Optimal. Leaf size=55 \[ \frac {16 a^4}{b (a-b x)}+\frac {32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac {b^2 x^3}{3} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {627, 43} \begin {gather*} \frac {16 a^4}{b (a-b x)}+\frac {32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac {b^2 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^4}{(a-b x)^2} \, dx\\ &=\int \left (17 a^2+6 a b x+b^2 x^2+\frac {16 a^4}{(a-b x)^2}-\frac {32 a^3}{a-b x}\right ) \, dx\\ &=17 a^2 x+3 a b x^2+\frac {b^2 x^3}{3}+\frac {16 a^4}{b (a-b x)}+\frac {32 a^3 \log (a-b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 56, normalized size = 1.02 \begin {gather*} -\frac {16 a^4}{b (b x-a)}+\frac {32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac {b^2 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6}{\left (a^2-b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 76, normalized size = 1.38 \begin {gather*} \frac {b^{4} x^{4} + 8 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} - 51 \, a^{3} b x - 48 \, a^{4} + 96 \, {\left (a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{3 \, {\left (b^{2} x - a b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 66, normalized size = 1.20 \begin {gather*} \frac {32 \, a^{3} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {16 \, a^{4}}{{\left (b x - a\right )} b} + \frac {b^{8} x^{3} + 9 \, a b^{7} x^{2} + 51 \, a^{2} b^{6} x}{3 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 56, normalized size = 1.02 \begin {gather*} \frac {b^{2} x^{3}}{3}+3 a b \,x^{2}-\frac {16 a^{4}}{\left (b x -a \right ) b}+\frac {32 a^{3} \ln \left (b x -a \right )}{b}+17 a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.39, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + 3 \, a b x^{2} - \frac {16 \, a^{4}}{b^{2} x - a b} + 17 \, a^{2} x + \frac {32 \, a^{3} \log \left (b x - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.39, size = 53, normalized size = 0.96 \begin {gather*} 17\,a^2\,x+\frac {b^2\,x^3}{3}+\frac {16\,a^4}{b\,\left (a-b\,x\right )}+\frac {32\,a^3\,\ln \left (a-b\,x\right )}{b}+3\,a\,b\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.28, size = 49, normalized size = 0.89 \begin {gather*} - \frac {16 a^{4}}{- a b + b^{2} x} + \frac {32 a^{3} \log {\left (- a + b x \right )}}{b} + 17 a^{2} x + 3 a b x^{2} + \frac {b^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________