
*(2*a^2*b^3*c^3 - 9*a^3*b^2*c^2*d + 12*a^4*b*c*d^2 - 5*a^5*d^3 + (2*a*b^4*c^3 -
9*a^2*b^3*c^2*d + 12*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x)*log(b*x + a))/(b^7*x + a*b^
6)

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Sympy [A]  time = 6.10854, size = 199, normalized size = 1.46 \[ \frac{a \left (a d - b c\right )^{2} \left (5 a d - 2 b c\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{a^{5} d^{3} - 3 a^{4} b c d^{2} + 3 a^{3} b^{2} c^{2} d - a^{2} b^{3} c^{3}}{a b^{6} + b^{7} x} + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x^{3} \left (2 a d^{3} - 3 b c d^{2}\right )}{3 b^{3}} + \frac{x^{2} \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{4}} - \frac{x \left (4 a^{3} d^{3} - 9 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - b^{3} c^{3}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(a*d - b*c)**2*(5*a*d - 2*b*c)*log(a + b*x)/b**6 + (a**5*d**3 - 3*a**4*b*c*d**
2 + 3*a**3*b**2*c**2*d - a**2*b**3*c**3)/(a*b**6 + b**7*x) + d**3*x**4/(4*b**2)
- x**3*(2*a*d**3 - 3*b*c*d**2)/(3*b**3) + x**2*(3*a**2*d**3 - 6*a*b*c*d**2 + 3*b
**2*c**2*d)/(2*b**4) - x*(4*a**3*d**3 - 9*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3
*c**3)/b**5

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GIAC/XCAS [A]  time = 0.280576, size = 386, normalized size = 2.84 \[ \frac{{\left (3 \, d^{3} + \frac{4 \,{\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac{6 \,{\left (3 \, b^{4} c^{2} d - 12 \, a b^{3} c d^{2} + 10 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{12 \,{\left (b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 18 \, a^{2} b^{4} c d^{2} - 10 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}{\left (b x + a\right )}^{4}}{12 \, b^{6}} + \frac{{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{6}} - \frac{\frac{a^{2} b^{7} c^{3}}{b x + a} - \frac{3 \, a^{3} b^{6} c^{2} d}{b x + a} + \frac{3 \, a^{4} b^{5} c d^{2}}{b x + a} - \frac{a^{5} b^{4} d^{3}}{b x + a}}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*d^3 + 4*(3*b^2*c*d^2 - 5*a*b*d^3)/((b*x + a)*b) + 6*(3*b^4*c^2*d - 12*a*
b^3*c*d^2 + 10*a^2*b^2*d^3)/((b*x + a)^2*b^2) + 12*(b^6*c^3 - 9*a*b^5*c^2*d + 18
*a^2*b^4*c*d^2 - 10*a^3*b^3*d^3)/((b*x + a)^3*b^3))*(b*x + a)^4/b^6 + (2*a*b^3*c
^3 - 9*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 - 5*a^4*d^3)*ln(abs(b*x + a)/((b*x + a)^2*
abs(b)))/b^6 - (a^2*b^7*c^3/(b*x + a) - 3*a^3*b^6*c^2*d/(b*x + a) + 3*a^4*b^5*c*
d^2/(b*x + a) - a^5*b^4*d^3/(b*x + a))/b^10