Optimal. Leaf size=86 \[ \frac {(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac {(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \[ \frac {(a+b x)^3 (-4 a B e+A b e+3 b B d)}{12 e (d+e x)^3 (b d-a e)^2}-\frac {(a+b x)^3 (B d-A e)}{4 e (d+e x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{(d+e x)^5} \, dx &=-\frac {(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac {(3 b B d+A b e-4 a B e) \int \frac {(a+b x)^2}{(d+e x)^4} \, dx}{4 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^3}{4 e (b d-a e) (d+e x)^4}+\frac {(3 b B d+A b e-4 a B e) (a+b x)^3}{12 e (b d-a e)^2 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 125, normalized size = 1.45 \[ -\frac {a^2 e^2 (3 A e+B (d+4 e x))+2 a b e \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b^2 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{12 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 187, normalized size = 2.17 \[ -\frac {12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} + {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B b^{2} d^{2} e + {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.21, size = 245, normalized size = 2.85 \[ -\frac {1}{12} \, {\left (\frac {12 \, B b^{2} e^{\left (-1\right )}}{x e + d} - \frac {18 \, B b^{2} d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac {12 \, B b^{2} d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B b^{2} d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {12 \, B a b}{{\left (x e + d\right )}^{2}} + \frac {6 \, A b^{2}}{{\left (x e + d\right )}^{2}} - \frac {16 \, B a b d}{{\left (x e + d\right )}^{3}} - \frac {8 \, A b^{2} d}{{\left (x e + d\right )}^{3}} + \frac {6 \, B a b d^{2}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A b^{2} d^{2}}{{\left (x e + d\right )}^{4}} + \frac {4 \, B a^{2} e}{{\left (x e + d\right )}^{3}} + \frac {8 \, A a b e}{{\left (x e + d\right )}^{3}} - \frac {3 \, B a^{2} d e}{{\left (x e + d\right )}^{4}} - \frac {6 \, A a b d e}{{\left (x e + d\right )}^{4}} + \frac {3 \, A a^{2} e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 166, normalized size = 1.93 \[ -\frac {B \,b^{2}}{\left (e x +d \right ) e^{4}}-\frac {\left (A b e +2 B a e -3 B b d \right ) b}{2 \left (e x +d \right )^{2} e^{4}}-\frac {2 A a b \,e^{2}-2 A d \,b^{2} e +B \,a^{2} e^{2}-4 B d a b e +3 B \,b^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {A \,a^{2} e^{3}-2 A d a b \,e^{2}+A \,d^{2} b^{2} e -B d \,a^{2} e^{2}+2 B \,d^{2} a b e -B \,b^{2} d^{3}}{4 \left (e x +d \right )^{4} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 187, normalized size = 2.17 \[ -\frac {12 \, B b^{2} e^{3} x^{3} + 3 \, B b^{2} d^{3} + 3 \, A a^{2} e^{3} + {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2} + 6 \, {\left (3 \, B b^{2} d e^{2} + {\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{2} + 4 \, {\left (3 \, B b^{2} d^{2} e + {\left (2 \, B a b + A b^{2}\right )} d e^{2} + {\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x}{12 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 184, normalized size = 2.14 \[ -\frac {\frac {B\,a^2\,d\,e^2+3\,A\,a^2\,e^3+2\,B\,a\,b\,d^2\,e+2\,A\,a\,b\,d\,e^2+3\,B\,b^2\,d^3+A\,b^2\,d^2\,e}{12\,e^4}+\frac {x\,\left (B\,a^2\,e^2+2\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+3\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{3\,e^3}+\frac {b\,x^2\,\left (A\,b\,e+2\,B\,a\,e+3\,B\,b\,d\right )}{2\,e^2}+\frac {B\,b^2\,x^3}{e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.50, size = 223, normalized size = 2.59 \[ \frac {- 3 A a^{2} e^{3} - 2 A a b d e^{2} - A b^{2} d^{2} e - B a^{2} d e^{2} - 2 B a b d^{2} e - 3 B b^{2} d^{3} - 12 B b^{2} e^{3} x^{3} + x^{2} \left (- 6 A b^{2} e^{3} - 12 B a b e^{3} - 18 B b^{2} d e^{2}\right ) + x \left (- 8 A a b e^{3} - 4 A b^{2} d e^{2} - 4 B a^{2} e^{3} - 8 B a b d e^{2} - 12 B b^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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