Optimal. Leaf size=28 \[ \frac {(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 37} \begin {gather*} \frac {(a+b x)^3}{3 (d+e x)^3 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 37
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx &=\int \frac {(a+b x)^2}{(d+e x)^4} \, dx\\ &=\frac {(a+b x)^3}{3 (b d-a e) (d+e x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 53, normalized size = 1.89 \begin {gather*} -\frac {a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^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^2+2 a b x+b^2 x^2}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.39, size = 84, normalized size = 3.00 \begin {gather*} -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.15, size = 58, normalized size = 2.07 \begin {gather*} -\frac {{\left (3 \, b^{2} x^{2} e^{2} + 3 \, b^{2} d x e + b^{2} d^{2} + 3 \, a b x e^{2} + a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 71, normalized size = 2.54 \begin {gather*} -\frac {b^{2}}{\left (e x +d \right ) e^{3}}-\frac {\left (a e -b d \right ) b}{\left (e x +d \right )^{2} e^{3}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.36, size = 84, normalized size = 3.00 \begin {gather*} -\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 80, normalized size = 2.86 \begin {gather*} -\frac {\frac {a^2\,e^2+a\,b\,d\,e+b^2\,d^2}{3\,e^3}+\frac {b^2\,x^2}{e}+\frac {b\,x\,\left (a\,e+b\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.61, size = 88, normalized size = 3.14 \begin {gather*} \frac {- a^{2} e^{2} - a b d e - b^{2} d^{2} - 3 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} - 3 b^{2} d e\right )}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________