Optimal. Leaf size=144 \[ \frac {e^2 \log (a+b x) (-4 a B e+A b e+3 b B d)}{b^5}-\frac {3 e (b d-a e) (-2 a B e+A b e+b B d)}{b^5 (a+b x)}-\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{2 b^5 (a+b x)^2}-\frac {(A b-a B) (b d-a e)^3}{3 b^5 (a+b x)^3}+\frac {B e^3 x}{b^4} \]
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Rubi [A] time = 0.15, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \begin {gather*} \frac {e^2 \log (a+b x) (-4 a B e+A b e+3 b B d)}{b^5}-\frac {3 e (b d-a e) (-2 a B e+A b e+b B d)}{b^5 (a+b x)}-\frac {(b d-a e)^2 (-4 a B e+3 A b e+b B d)}{2 b^5 (a+b x)^2}-\frac {(A b-a B) (b d-a e)^3}{3 b^5 (a+b x)^3}+\frac {B e^3 x}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(A+B x) (d+e x)^3}{(a+b x)^4} \, dx\\ &=\int \left (\frac {B e^3}{b^4}+\frac {(A b-a B) (b d-a e)^3}{b^4 (a+b x)^4}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^4 (a+b x)^3}+\frac {3 e (b d-a e) (b B d+A b e-2 a B e)}{b^4 (a+b x)^2}+\frac {e^2 (3 b B d+A b e-4 a B e)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {B e^3 x}{b^4}-\frac {(A b-a B) (b d-a e)^3}{3 b^5 (a+b x)^3}-\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{2 b^5 (a+b x)^2}-\frac {3 e (b d-a e) (b B d+A b e-2 a B e)}{b^5 (a+b x)}+\frac {e^2 (3 b B d+A b e-4 a B e) \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 217, normalized size = 1.51 \begin {gather*} -\frac {A b (b d-a e) \left (11 a^2 e^2+a b e (5 d+27 e x)+b^2 \left (2 d^2+9 d e x+18 e^2 x^2\right )\right )+B \left (26 a^4 e^3+3 a^3 b e^2 (18 e x-11 d)+3 a^2 b^2 e \left (2 d^2-27 d e x+6 e^2 x^2\right )+a b^3 \left (d^3+18 d^2 e x-54 d e^2 x^2-18 e^3 x^3\right )+3 b^4 x \left (d^3+6 d^2 e x-2 e^3 x^3\right )\right )-6 e^2 (a+b x)^3 \log (a+b x) (-4 a B e+A b e+3 b B d)}{6 b^5 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^3}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 427, normalized size = 2.97 \begin {gather*} \frac {6 \, B b^{4} e^{3} x^{4} + 18 \, B a b^{3} e^{3} x^{3} - {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 3 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} d^{2} e + 3 \, {\left (11 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{2} - {\left (26 \, B a^{4} - 11 \, A a^{3} b\right )} e^{3} - 18 \, {\left (B b^{4} d^{2} e - {\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} + {\left (B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} - 3 \, {\left (B b^{4} d^{3} + 3 \, {\left (2 \, B a b^{3} + A b^{4}\right )} d^{2} e - 3 \, {\left (9 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} + 9 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \, {\left (3 \, B a^{3} b d e^{2} - {\left (4 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (3 \, B b^{4} d e^{2} - {\left (4 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (3 \, B a b^{3} d e^{2} - {\left (4 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (3 \, B a^{2} b^{2} d e^{2} - {\left (4 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 266, normalized size = 1.85 \begin {gather*} \frac {B x e^{3}}{b^{4}} + \frac {{\left (3 \, B b d e^{2} - 4 \, B a e^{3} + A b e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {B a b^{3} d^{3} + 2 \, A b^{4} d^{3} + 6 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e - 33 \, B a^{3} b d e^{2} + 6 \, A a^{2} b^{2} d e^{2} + 26 \, B a^{4} e^{3} - 11 \, A a^{3} b e^{3} + 18 \, {\left (B b^{4} d^{2} e - 3 \, B a b^{3} d e^{2} + A b^{4} d e^{2} + 2 \, B a^{2} b^{2} e^{3} - A a b^{3} e^{3}\right )} x^{2} + 3 \, {\left (B b^{4} d^{3} + 6 \, B a b^{3} d^{2} e + 3 \, A b^{4} d^{2} e - 27 \, B a^{2} b^{2} d e^{2} + 6 \, A a b^{3} d e^{2} + 20 \, B a^{3} b e^{3} - 9 \, A a^{2} b^{2} e^{3}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 419, normalized size = 2.91 \begin {gather*} \frac {A \,a^{3} e^{3}}{3 \left (b x +a \right )^{3} b^{4}}-\frac {A \,a^{2} d \,e^{2}}{\left (b x +a \right )^{3} b^{3}}+\frac {A a \,d^{2} e}{\left (b x +a \right )^{3} b^{2}}-\frac {A \,d^{3}}{3 \left (b x +a \right )^{3} b}-\frac {B \,a^{4} e^{3}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {B \,a^{3} d \,e^{2}}{\left (b x +a \right )^{3} b^{4}}-\frac {B \,a^{2} d^{2} e}{\left (b x +a \right )^{3} b^{3}}+\frac {B a \,d^{3}}{3 \left (b x +a \right )^{3} b^{2}}-\frac {3 A \,a^{2} e^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {3 A a d \,e^{2}}{\left (b x +a \right )^{2} b^{3}}-\frac {3 A \,d^{2} e}{2 \left (b x +a \right )^{2} b^{2}}+\frac {2 B \,a^{3} e^{3}}{\left (b x +a \right )^{2} b^{5}}-\frac {9 B \,a^{2} d \,e^{2}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {3 B a \,d^{2} e}{\left (b x +a \right )^{2} b^{3}}-\frac {B \,d^{3}}{2 \left (b x +a \right )^{2} b^{2}}+\frac {3 A a \,e^{3}}{\left (b x +a \right ) b^{4}}-\frac {3 A d \,e^{2}}{\left (b x +a \right ) b^{3}}+\frac {A \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {6 B \,a^{2} e^{3}}{\left (b x +a \right ) b^{5}}+\frac {9 B a d \,e^{2}}{\left (b x +a \right ) b^{4}}-\frac {4 B a \,e^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {3 B \,d^{2} e}{\left (b x +a \right ) b^{3}}+\frac {3 B d \,e^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {B \,e^{3} x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 292, normalized size = 2.03 \begin {gather*} \frac {B e^{3} x}{b^{4}} - \frac {{\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} + 3 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} d^{2} e - 3 \, {\left (11 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{2} + {\left (26 \, B a^{4} - 11 \, A a^{3} b\right )} e^{3} + 18 \, {\left (B b^{4} d^{2} e - {\left (3 \, B a b^{3} - A b^{4}\right )} d e^{2} + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (B b^{4} d^{3} + 3 \, {\left (2 \, B a b^{3} + A b^{4}\right )} d^{2} e - 3 \, {\left (9 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} + {\left (20 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{6 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {{\left (3 \, B b d e^{2} - {\left (4 \, B a - A b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 301, normalized size = 2.09 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (A\,b\,e^3-4\,B\,a\,e^3+3\,B\,b\,d\,e^2\right )}{b^5}-\frac {\frac {26\,B\,a^4\,e^3-33\,B\,a^3\,b\,d\,e^2-11\,A\,a^3\,b\,e^3+6\,B\,a^2\,b^2\,d^2\,e+6\,A\,a^2\,b^2\,d\,e^2+B\,a\,b^3\,d^3+3\,A\,a\,b^3\,d^2\,e+2\,A\,b^4\,d^3}{6\,b}+x\,\left (10\,B\,a^3\,e^3-\frac {27\,B\,a^2\,b\,d\,e^2}{2}-\frac {9\,A\,a^2\,b\,e^3}{2}+3\,B\,a\,b^2\,d^2\,e+3\,A\,a\,b^2\,d\,e^2+\frac {B\,b^3\,d^3}{2}+\frac {3\,A\,b^3\,d^2\,e}{2}\right )+x^2\,\left (6\,B\,a^2\,b\,e^3-9\,B\,a\,b^2\,d\,e^2-3\,A\,a\,b^2\,e^3+3\,B\,b^3\,d^2\,e+3\,A\,b^3\,d\,e^2\right )}{a^3\,b^4+3\,a^2\,b^5\,x+3\,a\,b^6\,x^2+b^7\,x^3}+\frac {B\,e^3\,x}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.68, size = 337, normalized size = 2.34 \begin {gather*} \frac {B e^{3} x}{b^{4}} + \frac {11 A a^{3} b e^{3} - 6 A a^{2} b^{2} d e^{2} - 3 A a b^{3} d^{2} e - 2 A b^{4} d^{3} - 26 B a^{4} e^{3} + 33 B a^{3} b d e^{2} - 6 B a^{2} b^{2} d^{2} e - B a b^{3} d^{3} + x^{2} \left (18 A a b^{3} e^{3} - 18 A b^{4} d e^{2} - 36 B a^{2} b^{2} e^{3} + 54 B a b^{3} d e^{2} - 18 B b^{4} d^{2} e\right ) + x \left (27 A a^{2} b^{2} e^{3} - 18 A a b^{3} d e^{2} - 9 A b^{4} d^{2} e - 60 B a^{3} b e^{3} + 81 B a^{2} b^{2} d e^{2} - 18 B a b^{3} d^{2} e - 3 B b^{4} d^{3}\right )}{6 a^{3} b^{5} + 18 a^{2} b^{6} x + 18 a b^{7} x^{2} + 6 b^{8} x^{3}} - \frac {e^{2} \left (- A b e + 4 B a e - 3 B b d\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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