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

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

[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 = 0.10855, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 77} \[ -\frac{a^4 (A b-a B)}{b^6 (a+b x)}+\frac{a^2 x (3 A b-4 a B)}{b^5}-\frac{a^3 (4 A b-5 a B) \log (a+b x)}{b^6}-\frac{a x^2 (2 A b-3 a B)}{2 b^4}+\frac{x^3 (A b-2 a B)}{3 b^3}+\frac{B x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^4 (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{x^4 (A+B x)}{(a+b x)^2} \, dx\\ &=\int \left (-\frac{a^2 (-3 A b+4 a B)}{b^5}+\frac{a (-2 A b+3 a B) x}{b^4}+\frac{(A b-2 a B) x^2}{b^3}+\frac{B x^3}{b^2}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)^2}+\frac{a^3 (-4 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{a^2 (3 A b-4 a B) x}{b^5}-\frac{a (2 A b-3 a B) x^2}{2 b^4}+\frac{(A b-2 a B) x^3}{3 b^3}+\frac{B x^4}{4 b^2}-\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}\\ \end{align*}

Mathematica [A]  time = 0.053374, size = 107, normalized size = 0.95 \[ \frac{\frac{12 a^4 (a B-A b)}{a+b x}-12 a^2 b x (4 a B-3 A b)+12 a^3 (5 a B-4 A b) \log (a+b x)+6 a b^2 x^2 (3 a B-2 A b)+4 b^3 x^3 (A b-2 a B)+3 b^4 B x^4}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(A + B*x))/(a^2 + 2*a*b*x + b^2*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 = 0.006, size = 133, normalized size = 1.2 \begin{align*}{\frac{B{x}^{4}}{4\,{b}^{2}}}+{\frac{A{x}^{3}}{3\,{b}^{2}}}-{\frac{2\,aB{x}^{3}}{3\,{b}^{3}}}-{\frac{aA{x}^{2}}{{b}^{3}}}+{\frac{3\,{a}^{2}B{x}^{2}}{2\,{b}^{4}}}+3\,{\frac{{a}^{2}Ax}{{b}^{4}}}-4\,{\frac{{a}^{3}Bx}{{b}^{5}}}-{\frac{{a}^{4}A}{{b}^{5} \left ( bx+a \right ) }}+{\frac{B{a}^{5}}{{b}^{6} \left ( bx+a \right ) }}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) A}{{b}^{5}}}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) B}{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 1.02119, size = 166, normalized size = 1.47 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*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 = 1.18285, size = 352, normalized size = 3.12 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)/(b^2*x^2+2*a*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 = 0.633376, size = 114, normalized size = 1.01 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b**2*x**2+2*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 [A]  time = 1.11476, size = 170, normalized size = 1.5 \begin{align*} \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{B a^{5} - A a^{4} b}{{\left (b x + a\right )} b^{6}} + \frac{3 \, B b^{6} x^{4} - 8 \, B a b^{5} x^{3} + 4 \, A b^{6} x^{3} + 18 \, B a^{2} b^{4} x^{2} - 12 \, A a b^{5} x^{2} - 48 \, B a^{3} b^{3} x + 36 \, A a^{2} b^{4} x}{12 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*B*a^4 - 4*A*a^3*b)*log(abs(b*x + a))/b^6 + (B*a^5 - A*a^4*b)/((b*x + a)*b^6) + 1/12*(3*B*b^6*x^4 - 8*B*a*b^
5*x^3 + 4*A*b^6*x^3 + 18*B*a^2*b^4*x^2 - 12*A*a*b^5*x^2 - 48*B*a^3*b^3*x + 36*A*a^2*b^4*x)/b^8