Optimal. Leaf size=102 \[ -\frac{10 b^3}{a^6 (a+b x)}-\frac{2 b^3}{a^5 (a+b x)^2}-\frac{b^3}{3 a^4 (a+b x)^3}-\frac{10 b^2}{a^6 x}-\frac{20 b^3 \log (x)}{a^7}+\frac{20 b^3 \log (a+b x)}{a^7}+\frac{2 b}{a^5 x^2}-\frac{1}{3 a^4 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0529369, antiderivative size = 102, normalized size of antiderivative = 1., 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 = {44} \[ -\frac{10 b^3}{a^6 (a+b x)}-\frac{2 b^3}{a^5 (a+b x)^2}-\frac{b^3}{3 a^4 (a+b x)^3}-\frac{10 b^2}{a^6 x}-\frac{20 b^3 \log (x)}{a^7}+\frac{20 b^3 \log (a+b x)}{a^7}+\frac{2 b}{a^5 x^2}-\frac{1}{3 a^4 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 (a+b x)^4} \, dx &=\int \left (\frac{1}{a^4 x^4}-\frac{4 b}{a^5 x^3}+\frac{10 b^2}{a^6 x^2}-\frac{20 b^3}{a^7 x}+\frac{b^4}{a^4 (a+b x)^4}+\frac{4 b^4}{a^5 (a+b x)^3}+\frac{10 b^4}{a^6 (a+b x)^2}+\frac{20 b^4}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a^4 x^3}+\frac{2 b}{a^5 x^2}-\frac{10 b^2}{a^6 x}-\frac{b^3}{3 a^4 (a+b x)^3}-\frac{2 b^3}{a^5 (a+b x)^2}-\frac{10 b^3}{a^6 (a+b x)}-\frac{20 b^3 \log (x)}{a^7}+\frac{20 b^3 \log (a+b x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.0965544, size = 88, normalized size = 0.86 \[ -\frac{\frac{a \left (15 a^3 b^2 x^2+110 a^2 b^3 x^3-3 a^4 b x+a^5+150 a b^4 x^4+60 b^5 x^5\right )}{x^3 (a+b x)^3}-60 b^3 \log (a+b x)+60 b^3 \log (x)}{3 a^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 99, normalized size = 1. \begin{align*} -{\frac{1}{3\,{a}^{4}{x}^{3}}}+2\,{\frac{b}{{a}^{5}{x}^{2}}}-10\,{\frac{{b}^{2}}{{a}^{6}x}}-{\frac{{b}^{3}}{3\,{a}^{4} \left ( bx+a \right ) ^{3}}}-2\,{\frac{{b}^{3}}{{a}^{5} \left ( bx+a \right ) ^{2}}}-10\,{\frac{{b}^{3}}{{a}^{6} \left ( bx+a \right ) }}-20\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{7}}}+20\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06512, size = 158, normalized size = 1.55 \begin{align*} -\frac{60 \, b^{5} x^{5} + 150 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} - 3 \, a^{4} b x + a^{5}}{3 \,{\left (a^{6} b^{3} x^{6} + 3 \, a^{7} b^{2} x^{5} + 3 \, a^{8} b x^{4} + a^{9} x^{3}\right )}} + \frac{20 \, b^{3} \log \left (b x + a\right )}{a^{7}} - \frac{20 \, b^{3} \log \left (x\right )}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61725, size = 385, normalized size = 3.77 \begin{align*} -\frac{60 \, a b^{5} x^{5} + 150 \, a^{2} b^{4} x^{4} + 110 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} - 3 \, a^{5} b x + a^{6} - 60 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{5} + 3 \, a^{2} b^{4} x^{4} + a^{3} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{5} + 3 \, a^{2} b^{4} x^{4} + a^{3} b^{3} x^{3}\right )} \log \left (x\right )}{3 \,{\left (a^{7} b^{3} x^{6} + 3 \, a^{8} b^{2} x^{5} + 3 \, a^{9} b x^{4} + a^{10} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.00234, size = 114, normalized size = 1.12 \begin{align*} - \frac{a^{5} - 3 a^{4} b x + 15 a^{3} b^{2} x^{2} + 110 a^{2} b^{3} x^{3} + 150 a b^{4} x^{4} + 60 b^{5} x^{5}}{3 a^{9} x^{3} + 9 a^{8} b x^{4} + 9 a^{7} b^{2} x^{5} + 3 a^{6} b^{3} x^{6}} + \frac{20 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22387, size = 126, normalized size = 1.24 \begin{align*} \frac{20 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{7}} - \frac{20 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{7}} - \frac{60 \, b^{5} x^{5} + 150 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} - 3 \, a^{4} b x + a^{5}}{3 \,{\left (b x^{2} + a x\right )}^{3} a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]