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

Optimal. Leaf size=165 \[ -\frac{a^4 (b c-a d)^2}{b^7 (a+b x)}-\frac{2 a^3 (2 b c-3 a d) (b c-a d) \log (a+b x)}{b^7}+\frac{a^2 x (3 b c-5 a d) (b c-a d)}{b^6}-\frac{a x^2 (b c-2 a d) (b c-a d)}{b^5}+\frac{x^3 (b c-3 a d) (b c-a d)}{3 b^4}+\frac{d x^4 (b c-a d)}{2 b^3}+\frac{d^2 x^5}{5 b^2} \]

[Out]

(a^2*(3*b*c - 5*a*d)*(b*c - a*d)*x)/b^6 - (a*(b*c - 2*a*d)*(b*c - a*d)*x^2)/b^5
+ ((b*c - 3*a*d)*(b*c - a*d)*x^3)/(3*b^4) + (d*(b*c - a*d)*x^4)/(2*b^3) + (d^2*x
^5)/(5*b^2) - (a^4*(b*c - a*d)^2)/(b^7*(a + b*x)) - (2*a^3*(2*b*c - 3*a*d)*(b*c
- a*d)*Log[a + b*x])/b^7

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Rubi [A]  time = 0.424201, antiderivative size = 165, 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 \[ -\frac{a^4 (b c-a d)^2}{b^7 (a+b x)}-\frac{2 a^3 (2 b c-3 a d) (b c-a d) \log (a+b x)}{b^7}+\frac{a^2 x (3 b c-5 a d) (b c-a d)}{b^6}-\frac{a x^2 (b c-2 a d) (b c-a d)}{b^5}+\frac{x^3 (b c-3 a d) (b c-a d)}{3 b^4}+\frac{d x^4 (b c-a d)}{2 b^3}+\frac{d^2 x^5}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(3*b*c - 5*a*d)*(b*c - a*d)*x)/b^6 - (a*(b*c - 2*a*d)*(b*c - a*d)*x^2)/b^5
+ ((b*c - 3*a*d)*(b*c - a*d)*x^3)/(3*b^4) + (d*(b*c - a*d)*x^4)/(2*b^3) + (d^2*x
^5)/(5*b^2) - (a^4*(b*c - a*d)^2)/(b^7*(a + b*x)) - (2*a^3*(2*b*c - 3*a*d)*(b*c
- a*d)*Log[a + b*x])/b^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4} \left (a d - b c\right )^{2}}{b^{7} \left (a + b x\right )} - \frac{2 a^{3} \left (a d - b c\right ) \left (3 a d - 2 b c\right ) \log{\left (a + b x \right )}}{b^{7}} - \frac{2 a \left (a d - b c\right ) \left (2 a d - b c\right ) \int x\, dx}{b^{5}} + \frac{d^{2} x^{5}}{5 b^{2}} - \frac{d x^{4} \left (a d - b c\right )}{2 b^{3}} + \frac{x^{3} \left (a d - b c\right ) \left (3 a d - b c\right )}{3 b^{4}} + \frac{\left (a d - b c\right ) \left (5 a d - 3 b c\right ) \int a^{2}\, dx}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.187677, size = 183, normalized size = 1.11 \[ \frac{-\frac{30 a^4 (b c-a d)^2}{a+b x}-30 a b^2 x^2 \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+30 a^2 b x \left (5 a^2 d^2-8 a b c d+3 b^2 c^2\right )+10 b^3 x^3 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-60 a^3 \left (3 a^2 d^2-5 a b c d+2 b^2 c^2\right ) \log (a+b x)+15 b^4 d x^4 (b c-a d)+6 b^5 d^2 x^5}{30 b^7} \]

Antiderivative was successfully verified.

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

[Out]

(30*a^2*b*(3*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*x - 30*a*b^2*(b^2*c^2 - 3*a*b*c*d
+ 2*a^2*d^2)*x^2 + 10*b^3*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x^3 + 15*b^4*d*(b*c
- a*d)*x^4 + 6*b^5*d^2*x^5 - (30*a^4*(b*c - a*d)^2)/(a + b*x) - 60*a^3*(2*b^2*c^
2 - 5*a*b*c*d + 3*a^2*d^2)*Log[a + b*x])/(30*b^7)

