\(\int \frac {x^4 (A+B x)}{a+b x} \, dx\) [174]

Optimal result

Integrand size = 16, antiderivative size = 108 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=-\frac {a^3 (A b-a B) x}{b^5}+\frac {a^2 (A b-a B) x^2}{2 b^4}-\frac {a (A b-a B) x^3}{3 b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^5}{5 b}+\frac {a^4 (A b-a B) \log (a+b x)}{b^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=\frac {a^4 (A b-a B) \log (a+b x)}{b^6}-\frac {a^3 x (A b-a B)}{b^5}+\frac {a^2 x^2 (A b-a B)}{2 b^4}-\frac {a x^3 (A b-a B)}{3 b^3}+\frac {x^4 (A b-a B)}{4 b^2}+\frac {B x^5}{5 b} \]

[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 78

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*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=\frac {b x \left (60 a^4 B-30 a^3 b (2 A+B x)+10 a^2 b^2 x (3 A+2 B x)-5 a b^3 x^2 (4 A+3 B x)+3 b^4 x^3 (5 A+4 B x)\right )-60 a^4 (-A b+a B) \log (a+b x)}{60 b^6} \]

[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)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94

method	result	size
norman	\(-\frac {a^{3} \left (A b -B a \right ) x}{b^{5}}+\frac {a^{2} \left (A b -B a \right ) x^{2}}{2 b^{4}}-\frac {a \left (A b -B a \right ) x^{3}}{3 b^{3}}+\frac {\left (A b -B a \right ) x^{4}}{4 b^{2}}+\frac {B \,x^{5}}{5 b}+\frac {a^{4} \left (A b -B a \right ) \ln \left (b x +a \right )}{b^{6}}\)	\(101\)
default	\(-\frac {-\frac {1}{5} B \,b^{4} x^{5}-\frac {1}{4} A \,b^{4} x^{4}+\frac {1}{4} B a \,b^{3} x^{4}+\frac {1}{3} A a \,b^{3} x^{3}-\frac {1}{3} B \,a^{2} b^{2} x^{3}-\frac {1}{2} A \,a^{2} b^{2} x^{2}+\frac {1}{2} B \,a^{3} b \,x^{2}+A \,a^{3} b x -B \,a^{4} x}{b^{5}}+\frac {a^{4} \left (A b -B a \right ) \ln \left (b x +a \right )}{b^{6}}\)	\(115\)
risch	\(\frac {B \,x^{5}}{5 b}+\frac {A \,x^{4}}{4 b}-\frac {B a \,x^{4}}{4 b^{2}}-\frac {A a \,x^{3}}{3 b^{2}}+\frac {B \,a^{2} x^{3}}{3 b^{3}}+\frac {A \,a^{2} x^{2}}{2 b^{3}}-\frac {B \,a^{3} x^{2}}{2 b^{4}}-\frac {A \,a^{3} x}{b^{4}}+\frac {B \,a^{4} x}{b^{5}}+\frac {a^{4} \ln \left (b x +a \right ) A}{b^{5}}-\frac {a^{5} \ln \left (b x +a \right ) B}{b^{6}}\)	\(124\)
parallelrisch	\(\frac {12 b^{5} B \,x^{5}+15 A \,b^{5} x^{4}-15 B a \,b^{4} x^{4}-20 A a \,b^{4} x^{3}+20 B \,a^{2} b^{3} x^{3}+30 A \,a^{2} b^{3} x^{2}-30 B \,a^{3} b^{2} x^{2}+60 A \ln \left (b x +a \right ) a^{4} b -60 a^{3} b^{2} A x -60 B \ln \left (b x +a \right ) a^{5}+60 a^{4} b B x}{60 b^{6}}\)	\(124\)

[In]

int(x^4*(B*x+A)/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=\frac {12 \, B b^{5} x^{5} - 15 \, {\left (B a b^{4} - A b^{5}\right )} x^{4} + 20 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} x^{3} - 30 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} + 60 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} x - 60 \, {\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{60 \, b^{6}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=\frac {B x^{5}}{5 b} - \frac {a^{4} \left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{4} \left (\frac {A}{4 b} - \frac {B a}{4 b^{2}}\right ) + x^{3} \left (- \frac {A a}{3 b^{2}} + \frac {B a^{2}}{3 b^{3}}\right ) + x^{2} \left (\frac {A a^{2}}{2 b^{3}} - \frac {B a^{3}}{2 b^{4}}\right ) + x \left (- \frac {A a^{3}}{b^{4}} + \frac {B a^{4}}{b^{5}}\right ) \]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=\frac {12 \, B b^{4} x^{5} - 15 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} + 20 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 30 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 60 \, {\left (B a^{4} - A a^{3} b\right )} x}{60 \, b^{5}} - \frac {{\left (B a^{5} - A a^{4} b\right )} \log \left (b x + a\right )}{b^{6}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=\frac {12 \, B b^{4} x^{5} - 15 \, B a b^{3} x^{4} + 15 \, A b^{4} x^{4} + 20 \, B a^{2} b^{2} x^{3} - 20 \, A a b^{3} x^{3} - 30 \, B a^{3} b x^{2} + 30 \, A a^{2} b^{2} x^{2} + 60 \, B a^{4} x - 60 \, A a^{3} b x}{60 \, b^{5}} - \frac {{\left (B a^{5} - A a^{4} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 (A+B x)}{a+b x} \, dx=x^4\,\left (\frac {A}{4\,b}-\frac {B\,a}{4\,b^2}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^5-A\,a^4\,b\right )}{b^6}+\frac {B\,x^5}{5\,b}+\frac {a^2\,x^2\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{2\,b^2}-\frac {a\,x^3\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{3\,b}-\frac {a^3\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b^3} \]

[In]

int((x^4*(A + B*x))/(a + b*x),x)

[Out]

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