Optimal. Leaf size=101 \[ -\frac {(d+e x)^3 (A b-a B)}{3 b (a+b x)^3 (b d-a e)}-\frac {2 B e (b d-a e)}{b^4 (a+b x)}-\frac {B (b d-a e)^2}{2 b^4 (a+b x)^2}+\frac {B e^2 \log (a+b x)}{b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {27, 78, 43} \[ -\frac {(d+e x)^3 (A b-a B)}{3 b (a+b x)^3 (b d-a e)}-\frac {2 B e (b d-a e)}{b^4 (a+b x)}-\frac {B (b d-a e)^2}{2 b^4 (a+b x)^2}+\frac {B e^2 \log (a+b x)}{b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 43
Rule 78
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 )^2} \, dx &=\int \frac {(A+B x) (d+e x)^2}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^3}{3 b (b d-a e) (a+b x)^3}+\frac {B \int \frac {(d+e x)^2}{(a+b x)^3} \, dx}{b}\\ &=-\frac {(A b-a B) (d+e x)^3}{3 b (b d-a e) (a+b x)^3}+\frac {B \int \left (\frac {(b d-a e)^2}{b^2 (a+b x)^3}+\frac {2 e (b d-a e)}{b^2 (a+b x)^2}+\frac {e^2}{b^2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac {B (b d-a e)^2}{2 b^4 (a+b x)^2}-\frac {2 B e (b d-a e)}{b^4 (a+b x)}-\frac {(A b-a B) (d+e x)^3}{3 b (b d-a e) (a+b x)^3}+\frac {B e^2 \log (a+b x)}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 138, normalized size = 1.37 \[ \frac {-2 A b \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+B \left (11 a^3 e^2+a^2 b e (27 e x-4 d)-a b^2 \left (d^2+12 d e x-18 e^2 x^2\right )-3 b^3 d x (d+4 e x)\right )+6 B e^2 (a+b x)^3 \log (a+b x)}{6 b^4 (a+b x)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.52, size = 226, normalized size = 2.24 \[ -\frac {{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} + 2 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} d e - {\left (11 \, B a^{3} - 2 \, A a^{2} b\right )} e^{2} + 6 \, {\left (2 \, B b^{3} d e - {\left (3 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 3 \, {\left (B b^{3} d^{2} + 2 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d e - {\left (9 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{2}\right )} x - 6 \, {\left (B b^{3} e^{2} x^{3} + 3 \, B a b^{2} e^{2} x^{2} + 3 \, B a^{2} b e^{2} x + B a^{3} e^{2}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 161, normalized size = 1.59 \[ \frac {B e^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {6 \, {\left (2 \, B b^{2} d e - 3 \, B a b e^{2} + A b^{2} e^{2}\right )} x^{2} + 3 \, {\left (B b^{2} d^{2} + 4 \, B a b d e + 2 \, A b^{2} d e - 9 \, B a^{2} e^{2} + 2 \, A a b e^{2}\right )} x + \frac {B a b^{2} d^{2} + 2 \, A b^{3} d^{2} + 4 \, B a^{2} b d e + 2 \, A a b^{2} d e - 11 \, B a^{3} e^{2} + 2 \, A a^{2} b e^{2}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 251, normalized size = 2.49 \[ -\frac {A \,a^{2} e^{2}}{3 \left (b x +a \right )^{3} b^{3}}+\frac {2 A a d e}{3 \left (b x +a \right )^{3} b^{2}}-\frac {A \,d^{2}}{3 \left (b x +a \right )^{3} b}+\frac {B \,a^{3} e^{2}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {2 B \,a^{2} d e}{3 \left (b x +a \right )^{3} b^{3}}+\frac {B a \,d^{2}}{3 \left (b x +a \right )^{3} b^{2}}+\frac {A a \,e^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {A d e}{\left (b x +a \right )^{2} b^{2}}-\frac {3 B \,a^{2} e^{2}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {2 B a d e}{\left (b x +a \right )^{2} b^{3}}-\frac {B \,d^{2}}{2 \left (b x +a \right )^{2} b^{2}}-\frac {A \,e^{2}}{\left (b x +a \right ) b^{3}}+\frac {3 B a \,e^{2}}{\left (b x +a \right ) b^{4}}-\frac {2 B d e}{\left (b x +a \right ) b^{3}}+\frac {B \,e^{2} \ln \left (b x +a \right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.64, size = 189, normalized size = 1.87 \[ -\frac {{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} + 2 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} d e - {\left (11 \, B a^{3} - 2 \, A a^{2} b\right )} e^{2} + 6 \, {\left (2 \, B b^{3} d e - {\left (3 \, B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 3 \, {\left (B b^{3} d^{2} + 2 \, {\left (2 \, B a b^{2} + A b^{3}\right )} d e - {\left (9 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{2}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {B e^{2} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.25, size = 178, normalized size = 1.76 \[ \frac {B\,e^2\,\ln \left (a+b\,x\right )}{b^4}-\frac {\frac {-11\,B\,a^3\,e^2+4\,B\,a^2\,b\,d\,e+2\,A\,a^2\,b\,e^2+B\,a\,b^2\,d^2+2\,A\,a\,b^2\,d\,e+2\,A\,b^3\,d^2}{6\,b^4}+\frac {x\,\left (-9\,B\,a^2\,e^2+4\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{2\,b^3}+\frac {e\,x^2\,\left (A\,b\,e-3\,B\,a\,e+2\,B\,b\,d\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 5.77, size = 211, normalized size = 2.09 \[ \frac {B e^{2} \log {\left (a + b x \right )}}{b^{4}} + \frac {- 2 A a^{2} b e^{2} - 2 A a b^{2} d e - 2 A b^{3} d^{2} + 11 B a^{3} e^{2} - 4 B a^{2} b d e - B a b^{2} d^{2} + x^{2} \left (- 6 A b^{3} e^{2} + 18 B a b^{2} e^{2} - 12 B b^{3} d e\right ) + x \left (- 6 A a b^{2} e^{2} - 6 A b^{3} d e + 27 B a^{2} b e^{2} - 12 B a b^{2} d e - 3 B b^{3} d^{2}\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________