Optimal. Leaf size=65 \[ -\frac {2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac {(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac {e^2}{2 b^3 (a+b x)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 43} \begin {gather*} -\frac {2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac {(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac {e^2}{2 b^3 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^2}{(a+b x)^5} \, dx\\ &=\int \left (\frac {(b d-a e)^2}{b^2 (a+b x)^5}+\frac {2 e (b d-a e)}{b^2 (a+b x)^4}+\frac {e^2}{b^2 (a+b x)^3}\right ) \, dx\\ &=-\frac {(b d-a e)^2}{4 b^3 (a+b x)^4}-\frac {2 e (b d-a e)}{3 b^3 (a+b x)^3}-\frac {e^2}{2 b^3 (a+b x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 56, normalized size = 0.86 \begin {gather*} -\frac {a^2 e^2+2 a b e (d+2 e x)+b^2 \left (3 d^2+8 d e x+6 e^2 x^2\right )}{12 b^3 (a+b x)^4} \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) (d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 98, normalized size = 1.51 \begin {gather*} -\frac {6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 60, normalized size = 0.92 \begin {gather*} -\frac {6 \, b^{2} x^{2} e^{2} + 8 \, b^{2} d x e + 3 \, b^{2} d^{2} + 4 \, a b x e^{2} + 2 \, a b d e + a^{2} e^{2}}{12 \, {\left (b x + a\right )}^{4} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 71, normalized size = 1.09 \begin {gather*} -\frac {e^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {2 \left (a e -b d \right ) e}{3 \left (b x +a \right )^{3} b^{3}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{4 \left (b x +a \right )^{4} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.67, size = 98, normalized size = 1.51 \begin {gather*} -\frac {6 \, b^{2} e^{2} x^{2} + 3 \, b^{2} d^{2} + 2 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b^{2} d e + a b e^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 96, normalized size = 1.48 \begin {gather*} -\frac {\frac {a^2\,e^2+2\,a\,b\,d\,e+3\,b^2\,d^2}{12\,b^3}+\frac {e^2\,x^2}{2\,b}+\frac {e\,x\,\left (a\,e+2\,b\,d\right )}{3\,b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.80, size = 104, normalized size = 1.60 \begin {gather*} \frac {- a^{2} e^{2} - 2 a b d e - 3 b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (- 4 a b e^{2} - 8 b^{2} d e\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________