Optimal. Leaf size=187 \[ \frac {e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}+\frac {e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac {(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac {(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac {2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac {B e^4 (a+b x)^4}{4 b^6} \]
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Rubi [A] time = 0.26, antiderivative size = 187, 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} \[ \frac {e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}+\frac {e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}-\frac {(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac {2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac {(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac {B e^4 (a+b x)^4}{4 b^6} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^4}{(a+b x)^2} \, dx &=\int \left (\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^5}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)^2}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^5 (a+b x)}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)}{b^5}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^2}{b^5}+\frac {B e^4 (a+b x)^3}{b^5}\right ) \, dx\\ &=\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) x}{b^5}-\frac {(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac {e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^2}{b^6}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^3}{3 b^6}+\frac {B e^4 (a+b x)^4}{4 b^6}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e) \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 365, normalized size = 1.95 \[ \frac {-4 A b \left (3 a^4 e^4-3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a b^3 e \left (-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+b^4 \left (3 d^4-18 d^2 e^2 x^2-6 d e^3 x^3-e^4 x^4\right )\right )+B \left (12 a^5 e^4-48 a^4 b e^3 (d+e x)+6 a^3 b^2 e^2 \left (12 d^2+24 d e x-5 e^2 x^2\right )+2 a^2 b^3 e \left (-24 d^3-72 d^2 e x+48 d e^2 x^2+5 e^3 x^3\right )+a b^4 \left (12 d^4+48 d^3 e x-108 d^2 e^2 x^2-32 d e^3 x^3-5 e^4 x^4\right )+b^5 e x^2 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (a+b x) (b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{12 b^6 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 610, normalized size = 3.26 \[ \frac {3 \, B b^{5} e^{4} x^{5} + 12 \, {\left (B a b^{4} - A b^{5}\right )} d^{4} - 48 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 72 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 48 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + 12 \, {\left (B a^{5} - A a^{4} b\right )} e^{4} + {\left (16 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} e^{4}\right )} x^{4} + 2 \, {\left (18 \, B b^{5} d^{2} e^{2} - 4 \, {\left (4 \, B a b^{4} - 3 \, A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \, {\left (8 \, B b^{5} d^{3} e - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} d^{2} e^{2} + 4 \, {\left (4 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 12 \, {\left (4 \, B a b^{4} d^{3} e - 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} - {\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (B a b^{4} d^{4} - 4 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} + {\left (5 \, B a^{5} - 4 \, A a^{4} b\right )} e^{4} + {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.23, size = 522, normalized size = 2.79 \[ \frac {{\left (b x + a\right )}^{4} {\left (3 \, B e^{4} + \frac {4 \, {\left (4 \, B b^{2} d e^{3} - 5 \, B a b e^{4} + A b^{2} e^{4}\right )}}{{\left (b x + a\right )} b} + \frac {12 \, {\left (3 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} + 2 \, A b^{4} d e^{3} + 5 \, B a^{2} b^{2} e^{4} - 2 \, A a b^{3} e^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {24 \, {\left (2 \, B b^{6} d^{3} e - 9 \, B a b^{5} d^{2} e^{2} + 3 \, A b^{6} d^{2} e^{2} + 12 \, B a^{2} b^{4} d e^{3} - 6 \, A a b^{5} d e^{3} - 5 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}}{12 \, b^{6}} - \frac {{\left (B b^{4} d^{4} - 8 \, B a b^{3} d^{3} e + 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 16 \, B a^{3} b d e^{3} + 12 \, A a^{2} b^{2} d e^{3} + 5 \, B a^{4} e^{4} - 4 \, A a^{3} b e^{4}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} + \frac {\frac {B a b^{8} d^{4}}{b x + a} - \frac {A b^{9} d^{4}}{b x + a} - \frac {4 \, B a^{2} b^{7} d^{3} e}{b x + a} + \frac {4 \, A a b^{8} d^{3} e}{b x + a} + \frac {6 \, B