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

Optimal. Leaf size=94 \[ \frac{a^3 (A b-a B)}{2 b^5 (a+b x)^2}-\frac{a^2 (3 A b-4 a B)}{b^5 (a+b x)}-\frac{3 a (A b-2 a B) \log (a+b x)}{b^5}+\frac{x (A b-3 a B)}{b^4}+\frac{B x^2}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.204213, antiderivative size = 94, 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 \[ \frac{a^3 (A b-a B)}{2 b^5 (a+b x)^2}-\frac{a^2 (3 A b-4 a B)}{b^5 (a+b x)}-\frac{3 a (A b-2 a B) \log (a+b x)}{b^5}+\frac{x (A b-3 a B)}{b^4}+\frac{B x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0903696, size = 86, normalized size = 0.91 \[ \frac{\frac{a^3 (A b-a B)}{(a+b x)^2}+\frac{2 a^2 (4 a B-3 A b)}{a+b x}+2 b x (A b-3 a B)+6 a (2 a B-A b) \log (a+b x)+b^2 B x^2}{2 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 117, normalized size = 1.2 \[{\frac{B{x}^{2}}{2\,{b}^{3}}}+{\frac{Ax}{{b}^{3}}}-3\,{\frac{Bax}{{b}^{4}}}-3\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{4}}}+6\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{5}}}-3\,{\frac{A{a}^{2}}{ \left ( bx+a \right ){b}^{4}}}+4\,{\frac{{a}^{3}B}{ \left ( bx+a \right ){b}^{5}}}+{\frac{{a}^{3}A}{2\, \left ( bx+a \right ) ^{2}{b}^{4}}}-{\frac{B{a}^{4}}{2\, \left ( bx+a \right ) ^{2}{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)/(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.33581, size = 146, normalized size = 1.55 \[ \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac{B b x^{2} - 2 \,{\left (3 \, B a - A b\right )} x}{2 \, b^{4}} + \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.200986, size = 231, normalized size = 2.46 \[ \frac{B b^{4} x^{4} + 7 \, B a^{4} - 5 \, A a^{3} b - 2 \,{\left (2 \, B a b^{3} - A b^{4}\right )} x^{3} -{\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x + 6 \,{\left (2 \, B a^{4} - A a^{3} b +{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 2 \,{\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.20453, size = 105, normalized size = 1.12 \[ \frac{B x^{2}}{2 b^{3}} + \frac{3 a \left (- A b + 2 B a\right ) \log{\left (a + b x \right )}}{b^{5}} + \frac{- 5 A a^{3} b + 7 B a^{4} + x \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} - \frac{x \left (- A b + 3 B a\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.289922, size = 135, normalized size = 1.44 \[ \frac{3 \,{\left (2 \, B a^{2} - A a b\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{B b^{3} x^{2} - 6 \, B a b^{2} x + 2 \, A b^{3} x}{2 \, b^{6}} + \frac{7 \, B a^{4} - 5 \, A a^{3} b + 2 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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