3.162 \(\int \frac{x^4 (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=116 \[ -\frac{a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac{a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac{2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{x^2 (A b-3 a B)}{2 b^4}+\frac{B x^3}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.272835, antiderivative size = 116, 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 \[ -\frac{a^4 (A b-a B)}{2 b^6 (a+b x)^2}+\frac{a^3 (4 A b-5 a B)}{b^6 (a+b x)}+\frac{2 a^2 (3 A b-5 a B) \log (a+b x)}{b^6}-\frac{3 a x (A b-2 a B)}{b^5}+\frac{x^2 (A b-3 a B)}{2 b^4}+\frac{B x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x))/(a + b*x)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)/(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.126032, size = 108, normalized size = 0.93 \[ \frac{\frac{3 a^4 (a B-A b)}{(a+b x)^2}+\frac{6 a^3 (4 A b-5 a B)}{a+b x}-12 a^2 (5 a B-3 A b) \log (a+b x)+3 b^2 x^2 (A b-3 a B)+18 a b x (2 a B-A b)+2 b^3 B x^3}{6 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x))/(a + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.017, size = 142, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,{b}^{3}}}+{\frac{A{x}^{2}}{2\,{b}^{3}}}-{\frac{3\,B{x}^{2}a}{2\,{b}^{4}}}-3\,{\frac{aAx}{{b}^{4}}}+6\,{\frac{{a}^{2}Bx}{{b}^{5}}}+6\,{\frac{{a}^{2}\ln \left ( bx+a \right ) A}{{b}^{5}}}-10\,{\frac{{a}^{3}\ln \left ( bx+a \right ) B}{{b}^{6}}}+4\,{\frac{A{a}^{3}}{ \left ( bx+a \right ){b}^{5}}}-5\,{\frac{B{a}^{4}}{ \left ( bx+a \right ){b}^{6}}}-{\frac{{a}^{4}A}{2\, \left ( bx+a \right ) ^{2}{b}^{5}}}+{\frac{B{a}^{5}}{2\, \left ( bx+a \right ) ^{2}{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)/(b*x+a)^3,x)

[Out]

1/3*B*x^3/b^3+1/2/b^3*A*x^2-3/2/b^4*B*x^2*a-3/b^4*a*A*x+6/b^5*a^2*B*x+6*a^2/b^5*
ln(b*x+a)*A-10*a^3/b^6*ln(b*x+a)*B+4*a^3/(b*x+a)/b^5*A-5*a^4/(b*x+a)/b^6*B-1/2*a
^4/(b*x+a)^2/b^5*A+1/2*a^5/(b*x+a)^2/b^6*B

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Maxima [A]  time = 1.33707, size = 180, normalized size = 1.55 \[ -\frac{9 \, B a^{5} - 7 \, A a^{4} b + 2 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac{2 \, B b^{2} x^{3} - 3 \,{\left (3 \, B a b - A b^{2}\right )} x^{2} + 18 \,{\left (2 \, B a^{2} - A a b\right )} x}{6 \, b^{5}} - \frac{2 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/(b*x + a)^3,x, algorithm="maxima")

[Out]

-1/2*(9*B*a^5 - 7*A*a^4*b + 2*(5*B*a^4*b - 4*A*a^3*b^2)*x)/(b^8*x^2 + 2*a*b^7*x
+ a^2*b^6) + 1/6*(2*B*b^2*x^3 - 3*(3*B*a*b - A*b^2)*x^2 + 18*(2*B*a^2 - A*a*b)*x
)/b^5 - 2*(5*B*a^3 - 3*A*a^2*b)*log(b*x + a)/b^6

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Fricas [A]  time = 0.201878, size = 266, normalized size = 2.29 \[ \frac{2 \, B b^{5} x^{5} - 27 \, B a^{5} + 21 \, A a^{4} b -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 3 \,{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} + 6 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x - 12 \,{\left (5 \, B a^{5} - 3 \, A a^{4} b +{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 2 \,{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/(b*x + a)^3,x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*x^5 - 27*B*a^5 + 21*A*a^4*b - (5*B*a*b^4 - 3*A*b^5)*x^4 + 4*(5*B*a^
2*b^3 - 3*A*a*b^4)*x^3 + 3*(21*B*a^3*b^2 - 11*A*a^2*b^3)*x^2 + 6*(B*a^4*b + A*a^
3*b^2)*x - 12*(5*B*a^5 - 3*A*a^4*b + (5*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 2*(5*B*a^
4*b - 3*A*a^3*b^2)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)

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Sympy [A]  time = 5.73194, size = 131, normalized size = 1.13 \[ \frac{B x^{3}}{3 b^{3}} - \frac{2 a^{2} \left (- 3 A b + 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{- 7 A a^{4} b + 9 B a^{5} + x \left (- 8 A a^{3} b^{2} + 10 B a^{4} b\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac{x^{2} \left (- A b + 3 B a\right )}{2 b^{4}} + \frac{x \left (- 3 A a b + 6 B a^{2}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(b*x+a)**3,x)

[Out]

B*x**3/(3*b**3) - 2*a**2*(-3*A*b + 5*B*a)*log(a + b*x)/b**6 - (-7*A*a**4*b + 9*B
*a**5 + x*(-8*A*a**3*b**2 + 10*B*a**4*b))/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**
2) - x**2*(-A*b + 3*B*a)/(2*b**4) + x*(-3*A*a*b + 6*B*a**2)/b**5

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GIAC/XCAS [A]  time = 0.247127, size = 169, normalized size = 1.46 \[ -\frac{2 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{9 \, B a^{5} - 7 \, A a^{4} b + 2 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, B b^{6} x^{3} - 9 \, B a b^{5} x^{2} + 3 \, A b^{6} x^{2} + 36 \, B a^{2} b^{4} x - 18 \, A a b^{5} x}{6 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/(b*x + a)^3,x, algorithm="giac")

[Out]

-2*(5*B*a^3 - 3*A*a^2*b)*ln(abs(b*x + a))/b^6 - 1/2*(9*B*a^5 - 7*A*a^4*b + 2*(5*
B*a^4*b - 4*A*a^3*b^2)*x)/((b*x + a)^2*b^6) + 1/6*(2*B*b^6*x^3 - 9*B*a*b^5*x^2 +
3*A*b^6*x^2 + 36*B*a^2*b^4*x - 18*A*a*b^5*x)/b^9