Optimal. Leaf size=98 \[ \frac {(b d-a e)^4 \log (d+e x)}{e^5}-\frac {b x (b d-a e)^3}{e^4}+\frac {(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac {(a+b x)^3 (b d-a e)}{3 e^2}+\frac {(a+b x)^4}{4 e} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {b x (b d-a e)^3}{e^4}+\frac {(a+b x)^2 (b d-a e)^2}{2 e^3}-\frac {(a+b x)^3 (b d-a e)}{3 e^2}+\frac {(b d-a e)^4 \log (d+e x)}{e^5}+\frac {(a+b x)^4}{4 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx &=\int \frac {(a+b x)^4}{d+e x} \, dx\\ &=\int \left (-\frac {b (b d-a e)^3}{e^4}+\frac {b (b d-a e)^2 (a+b x)}{e^3}-\frac {b (b d-a e) (a+b x)^2}{e^2}+\frac {b (a+b x)^3}{e}+\frac {(-b d+a e)^4}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {b (b d-a e)^3 x}{e^4}+\frac {(b d-a e)^2 (a+b x)^2}{2 e^3}-\frac {(b d-a e) (a+b x)^3}{3 e^2}+\frac {(a+b x)^4}{4 e}+\frac {(b d-a e)^4 \log (d+e x)}{e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 115, normalized size = 1.17 \begin {gather*} \frac {b e x \left (48 a^3 e^3+36 a^2 b e^2 (e x-2 d)+8 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 (b d-a e)^4 \log (d+e x)}{12 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{d+e x} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 179, normalized size = 1.83 \begin {gather*} \frac {3 \, b^{4} e^{4} x^{4} - 4 \, {\left (b^{4} d e^{3} - 4 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} - 12 \, {\left (b^{4} d^{3} e - 4 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} - 4 \, a^{3} b e^{4}\right )} x + 12 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 176, normalized size = 1.80 \begin {gather*} {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, b^{4} x^{4} e^{3} - 4 \, b^{4} d x^{3} e^{2} + 6 \, b^{4} d^{2} x^{2} e - 12 \, b^{4} d^{3} x + 16 \, a b^{3} x^{3} e^{3} - 24 \, a b^{3} d x^{2} e^{2} + 48 \, a b^{3} d^{2} x e + 36 \, a^{2} b^{2} x^{2} e^{3} - 72 \, a^{2} b^{2} d x e^{2} + 48 \, a^{3} b x e^{3}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 209, normalized size = 2.13 \begin {gather*} \frac {b^{4} x^{4}}{4 e}+\frac {4 a \,b^{3} x^{3}}{3 e}-\frac {b^{4} d \,x^{3}}{3 e^{2}}+\frac {3 a^{2} b^{2} x^{2}}{e}-\frac {2 a \,b^{3} d \,x^{2}}{e^{2}}+\frac {b^{4} d^{2} x^{2}}{2 e^{3}}+\frac {a^{4} \ln \left (e x +d \right )}{e}-\frac {4 a^{3} b d \ln \left (e x +d \right )}{e^{2}}+\frac {4 a^{3} b x}{e}+\frac {6 a^{2} b^{2} d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {6 a^{2} b^{2} d x}{e^{2}}-\frac {4 a \,b^{3} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {4 a \,b^{3} d^{2} x}{e^{3}}+\frac {b^{4} d^{4} \ln \left (e x +d \right )}{e^{5}}-\frac {b^{4} d^{3} x}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.33, size = 177, normalized size = 1.81 \begin {gather*} \frac {3 \, b^{4} e^{3} x^{4} - 4 \, {\left (b^{4} d e^{2} - 4 \, a b^{3} e^{3}\right )} x^{3} + 6 \, {\left (b^{4} d^{2} e - 4 \, a b^{3} d e^{2} + 6 \, a^{2} b^{2} e^{3}\right )} x^{2} - 12 \, {\left (b^{4} d^{3} - 4 \, a b^{3} d^{2} e + 6 \, a^{2} b^{2} d e^{2} - 4 \, a^{3} b e^{3}\right )} x}{12 \, e^{4}} + \frac {{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \log \left (e x + d\right )}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 189, normalized size = 1.93 \begin {gather*} x^3\,\left (\frac {4\,a\,b^3}{3\,e}-\frac {b^4\,d}{3\,e^2}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {4\,a\,b^3}{e}-\frac {b^4\,d}{e^2}\right )}{e}-\frac {6\,a^2\,b^2}{e}\right )}{e}+\frac {4\,a^3\,b}{e}\right )-x^2\,\left (\frac {d\,\left (\frac {4\,a\,b^3}{e}-\frac {b^4\,d}{e^2}\right )}{2\,e}-\frac {3\,a^2\,b^2}{e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{e^5}+\frac {b^4\,x^4}{4\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.40, size = 136, normalized size = 1.39 \begin {gather*} \frac {b^{4} x^{4}}{4 e} + x^{3} \left (\frac {4 a b^{3}}{3 e} - \frac {b^{4} d}{3 e^{2}}\right ) + x^{2} \left (\frac {3 a^{2} b^{2}}{e} - \frac {2 a b^{3} d}{e^{2}} + \frac {b^{4} d^{2}}{2 e^{3}}\right ) + x \left (\frac {4 a^{3} b}{e} - \frac {6 a^{2} b^{2} d}{e^{2}} + \frac {4 a b^{3} d^{2}}{e^{3}} - \frac {b^{4} d^{3}}{e^{4}}\right ) + \frac {\left (a e - b d\right )^{4} \log {\left (d + e x \right )}}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________