3.2.70 \(\int \frac {x^4}{(a+b x)^2} \, dx\) [170]

Optimal. Leaf size=58 \[ \frac {3 a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{3 b^2}-\frac {a^4}{b^5 (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \begin {gather*} -\frac {a^4}{b^5 (a+b x)}-\frac {4 a^3 \log (a+b x)}{b^5}+\frac {3 a^2 x}{b^4}-\frac {a x^2}{b^3}+\frac {x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b*x)^2,x]

[Out]

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

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Mathematica [A]
time = 0.01, size = 54, normalized size = 0.93 \begin {gather*} \frac {9 a^2 b x-3 a b^2 x^2+b^3 x^3-\frac {3 a^4}{a+b x}-12 a^3 \log (a+b x)}{3 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b*x)^2,x]

[Out]

(9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 - (3*a^4)/(a + b*x) - 12*a^3*Log[a + b*x])/(3*b^5)

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Mathics [A]
time = 2.13, size = 72, normalized size = 1.24 \begin {gather*} \frac {-4 a^3 \text {Log}\left [a+b x\right ] \left (a+b x\right )-a^4+3 a^2 b x \left (a+b x\right )-a b^2 x^2 \left (a+b x\right )+\frac {b^3 x^3 \left (a+b x\right )}{3}}{b^5 \left (a+b x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^4/(a + b*x)^2,x]')

[Out]

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

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Maple [A]
time = 0.08, size = 57, normalized size = 0.98

method	result	size
default	\(\frac {\frac {1}{3} b^{2} x^{3}-a b \,x^{2}+3 a^{2} x}{b^{4}}-\frac {a^{4}}{b^{5} \left (b x +a \right )}-\frac {4 a^{3} \ln \left (b x +a \right )}{b^{5}}\)	\(57\)
risch	\(\frac {3 a^{2} x}{b^{4}}-\frac {a \,x^{2}}{b^{3}}+\frac {x^{3}}{3 b^{2}}-\frac {a^{4}}{b^{5} \left (b x +a \right )}-\frac {4 a^{3} \ln \left (b x +a \right )}{b^{5}}\)	\(57\)
norman	\(\frac {\frac {x^{4}}{3 b}-\frac {2 a \,x^{3}}{3 b^{2}}-\frac {4 a^{4}}{b^{5}}+\frac {2 a^{2} x^{2}}{b^{3}}}{b x +a}-\frac {4 a^{3} \ln \left (b x +a \right )}{b^{5}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 59, normalized size = 1.02 \begin {gather*} -\frac {a^{4}}{b^{6} x + a b^{5}} - \frac {4 \, a^{3} \log \left (b x + a\right )}{b^{5}} + \frac {b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{3 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

-a^4/(b^6*x + a*b^5) - 4*a^3*log(b*x + a)/b^5 + 1/3*(b^2*x^3 - 3*a*b*x^2 + 9*a^2*x)/b^4

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Fricas [A]
time = 0.31, size = 73, normalized size = 1.26 \begin {gather*} \frac {b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \, {\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{6} x + a b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a^4)*log(b*x + a))/(b^6*x + a*b
^5)

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Sympy [A]
time = 0.10, size = 54, normalized size = 0.93 \begin {gather*} - \frac {a^{4}}{a b^{5} + b^{6} x} - \frac {4 a^{3} \log {\left (a + b x \right )}}{b^{5}} + \frac {3 a^{2} x}{b^{4}} - \frac {a x^{2}}{b^{3}} + \frac {x^{3}}{3 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**2,x)

[Out]

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

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Giac [A]
time = 0.00, size = 67, normalized size = 1.16 \begin {gather*} \frac {\frac {1}{3} x^{3} b^{4}-x^{2} b^{3} a+3 x b^{2} a^{2}}{b^{6}}-\frac {a^{4}}{b^{5} \left (x b+a\right )}-\frac {4 a^{3} \ln \left |x b+a\right |}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2,x)

[Out]

-4*a^3*log(abs(b*x + a))/b^5 - a^4/((b*x + a)*b^5) + 1/3*(b^4*x^3 - 3*a*b^3*x^2 + 9*a^2*b^2*x)/b^6

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Mupad [B]
time = 0.07, size = 62, normalized size = 1.07 \begin {gather*} \frac {x^3}{3\,b^2}-\frac {4\,a^3\,\ln \left (a+b\,x\right )}{b^5}-\frac {a\,x^2}{b^3}+\frac {3\,a^2\,x}{b^4}-\frac {a^4}{b\,\left (x\,b^5+a\,b^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b*x)^2,x)

[Out]

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

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