Optimal. Leaf size=106 \[ -\frac {(b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 (d+e x)}+\frac {(b d-a e)^2 (B d-A e)}{2 e^4 (d+e x)^2}-\frac {b \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^4}+\frac {b^2 B x}{e^3} \]
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Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ -\frac {(b d-a e) (-a B e-2 A b e+3 b B d)}{e^4 (d+e x)}+\frac {(b d-a e)^2 (B d-A e)}{2 e^4 (d+e x)^2}-\frac {b \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^4}+\frac {b^2 B x}{e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^3} \, dx &=\int \left (\frac {b^2 B}{e^3}+\frac {(-b d+a e)^2 (-B d+A e)}{e^3 (d+e x)^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e)}{e^3 (d+e x)^2}+\frac {b (-3 b B d+A b e+2 a B e)}{e^3 (d+e x)}\right ) \, dx\\ &=\frac {b^2 B x}{e^3}+\frac {(b d-a e)^2 (B d-A e)}{2 e^4 (d+e x)^2}-\frac {(b d-a e) (3 b B d-2 A b e-a B e)}{e^4 (d+e x)}-\frac {b (3 b B d-A b e-2 a B e) \log (d+e x)}{e^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 143, normalized size = 1.35 \[ -\frac {a^2 e^2 (A e+B (d+2 e x))+2 a b e (A e (d+2 e x)-B d (3 d+4 e x))+2 b (d+e x)^2 \log (d+e x) (-2 a B e-A b e+3 b B d)-\left (b^2 \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )\right )}{2 e^4 (d+e x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 247, normalized size = 2.33 \[ \frac {2 \, B b^{2} e^{3} x^{3} + 4 \, B b^{2} d e^{2} x^{2} - 5 \, B b^{2} d^{3} - A a^{2} e^{3} + 3 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e - {\left (B a^{2} + 2 \, A a b\right )} d e^{2} - 2 \, {\left (2 \, B b^{2} d^{2} e - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x - 2 \, {\left (3 \, B b^{2} d^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (3 \, B b^{2} d e^{2} - {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 2 \, {\left (3 \, B b^{2} d^{2} e - {\left (2 \, B a b + A b^{2}\right )} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 156, normalized size = 1.47 \[ B b^{2} x e^{\left (-3\right )} - {\left (3 \, B b^{2} d - 2 \, B a b e - A b^{2} e\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (5 \, B b^{2} d^{3} - 6 \, B a b d^{2} e - 3 \, A b^{2} d^{2} e + B a^{2} d e^{2} + 2 \, A a b d e^{2} + A a^{2} e^{3} + 2 \, {\left (3 \, B b^{2} d^{2} e - 4 \, B a b d e^{2} - 2 \, A b^{2} d e^{2} + B a^{2} e^{3} + 2 \, A a b e^{3}\right )} x\right )} e^{\left (-4\right )}}{2 \, {\left (x e + d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 242, normalized size = 2.28 \[ -\frac {A \,a^{2}}{2 \left (e x +d \right )^{2} e}+\frac {A a b d}{\left (e x +d \right )^{2} e^{2}}-\frac {A \,b^{2} d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {B \,a^{2} d}{2 \left (e x +d \right )^{2} e^{2}}-\frac {B a b \,d^{2}}{\left (e x +d \right )^{2} e^{3}}+\frac {B \,b^{2} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {2 A a b}{\left (e x +d \right ) e^{2}}+\frac {2 A \,b^{2} d}{\left (e x +d \right ) e^{3}}+\frac {A \,b^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {B \,a^{2}}{\left (e x +d \right ) e^{2}}+\frac {4 B a b d}{\left (e x +d \right ) e^{3}}+\frac {2 B a b \ln \left (e x +d \right )}{e^{3}}-\frac {3 B \,b^{2} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 B \,b^{2} d \ln \left (e x +d \right )}{e^{4}}+\frac {B \,b^{2} x}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 166, normalized size = 1.57 \[ \frac {B b^{2} x}{e^{3}} - \frac {5 \, B b^{2} d^{3} + A a^{2} e^{3} - 3 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 2 \, {\left (3 \, B b^{2} d^{2} e - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{2 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} - \frac {{\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} \log \left (e x + d\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 170, normalized size = 1.60 \[ \frac {\ln \left (d+e\,x\right )\,\left (A\,b^2\,e-3\,B\,b^2\,d+2\,B\,a\,b\,e\right )}{e^4}-\frac {x\,\left (B\,a^2\,e^2-4\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+3\,B\,b^2\,d^2-2\,A\,b^2\,d\,e\right )+\frac {B\,a^2\,d\,e^2+A\,a^2\,e^3-6\,B\,a\,b\,d^2\,e+2\,A\,a\,b\,d\,e^2+5\,B\,b^2\,d^3-3\,A\,b^2\,d^2\,e}{2\,e}}{d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2}+\frac {B\,b^2\,x}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.67, size = 187, normalized size = 1.76 \[ \frac {B b^{2} x}{e^{3}} + \frac {b \left (A b e + 2 B a e - 3 B b d\right ) \log {\left (d + e x \right )}}{e^{4}} + \frac {- A a^{2} e^{3} - 2 A a b d e^{2} + 3 A b^{2} d^{2} e - B a^{2} d e^{2} + 6 B a b d^{2} e - 5 B b^{2} d^{3} + x \left (- 4 A a b e^{3} + 4 A b^{2} d e^{2} - 2 B a^{2} e^{3} + 8 B a b d e^{2} - 6 B b^{2} d^{2} e\right )}{2 d^{2} e^{4} + 4 d e^{5} x + 2 e^{6} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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