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Maple [A]  time = 0.014, size = 247, normalized size = 1.5 \[{\frac{{d}^{2}{x}^{5}}{5\,{b}^{2}}}-{\frac{{x}^{4}a{d}^{2}}{2\,{b}^{3}}}+{\frac{{x}^{4}cd}{2\,{b}^{2}}}+{\frac{{x}^{3}{a}^{2}{d}^{2}}{{b}^{4}}}-{\frac{4\,{x}^{3}acd}{3\,{b}^{3}}}+{\frac{{x}^{3}{c}^{2}}{3\,{b}^{2}}}-2\,{\frac{{x}^{2}{a}^{3}{d}^{2}}{{b}^{5}}}+3\,{\frac{{a}^{2}{x}^{2}cd}{{b}^{4}}}-{\frac{{x}^{2}a{c}^{2}}{{b}^{3}}}+5\,{\frac{{a}^{4}{d}^{2}x}{{b}^{6}}}-8\,{\frac{{a}^{3}cdx}{{b}^{5}}}+3\,{\frac{{a}^{2}{c}^{2}x}{{b}^{4}}}-6\,{\frac{{a}^{5}\ln \left ( bx+a \right ){d}^{2}}{{b}^{7}}}+10\,{\frac{{a}^{4}\ln \left ( bx+a \right ) cd}{{b}^{6}}}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ){c}^{2}}{{b}^{5}}}-{\frac{{a}^{6}{d}^{2}}{ \left ( bx+a \right ){b}^{7}}}+2\,{\frac{{a}^{5}cd}{ \left ( bx+a \right ){b}^{6}}}-{\frac{{a}^{4}{c}^{2}}{ \left ( bx+a \right ){b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x+c)^2/(b*x+a)^2,x)

[Out]

1/5*d^2*x^5/b^2-1/2/b^3*x^4*a*d^2+1/2/b^2*x^4*c*d+1/b^4*x^3*a^2*d^2-4/3/b^3*x^3*
a*c*d+1/3/b^2*x^3*c^2-2/b^5*x^2*a^3*d^2+3/b^4*x^2*a^2*c*d-1/b^3*x^2*a*c^2+5/b^6*
a^4*d^2*x-8/b^5*a^3*c*d*x+3/b^4*a^2*c^2*x-6*a^5/b^7*ln(b*x+a)*d^2+10*a^4/b^6*ln(
b*x+a)*c*d-4*a^3/b^5*ln(b*x+a)*c^2-a^6/(b*x+a)/b^7*d^2+2*a^5/(b*x+a)/b^6*c*d-a^4
/(b*x+a)/b^5*c^2

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Maxima [A]  time = 1.33416, size = 290, normalized size = 1.76 \[ -\frac{a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}}{b^{8} x + a b^{7}} + \frac{6 \, b^{4} d^{2} x^{5} + 15 \,{\left (b^{4} c d - a b^{3} d^{2}\right )} x^{4} + 10 \,{\left (b^{4} c^{2} - 4 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{3} - 30 \,{\left (a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{2} + 30 \,{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} x}{30 \, b^{6}} - \frac{2 \,{\left (2 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)/(b^8*x + a*b^7) + 1/30*(6*b^4*d^2*x^5 + 1
5*(b^4*c*d - a*b^3*d^2)*x^4 + 10*(b^4*c^2 - 4*a*b^3*c*d + 3*a^2*b^2*d^2)*x^3 - 3
0*(a*b^3*c^2 - 3*a^2*b^2*c*d + 2*a^3*b*d^2)*x^2 + 30*(3*a^2*b^2*c^2 - 8*a^3*b*c*
d + 5*a^4*d^2)*x)/b^6 - 2*(2*a^3*b^2*c^2 - 5*a^4*b*c*d + 3*a^5*d^2)*log(b*x + a)
/b^7

