3.3.14 \(\int \frac {x^5}{a x^2+b x^3} \, dx\) [214]

Optimal. Leaf size=44 \[ \frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b}-\frac {a^3 \log (a+b x)}{b^4} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45} \begin {gather*} -\frac {a^3 \log (a+b x)}{b^4}+\frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/(a*x^2 + b*x^3),x]

[Out]

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

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])

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 44, normalized size = 1.00 \begin {gather*} \frac {a^2 x}{b^3}-\frac {a x^2}{2 b^2}+\frac {x^3}{3 b}-\frac {a^3 \log (a+b x)}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a*x^2 + b*x^3),x]

[Out]

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

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Maple [A]
time = 0.35, size = 41, normalized size = 0.93

method	result	size
default	\(\frac {\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x}{b^{3}}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}\)	\(41\)
risch	\(\frac {a^{2} x}{b^{3}}-\frac {a \,x^{2}}{2 b^{2}}+\frac {x^{3}}{3 b}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}\)	\(41\)
norman	\(\frac {\frac {a^{2} x^{2}}{b^{3}}+\frac {x^{4}}{3 b}-\frac {a \,x^{3}}{2 b^{2}}}{x}-\frac {a^{3} \ln \left (b x +a \right )}{b^{4}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^3+a*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 42, normalized size = 0.95 \begin {gather*} -\frac {a^{3} \log \left (b x + a\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.43, size = 41, normalized size = 0.93 \begin {gather*} \frac {2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )}{6 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.04, size = 37, normalized size = 0.84 \begin {gather*} - \frac {a^{3} \log {\left (a + b x \right )}}{b^{4}} + \frac {a^{2} x}{b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{3}}{3 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**3+a*x**2),x)

[Out]

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

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Giac [A]
time = 1.74, size = 43, normalized size = 0.98 \begin {gather*} -\frac {a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{6 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^3+a*x^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 40, normalized size = 0.91 \begin {gather*} \frac {x^3}{3\,b}-\frac {a^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {a\,x^2}{2\,b^2}+\frac {a^2\,x}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a*x^2 + b*x^3),x)

[Out]

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

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