Optimal. Leaf size=199 \[ -\frac {b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac {e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac {e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac {3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac {3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \]
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Rubi [A] time = 0.22, antiderivative size = 199, 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} \[ -\frac {b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac {e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac {e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac {3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac {3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx &=\int \left (\frac {b^2 (A b-a B)}{(b d-a e)^3 (a+b x)^3}+\frac {b^2 (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)^2}+\frac {3 b^2 e (-b B d+2 A b e-a B e)}{(b d-a e)^5 (a+b x)}-\frac {e^2 (-B d+A e)}{(b d-a e)^3 (d+e x)^3}-\frac {e^2 (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (d+e x)^2}-\frac {3 b e^2 (-b B d+2 A b e-a B e)}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac {b (A b-a B)}{2 (b d-a e)^3 (a+b x)^2}-\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)}-\frac {e (B d-A e)}{2 (b d-a e)^3 (d+e x)^2}-\frac {e (2 b B d-3 A b e+a B e)}{(b d-a e)^4 (d+e x)}-\frac {3 b e (b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^5}+\frac {3 b e (b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^5}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 185, normalized size = 0.93 \[ \frac {-\frac {b (A b-a B) (b d-a e)^2}{(a+b x)^2}+\frac {e (b d-a e)^2 (A e-B d)}{(d+e x)^2}-\frac {2 b (b d-a e) (2 a B e-3 A b e+b B d)}{a+b x}+\frac {2 e (b d-a e) (-a B e+3 A b e-2 b B d)}{d+e x}-6 b e \log (a+b x) (a B e-2 A b e+b B d)+6 b e \log (d+e x) (a B e-2 A b e+b B d)}{2 (b d-a e)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 1215, normalized size = 6.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 513, normalized size = 2.58 \[ -\frac {3 \, {\left (B b^{3} d e + B a b^{2} e^{2} - 2 \, A b^{3} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac {3 \, {\left (B b^{2} d e^{2} + B a b e^{3} - 2 \, A b^{2} e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {6 \, B b^{3} d x^{3} e^{2} + 9 \, B b^{3} d^{2} x^{2} e + 2 \, B b^{3} d^{3} x + 6 \, B a b^{2} x^{3} e^{3} - 12 \, A b^{3} x^{3} e^{3} + 18 \, B a b^{2} d x^{2} e^{2} - 18 \, A b^{3} d x^{2} e^{2} + 16 \, B a b^{2} d^{2} x e - 4 \, A b^{3} d^{2} x e + B a b^{2} d^{3} + A b^{3} d^{3} + 9 \, B a^{2} b x^{2} e^{3} - 18 \, A a b^{2} x^{2} e^{3} + 16 \, B a^{2} b d x e^{2} - 28 \, A a b^{2} d x e^{2} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + 2 \, B a^{3} x e^{3} - 4 \, A a^{2} b x e^{3} + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 380, normalized size = 1.91 \[ -\frac {6 A \,b^{2} e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}+\frac {6 A \,b^{2} e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 B a b \,e^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {3 B a b \,e^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 B \,b^{2} d e \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {3 B \,b^{2} d e \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}+\frac {3 A \,b^{2} e}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {3 A b \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {2 B a b e}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {B a \,e^{2}}{\left (a e -b d \right )^{4} \left (e x +d \right )}-\frac {B \,b^{2} d}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {2 B b d e}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {A \,b^{2}}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {A \,e^{2}}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}-\frac {B a b}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}+\frac {B d e}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.74, size = 745, normalized size = 3.74 \[ -\frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {A a^{3} e^{3} + {\left (B a b^{2} + A b^{3}\right )} d^{3} + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 7 \, A a^{2} b\right )} d e^{2} + 6 \, {\left (B b^{3} d e^{2} + {\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 9 \, {\left (B b^{3} d^{2} e + 2 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + {\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B b^{3} d^{3} + 2 \, {\left (4 \, B a b^{2} - A b^{3}\right )} d^{2} e + 2 \, {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x}{2 \, {\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} + {\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \, {\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 726, normalized size = 3.65 \[ \frac {2\,\mathrm {atanh}\left (\frac {\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )\,\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+2\,b\,e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5\,\left (-6\,A\,b^2\,e^2+3\,B\,d\,b^2\,e+3\,B\,a\,b\,e^2\right )}\right )\,\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {\frac {B\,a^3\,d\,e^2+A\,a^3\,e^3+10\,B\,a^2\,b\,d^2\,e-7\,A\,a^2\,b\,d\,e^2+B\,a\,b^2\,d^3-7\,A\,a\,b^2\,d^2\,e+A\,b^3\,d^3}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {9\,x^2\,\left (d\,b^2\,e+a\,b\,e^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {x\,\left (a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {3\,b^2\,e^2\,x^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{x\,\left (2\,e\,a^2\,d+2\,b\,a\,d^2\right )+x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )+x^3\,\left (2\,d\,b^2\,e+2\,a\,b\,e^2\right )+a^2\,d^2+b^2\,e^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.67, size = 1431, normalized size = 7.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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