\(\int \frac {x^4}{a+b x} \, dx\) [158]

Optimal result

Integrand size = 11, antiderivative size = 57 \[ \int \frac {x^4}{a+b x} \, dx=-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b}+\frac {a^4 \log (a+b x)}{b^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^4}{a+b x} \, dx=\frac {a^4 \log (a+b x)}{b^5}-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b} \]

[In]

Int[x^4/(a + b*x),x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{b^2}+\frac {x^3}{b}+\frac {a^4}{b^4 (a+b x)}\right ) \, dx \\ & = -\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b}+\frac {a^4 \log (a+b x)}{b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{a+b x} \, dx=-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{2 b^3}-\frac {a x^3}{3 b^2}+\frac {x^4}{4 b}+\frac {a^4 \log (a+b x)}{b^5} \]

[In]

Integrate[x^4/(a + b*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {-\frac {1}{4} b^{3} x^{4}+\frac {1}{3} a \,b^{2} x^{3}-\frac {1}{2} a^{2} b \,x^{2}+a^{3} x}{b^{4}}+\frac {a^{4} \ln \left (b x +a \right )}{b^{5}}\)	\(52\)
norman	\(-\frac {a^{3} x}{b^{4}}+\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a \,x^{3}}{3 b^{2}}+\frac {x^{4}}{4 b}+\frac {a^{4} \ln \left (b x +a \right )}{b^{5}}\)	\(52\)
risch	\(-\frac {a^{3} x}{b^{4}}+\frac {a^{2} x^{2}}{2 b^{3}}-\frac {a \,x^{3}}{3 b^{2}}+\frac {x^{4}}{4 b}+\frac {a^{4} \ln \left (b x +a \right )}{b^{5}}\)	\(52\)
parallelrisch	\(\frac {3 b^{4} x^{4}-4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )-12 a^{3} b x}{12 b^{5}}\)	\(53\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {x^4}{a+b x} \, dx=\frac {3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{a+b x} \, dx=\frac {a^{4} \log {\left (a + b x \right )}}{b^{5}} - \frac {a^{3} x}{b^{4}} + \frac {a^{2} x^{2}}{2 b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {x^{4}}{4 b} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {x^4}{a+b x} \, dx=\frac {a^{4} \log \left (b x + a\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \]

[In]

integrate(x^4/(b*x+a),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {x^4}{a+b x} \, dx=\frac {a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {x^4}{a+b x} \, dx=\frac {x^4}{4\,b}+\frac {a^4\,\ln \left (a+b\,x\right )}{b^5}-\frac {a\,x^3}{3\,b^2}-\frac {a^3\,x}{b^4}+\frac {a^2\,x^2}{2\,b^3} \]

[In]

int(x^4/(a + b*x),x)

[Out]

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