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

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

[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 = 0.0809364, 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, Rules used = {77} \[ \frac{a^3 (A b-a B)}{2 b^5 (a+b x)^2}-\frac{a^2 (3 A b-4 a B)}{b^5 (a+b x)}+\frac{x (A b-3 a B)}{b^4}-\frac{3 a (A b-2 a B) \log (a+b x)}{b^5}+\frac{B x^2}{2 b^3} \]

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

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

Mathematica [A]  time = 0.0535222, 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 = 0.007, size = 117, normalized size = 1.2 \begin{align*}{\frac{B{x}^{2}}{2\,{b}^{3}}}+{\frac{Ax}{{b}^{3}}}-3\,{\frac{Bax}{{b}^{4}}}-3\,{\frac{A{a}^{2}}{{b}^{4} \left ( bx+a \right ) }}+4\,{\frac{B{a}^{3}}{{b}^{5} \left ( bx+a \right ) }}+{\frac{{a}^{3}A}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{4}}{2\,{b}^{5} \left ( bx+a \right ) ^{2}}}-3\,{\frac{a\ln \left ( bx+a \right ) A}{{b}^{4}}}+6\,{\frac{{a}^{2}\ln \left ( bx+a \right ) B}{{b}^{5}}} \end{align*}

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

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Maxima [A]  time = 1.19136, size = 146, normalized size = 1.55 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)/(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 = 1.8851, size = 350, normalized size = 3.72 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(B*x+A)/(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 = 1.06988, size = 105, normalized size = 1.12 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.16256, size = 135, normalized size = 1.44 \begin{align*} \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(2*B*a^2 - A*a*b)*log(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)