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Fricas [A]  time = 0.214596, size = 386, normalized size = 2.34 \[ \frac{6 \, b^{6} d^{2} x^{6} - 30 \, a^{4} b^{2} c^{2} + 60 \, a^{5} b c d - 30 \, a^{6} d^{2} + 3 \,{\left (5 \, b^{6} c d - 3 \, a b^{5} d^{2}\right )} x^{5} + 5 \,{\left (2 \, b^{6} c^{2} - 5 \, a b^{5} c d + 3 \, a^{2} b^{4} d^{2}\right )} x^{4} - 10 \,{\left (2 \, a b^{5} c^{2} - 5 \, a^{2} b^{4} c d + 3 \, a^{3} b^{3} d^{2}\right )} x^{3} + 30 \,{\left (2 \, a^{2} b^{4} c^{2} - 5 \, a^{3} b^{3} c d + 3 \, a^{4} b^{2} d^{2}\right )} x^{2} + 30 \,{\left (3 \, a^{3} b^{3} c^{2} - 8 \, a^{4} b^{2} c d + 5 \, a^{5} b d^{2}\right )} x - 60 \,{\left (2 \, a^{4} b^{2} c^{2} - 5 \, a^{5} b c d + 3 \, a^{6} d^{2} +{\left (2 \, a^{3} b^{3} c^{2} - 5 \, a^{4} b^{2} c d + 3 \, a^{5} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{30 \,{\left (b^{8} x + a b^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(6*b^6*d^2*x^6 - 30*a^4*b^2*c^2 + 60*a^5*b*c*d - 30*a^6*d^2 + 3*(5*b^6*c*d
- 3*a*b^5*d^2)*x^5 + 5*(2*b^6*c^2 - 5*a*b^5*c*d + 3*a^2*b^4*d^2)*x^4 - 10*(2*a*b
^5*c^2 - 5*a^2*b^4*c*d + 3*a^3*b^3*d^2)*x^3 + 30*(2*a^2*b^4*c^2 - 5*a^3*b^3*c*d
+ 3*a^4*b^2*d^2)*x^2 + 30*(3*a^3*b^3*c^2 - 8*a^4*b^2*c*d + 5*a^5*b*d^2)*x - 60*(
2*a^4*b^2*c^2 - 5*a^5*b*c*d + 3*a^6*d^2 + (2*a^3*b^3*c^2 - 5*a^4*b^2*c*d + 3*a^5
*b*d^2)*x)*log(b*x + a))/(b^8*x + a*b^7)

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Sympy [A]  time = 5.63829, size = 201, normalized size = 1.22 \[ - \frac{2 a^{3} \left (a d - b c\right ) \left (3 a d - 2 b c\right ) \log{\left (a + b x \right )}}{b^{7}} - \frac{a^{6} d^{2} - 2 a^{5} b c d + a^{4} b^{2} c^{2}}{a b^{7} + b^{8} x} + \frac{d^{2} x^{5}}{5 b^{2}} - \frac{x^{4} \left (a d^{2} - b c d\right )}{2 b^{3}} + \frac{x^{3} \left (3 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{3 b^{4}} - \frac{x^{2} \left (2 a^{3} d^{2} - 3 a^{2} b c d + a b^{2} c^{2}\right )}{b^{5}} + \frac{x \left (5 a^{4} d^{2} - 8 a^{3} b c d + 3 a^{2} b^{2} c^{2}\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**3*(a*d - b*c)*(3*a*d - 2*b*c)*log(a + b*x)/b**7 - (a**6*d**2 - 2*a**5*b*c*
d + a**4*b**2*c**2)/(a*b**7 + b**8*x) + d**2*x**5/(5*b**2) - x**4*(a*d**2 - b*c*
d)/(2*b**3) + x**3*(3*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(3*b**4) - x**2*(2*a**3
*d**2 - 3*a**2*b*c*d + a*b**2*c**2)/b**5 + x*(5*a**4*d**2 - 8*a**3*b*c*d + 3*a**
2*b**2*c**2)/b**6

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GIAC/XCAS [A]  time = 0.386624, size = 378, normalized size = 2.29 \[ \frac{{\left (6 \, d^{2} + \frac{15 \,{\left (b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac{10 \,{\left (b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{60 \,{\left (a b^{5} c^{2} - 5 \, a^{2} b^{4} c d + 5 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{30 \,{\left (6 \, a^{2} b^{6} c^{2} - 20 \, a^{3} b^{5} c d + 15 \, a^{4} b^{4} d^{2}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )}{\left (b x + a\right )}^{5}}{30 \, b^{7}} + \frac{2 \,{\left (2 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{7}} - \frac{\frac{a^{4} b^{7} c^{2}}{b x + a} - \frac{2 \, a^{5} b^{6} c d}{b x + a} + \frac{a^{6} b^{5} d^{2}}{b x + a}}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(6*d^2 + 15*(b^2*c*d - 3*a*b*d^2)/((b*x + a)*b) + 10*(b^4*c^2 - 10*a*b^3*c*
d + 15*a^2*b^2*d^2)/((b*x + a)^2*b^2) - 60*(a*b^5*c^2 - 5*a^2*b^4*c*d + 5*a^3*b^
3*d^2)/((b*x + a)^3*b^3) + 30*(6*a^2*b^6*c^2 - 20*a^3*b^5*c*d + 15*a^4*b^4*d^2)/
((b*x + a)^4*b^4))*(b*x + a)^5/b^7 + 2*(2*a^3*b^2*c^2 - 5*a^4*b*c*d + 3*a^5*d^2)
*ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^7 - (a^4*b^7*c^2/(b*x + a) - 2*a^5*b^6*
c*d/(b*x + a) + a^6*b^5*d^2/(b*x + a))/b^12