Optimal. Leaf size=129 \[ -\frac {(a+b x)^4 (B d-A e)}{4 e (d+e x)^4 (b d-a e)}+\frac {3 b^2 B (b d-a e)}{e^5 (d+e x)}-\frac {3 b B (b d-a e)^2}{2 e^5 (d+e x)^2}+\frac {B (b d-a e)^3}{3 e^5 (d+e x)^3}+\frac {b^3 B \log (d+e x)}{e^5} \]
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Rubi [A] time = 0.11, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 43} \[ -\frac {(a+b x)^4 (B d-A e)}{4 e (d+e x)^4 (b d-a e)}+\frac {3 b^2 B (b d-a e)}{e^5 (d+e x)}-\frac {3 b B (b d-a e)^2}{2 e^5 (d+e x)^2}+\frac {B (b d-a e)^3}{3 e^5 (d+e x)^3}+\frac {b^3 B \log (d+e x)}{e^5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^5} \, dx &=-\frac {(B d-A e) (a+b x)^4}{4 e (b d-a e) (d+e x)^4}+\frac {B \int \frac {(a+b x)^3}{(d+e x)^4} \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^4}{4 e (b d-a e) (d+e x)^4}+\frac {B \int \left (\frac {(-b d+a e)^3}{e^3 (d+e x)^4}+\frac {3 b (b d-a e)^2}{e^3 (d+e x)^3}-\frac {3 b^2 (b d-a e)}{e^3 (d+e x)^2}+\frac {b^3}{e^3 (d+e x)}\right ) \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^4}{4 e (b d-a e) (d+e x)^4}+\frac {B (b d-a e)^3}{3 e^5 (d+e x)^3}-\frac {3 b B (b d-a e)^2}{2 e^5 (d+e x)^2}+\frac {3 b^2 B (b d-a e)}{e^5 (d+e x)}+\frac {b^3 B \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 222, normalized size = 1.72 \[ \frac {-a^3 e^3 (3 A e+B (d+4 e x))-3 a^2 b e^2 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )-3 a b^2 e \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+b^3 \left (B d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-3 A e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 b^3 B (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 354, normalized size = 2.74 \[ \frac {25 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 12 \, {\left (4 \, B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 18 \, {\left (6 \, B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} - {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 4 \, {\left (22 \, B b^{3} d^{3} e - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} - 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x + 12 \, {\left (B b^{3} e^{4} x^{4} + 4 \, B b^{3} d e^{3} x^{3} + 6 \, B b^{3} d^{2} e^{2} x^{2} + 4 \, B b^{3} d^{3} e x + B b^{3} d^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 447, normalized size = 3.47 \[ -B b^{3} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{12} \, {\left (\frac {48 \, B b^{3} d e^{15}}{x e + d} - \frac {36 \, B b^{3} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac {16 \, B b^{3} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac {3 \, B b^{3} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac {36 \, B a b^{2} e^{16}}{x e + d} - \frac {12 \, A b^{3} e^{16}}{x e + d} + \frac {54 \, B a b^{2} d e^{16}}{{\left (x e + d\right )}^{2}} + \frac {18 \, A b^{3} d e^{16}}{{\left (x e + d\right )}^{2}} - \frac {36 \, B a b^{2} d^{2} e^{16}}{{\left (x e + d\right )}^{3}} - \frac {12 \, A b^{3} d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac {9 \, B a b^{2} d^{3} e^{16}}{{\left (x e + d\right )}^{4}} + \frac {3 \, A b^{3} d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac {18 \, B a^{2} b e^{17}}{{\left (x e + d\right )}^{2}} - \frac {18 \, A a b^{2} e^{17}}{{\left (x e + d\right )}^{2}} + \frac {24 \, B a^{2} b d e^{17}}{{\left (x e + d\right )}^{3}} + \frac {24 \, A a b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} - \frac {9 \, B a^{2} b d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac {9 \, A a b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac {4 \, B a^{3} e^{18}}{{\left (x e + d\right )}^{3}} - \frac {12 \, A a^{2} b e^{18}}{{\left (x e + d\right )}^{3}} + \frac {3 \, B a^{3} d