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

Optimal. Leaf size=108 \[ \frac{a^2 x^2 (A b-a B)}{2 b^4}-\frac{a^3 x (A b-a B)}{b^5}+\frac{a^4 (A b-a B) \log (a+b x)}{b^6}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{B x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.0909027, antiderivative size = 108, 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} \[ \frac{a^2 x^2 (A b-a B)}{2 b^4}-\frac{a^3 x (A b-a B)}{b^5}+\frac{a^4 (A b-a B) \log (a+b x)}{b^6}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x^3 (A b-a B)}{3 b^3}+\frac{B x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(A + B*x))/(a + b*x),x]

[Out]

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

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+b x} \, dx &=\int \left (\frac{a^3 (-A b+a B)}{b^5}-\frac{a^2 (-A b+a B) x}{b^4}+\frac{a (-A b+a B) x^2}{b^3}+\frac{(A b-a B) x^3}{b^2}+\frac{B x^4}{b}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{a^3 (A b-a B) x}{b^5}+\frac{a^2 (A b-a B) x^2}{2 b^4}-\frac{a (A b-a B) x^3}{3 b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^5}{5 b}+\frac{a^4 (A b-a B) \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0365787, size = 100, normalized size = 0.93 \[ \frac{b x \left (10 a^2 b^2 x (3 A+2 B x)-30 a^3 b (2 A+B x)+60 a^4 B-5 a b^3 x^2 (4 A+3 B x)+3 b^4 x^3 (5 A+4 B x)\right )-60 a^4 (a B-A b) \log (a+b x)}{60 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(60*a^4*B - 30*a^3*b*(2*A + B*x) + 10*a^2*b^2*x*(3*A + 2*B*x) - 5*a*b^3*x^2*(4*A + 3*B*x) + 3*b^4*x^3*(5*
A + 4*B*x)) - 60*a^4*(-(A*b) + a*B)*Log[a + b*x])/(60*b^6)

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Maple [A]  time = 0.003, size = 124, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{5\,b}}+{\frac{A{x}^{4}}{4\,b}}-{\frac{B{x}^{4}a}{4\,{b}^{2}}}-{\frac{aA{x}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{3}{a}^{2}}{3\,{b}^{3}}}+{\frac{{a}^{2}A{x}^{2}}{2\,{b}^{3}}}-{\frac{B{x}^{2}{a}^{3}}{2\,{b}^{4}}}-{\frac{{a}^{3}Ax}{{b}^{4}}}+{\frac{B{a}^{4}x}{{b}^{5}}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) A}{{b}^{5}}}-{\frac{{a}^{5}\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*x+a),x)

[Out]

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

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Maxima [A]  time = 1.0333, size = 157, normalized size = 1.45 \begin{align*} \frac{12 \, B b^{4} x^{5} - 15 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 20 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 30 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 60 \,{\left (B a^{4} - A a^{3} b\right )} x}{60 \, b^{5}} - \frac{{\left (B a^{5} - A a^{4} 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*x+a),x, algorithm="maxima")

[Out]

1/60*(12*B*b^4*x^5 - 15*(B*a*b^3 - A*b^4)*x^4 + 20*(B*a^2*b^2 - A*a*b^3)*x^3 - 30*(B*a^3*b - A*a^2*b^2)*x^2 +
60*(B*a^4 - A*a^3*b)*x)/b^5 - (B*a^5 - A*a^4*b)*log(b*x + a)/b^6

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Fricas [A]  time = 1.41963, size = 244, normalized size = 2.26 \begin{align*} \frac{12 \, B b^{5} x^{5} - 15 \,{\left (B a b^{4} - A b^{5}\right )} x^{4} + 20 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 30 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 60 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} x - 60 \,{\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^5*x^5 - 15*(B*a*b^4 - A*b^5)*x^4 + 20*(B*a^2*b^3 - A*a*b^4)*x^3 - 30*(B*a^3*b^2 - A*a^2*b^3)*x^2
+ 60*(B*a^4*b - A*a^3*b^2)*x - 60*(B*a^5 - A*a^4*b)*log(b*x + a))/b^6

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Sympy [A]  time = 0.478746, size = 99, normalized size = 0.92 \begin{align*} \frac{B x^{5}}{5 b} - \frac{a^{4} \left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x^{3} \left (- A a b + B a^{2}\right )}{3 b^{3}} - \frac{x^{2} \left (- A a^{2} b + B a^{3}\right )}{2 b^{4}} + \frac{x \left (- A a^{3} b + B a^{4}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.19186, size = 161, normalized size = 1.49 \begin{align*} \frac{12 \, B b^{4} x^{5} - 15 \, B a b^{3} x^{4} + 15 \, A b^{4} x^{4} + 20 \, B a^{2} b^{2} x^{3} - 20 \, A a b^{3} x^{3} - 30 \, B a^{3} b x^{2} + 30 \, A a^{2} b^{2} x^{2} + 60 \, B a^{4} x - 60 \, A a^{3} b x}{60 \, b^{5}} - \frac{{\left (B a^{5} - A a^{4} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*B*b^4*x^5 - 15*B*a*b^3*x^4 + 15*A*b^4*x^4 + 20*B*a^2*b^2*x^3 - 20*A*a*b^3*x^3 - 30*B*a^3*b*x^2 + 30*A
*a^2*b^2*x^2 + 60*B*a^4*x - 60*A*a^3*b*x)/b^5 - (B*a^5 - A*a^4*b)*log(abs(b*x + a))/b^6