3.153 \(\int \frac{x^4 (A+B x)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=113 \[ -\frac{a^4 (A b-a B)}{b^6 (a+b x)}-\frac{a^3 (4 A b-5 a B) \log (a+b x)}{b^6}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.261204, antiderivative size = 113, 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 \[ -\frac{a^4 (A b-a B)}{b^6 (a+b x)}-\frac{a^3 (4 A b-5 a B) \log (a+b x)}{b^6}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*(3*A*b - 4*a*B)*x)/b^5 - (a*(2*A*b - 3*a*B)*x^2)/(2*b^4) + ((A*b - 2*a*B)*x
^3)/(3*b^3) + (B*x^4)/(4*b^2) - (a^4*(A*b - a*B))/(b^6*(a + b*x)) - (a^3*(4*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. \[ \frac{B x^{4}}{4 b^{2}} - \frac{a^{4} \left (A b - B a\right )}{b^{6} \left (a + b x\right )} - \frac{a^{3} \left (4 A b - 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{a \left (2 A b - 3 B a\right ) \int x\, dx}{b^{4}} + \frac{x^{3} \left (A b - 2 B a\right )}{3 b^{3}} + \frac{\left (3 A b - 4 B a\right ) \int a^{2}\, dx}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)/(b*x+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.115586, size = 107, normalized size = 0.95 \[ \frac{\frac{12 a^4 (a B-A b)}{a+b x}+12 a^3 (5 a B-4 A b) \log (a+b x)-12 a^2 b x (4 a B-3 A b)+4 b^3 x^3 (A b-2 a B)+6 a b^2 x^2 (3 a B-2 A b)+3 b^4 B x^4}{12 b^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 133, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,{b}^{2}}}+{\frac{A{x}^{3}}{3\,{b}^{2}}}-{\frac{2\,B{x}^{3}a}{3\,{b}^{3}}}-{\frac{aA{x}^{2}}{{b}^{3}}}+{\frac{3\,B{x}^{2}{a}^{2}}{2\,{b}^{4}}}+3\,{\frac{{a}^{2}Ax}{{b}^{4}}}-4\,{\frac{{a}^{3}Bx}{{b}^{5}}}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) A}{{b}^{5}}}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) B}{{b}^{6}}}-{\frac{{a}^{4}A}{ \left ( bx+a \right ){b}^{5}}}+{\frac{B{a}^{5}}{ \left ( bx+a \right ){b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.32863, size = 166, normalized size = 1.47 \[ \frac{B a^{5} - A a^{4} b}{b^{7} x + a b^{6}} + \frac{3 \, B b^{3} x^{4} - 4 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{3} + 6 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2} - 12 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} x}{12 \, b^{5}} + \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.205272, size = 221, normalized size = 1.96 \[ \frac{3 \, B b^{5} x^{5} + 12 \, B a^{5} - 12 \, A a^{4} b -{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} x^{4} + 2 \,{\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{3} - 6 \,{\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{2} - 12 \,{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x + 12 \,{\left (5 \, B a^{5} - 4 \, A a^{4} b +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.9557, size = 114, normalized size = 1.01 \[ \frac{B x^{4}}{4 b^{2}} + \frac{a^{3} \left (- 4 A b + 5 B a\right ) \log{\left (a + b x \right )}}{b^{6}} + \frac{- A a^{4} b + B a^{5}}{a b^{6} + b^{7} x} - \frac{x^{3} \left (- A b + 2 B a\right )}{3 b^{3}} + \frac{x^{2} \left (- 2 A a b + 3 B a^{2}\right )}{2 b^{4}} - \frac{x \left (- 3 A a^{2} b + 4 B a^{3}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(b*x+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.255956, size = 236, normalized size = 2.09 \[ \frac{{\left (b x + a\right )}^{4}{\left (3 \, B - \frac{4 \,{\left (5 \, B a b - A b^{2}\right )}}{{\left (b x + a\right )} b} + \frac{12 \,{\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{24 \,{\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}}{12 \, b^{6}} - \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{6}} + \frac{\frac{B a^{5} b^{4}}{b x + a} - \frac{A a^{4} b^{5}}{b x + a}}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(b*x + a)^4*(3*B - 4*(5*B*a*b - A*b^2)/((b*x + a)*b) + 12*(5*B*a^2*b^2 - 2*
A*a*b^3)/((b*x + a)^2*b^2) - 24*(5*B*a^3*b^3 - 3*A*a^2*b^4)/((b*x + a)^3*b^3))/b
^6 - (5*B*a^4 - 4*A*a^3*b)*ln(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^6 + (B*a^5*b^
4/(b*x + a) - A*a^4*b^5/(b*x + a))/b^10