e^{18}}{{\left (x e + d\right )}^{4}} + \frac {9 \, A a^{2} b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac {3 \, A a^{3} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 430, normalized size = 3.33 \[ -\frac {A \,a^{3}}{4 \left (e x +d \right )^{4} e}+\frac {3 A \,a^{2} b d}{4 \left (e x +d \right )^{4} e^{2}}-\frac {3 A a \,b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {A \,b^{3} d^{3}}{4 \left (e x +d \right )^{4} e^{4}}+\frac {B \,a^{3} d}{4 \left (e x +d \right )^{4} e^{2}}-\frac {3 B \,a^{2} b \,d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {3 B a \,b^{2} d^{3}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {B \,b^{3} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}-\frac {A \,a^{2} b}{\left (e x +d \right )^{3} e^{2}}+\frac {2 A a \,b^{2} d}{\left (e x +d \right )^{3} e^{3}}-\frac {A \,b^{3} d^{2}}{\left (e x +d \right )^{3} e^{4}}-\frac {B \,a^{3}}{3 \left (e x +d \right )^{3} e^{2}}+\frac {2 B \,a^{2} b d}{\left (e x +d \right )^{3} e^{3}}-\frac {3 B a \,b^{2} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 B \,b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {3 A a \,b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 A \,b^{3} d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 B \,a^{2} b}{2 \left (e x +d \right )^{2} e^{3}}+\frac {9 B a \,b^{2} d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 B \,b^{3} d^{2}}{\left (e x +d \right )^{2} e^{5}}-\frac {A \,b^{3}}{\left (e x +d \right ) e^{4}}-\frac {3 B a \,b^{2}}{\left (e x +d \right ) e^{4}}+\frac {4 B \,b^{3} d}{\left (e x +d \right ) e^{5}}+\frac {B \,b^{3} \ln \left (e x +d \right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.52, size = 302, normalized size = 2.34 \[ \frac {25 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 12 \, {\left (4 \, B b^{3} d e^{3} - {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 18 \, {\left (6 \, B b^{3} d^{2} e^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} - {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 4 \, {\left (22 \, B b^{3} d^{3} e - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} - 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac {B b^{3} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 303, normalized size = 2.35 \[ \frac {B\,b^3\,\ln \left (d+e\,x\right )}{e^5}-\frac {\frac {B\,a^3\,d\,e^3+3\,A\,a^3\,e^4+3\,B\,a^2\,b\,d^2\,e^2+3\,A\,a^2\,b\,d\,e^3+9\,B\,a\,b^2\,d^3\,e+3\,A\,a\,b^2\,d^2\,e^2-25\,B\,b^3\,d^4+3\,A\,b^3\,d^3\,e}{12\,e^5}+\frac {x\,\left (B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+9\,B\,a\,b^2\,d^2\,e+3\,A\,a\,b^2\,d\,e^2-22\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{3\,e^4}+\frac {3\,x^2\,\left (B\,a^2\,b\,e^2+3\,B\,a\,b^2\,d\,e+A\,a\,b^2\,e^2-6\,B\,b^3\,d^2+A\,b^3\,d\,e\right )}{2\,e^3}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e-4\,B\,b\,d\right )}{e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 32.19, size = 359, normalized size = 2.78 \[ \frac {B b^{3} \log {\left (d + e x \right )}}{e^{5}} + \frac {- 3 A a^{3} e^{4} - 3 A a^{2} b d e^{3} - 3 A a b^{2} d^{2} e^{2} - 3 A b^{3} d^{3} e - B a^{3} d e^{3} - 3 B a^{2} b d^{2} e^{2} - 9 B a b^{2} d^{3} e + 25 B b^{3} d^{4} + x^{3} \left (- 12 A b^{3} e^{4} - 36 B a b^{2} e^{4} + 48 B b^{3} d e^{3}\right ) + x^{2} \left (- 18 A a b^{2} e^{4} - 18 A b^{3} d e^{3} - 18 B a^{2} b e^{4} - 54 B a b^{2} d e^{3} + 108 B b^{3} d^{2} e^{2}\right ) + x \left (- 12 A a^{2} b e^{4} - 12 A a b^{2} d e^{3} - 12 A b^{3} d^{2} e^{2} - 4 B a^{3} e^{4} - 12 B a^{2} b d e^{3} - 36 B a b^{2} d^{2} e^{2} + 88 B b^{3} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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