a^{3} b^{6} d^{2} e^{2}}{b x + a} - \frac {6 \, A a^{2} b^{7} d^{2} e^{2}}{b x + a} - \frac {4 \, B a^{4} b^{5} d e^{3}}{b x + a} + \frac {4 \, A a^{3} b^{6} d e^{3}}{b x + a} + \frac {B a^{5} b^{4} e^{4}}{b x + a} - \frac {A a^{4} b^{5} e^{4}}{b x + a}}{b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 564, normalized size = 3.02 \[ \frac {B \,e^{4} x^{4}}{4 b^{2}}+\frac {A \,e^{4} x^{3}}{3 b^{2}}-\frac {2 B a \,e^{4} x^{3}}{3 b^{3}}+\frac {4 B d \,e^{3} x^{3}}{3 b^{2}}-\frac {A a \,e^{4} x^{2}}{b^{3}}+\frac {2 A d \,e^{3} x^{2}}{b^{2}}+\frac {3 B \,a^{2} e^{4} x^{2}}{2 b^{4}}-\frac {4 B a d \,e^{3} x^{2}}{b^{3}}+\frac {3 B \,d^{2} e^{2} x^{2}}{b^{2}}-\frac {A \,a^{4} e^{4}}{\left (b x +a \right ) b^{5}}+\frac {4 A \,a^{3} d \,e^{3}}{\left (b x +a \right ) b^{4}}-\frac {4 A \,a^{3} e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {6 A \,a^{2} d^{2} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {12 A \,a^{2} d \,e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 A \,a^{2} e^{4} x}{b^{4}}+\frac {4 A a \,d^{3} e}{\left (b x +a \right ) b^{2}}-\frac {12 A a \,d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {8 A a d \,e^{3} x}{b^{3}}-\frac {A \,d^{4}}{\left (b x +a \right ) b}+\frac {4 A \,d^{3} e \ln \left (b x +a \right )}{b^{2}}+\frac {6 A \,d^{2} e^{2} x}{b^{2}}+\frac {B \,a^{5} e^{4}}{\left (b x +a \right ) b^{6}}-\frac {4 B \,a^{4} d \,e^{3}}{\left (b x +a \right ) b^{5}}+\frac {5 B \,a^{4} e^{4} \ln \left (b x +a \right )}{b^{6}}+\frac {6 B \,a^{3} d^{2} e^{2}}{\left (b x +a \right ) b^{4}}-\frac {16 B \,a^{3} d \,e^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {4 B \,a^{3} e^{4} x}{b^{5}}-\frac {4 B \,a^{2} d^{3} e}{\left (b x +a \right ) b^{3}}+\frac {18 B \,a^{2} d^{2} e^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {12 B \,a^{2} d \,e^{3} x}{b^{4}}+\frac {B a \,d^{4}}{\left (b x +a \right ) b^{2}}-\frac {8 B a \,d^{3} e \ln \left (b x +a \right )}{b^{3}}-\frac {12 B a \,d^{2} e^{2} x}{b^{3}}+\frac {B \,d^{4} \ln \left (b x +a \right )}{b^{2}}+\frac {4 B \,d^{3} e x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 411, normalized size = 2.20 \[ \frac {{\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (B a^{5} - A a^{4} b\right )} e^{4}}{b^{7} x + a b^{6}} + \frac {3 \, B b^{3} e^{4} x^{4} + 4 \, {\left (4 \, B b^{3} d e^{3} - {\left (2 \, B a b^{2} - A b^{3}\right )} e^{4}\right )} x^{3} + 6 \, {\left (6 \, B b^{3} d^{2} e^{2} - 4 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d e^{3} + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{4}\right )} x^{2} + 12 \, {\left (4 \, B b^{3} d^{3} e - 6 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 4 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{3} - {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} x}{12 \, b^{5}} + \frac {{\left (B b^{4} d^{4} - 4 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 486, normalized size = 2.60 \[ x^3\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{3\,b^2}-\frac {2\,B\,a\,e^4}{3\,b^3}\right )-x^2\,\left (\frac {a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^2}-\frac {2\,B\,a\,e^4}{b^3}\right )}{b}-\frac {d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^4}{2\,b^4}\right )+x\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^2}-\frac {2\,B\,a\,e^4}{b^3}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^2}+\frac {B\,a^2\,e^4}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^2}-\frac {2\,B\,a\,e^4}{b^3}\right )}{b^2}+\frac {2\,d^2\,e\,\left (3\,A\,e+2\,B\,d\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (5\,B\,a^4\,e^4-16\,B\,a^3\,b\,d\,e^3-4\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3-8\,B\,a\,b^3\,d^3\,e-12\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{b^6}-\frac {-B\,a^5\,e^4+4\,B\,a^4\,b\,d\,e^3+A\,a^4\,b\,e^4-6\,B\,a^3\,b^2\,d^2\,e^2-4\,A\,a^3\,b^2\,d\,e^3+4\,B\,a^2\,b^3\,d^3\,e+6\,A\,a^2\,b^3\,d^2\,e^2-B\,a\,b^4\,d^4-4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {B\,e^4\,x^4}{4\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.26, size = 396, normalized size = 2.12 \[ \frac {B e^{4} x^{4}}{4 b^{2}} + x^{3} \left (\frac {A e^{4}}{3 b^{2}} - \frac {2 B a e^{4}}{3 b^{3}} + \frac {4 B d e^{3}}{3 b^{2}}\right ) + x^{2} \left (- \frac {A a e^{4}}{b^{3}} + \frac {2 A d e^{3}}{b^{2}} + \frac {3 B a^{2} e^{4}}{2 b^{4}} - \frac {4 B a d e^{3}}{b^{3}} + \frac {3 B d^{2} e^{2}}{b^{2}}\right ) + x \left (\frac {3 A a^{2} e^{4}}{b^{4}} - \frac {8 A a d e^{3}}{b^{3}} + \frac {6 A d^{2} e^{2}}{b^{2}} - \frac {4 B a^{3} e^{4}}{b^{5}} + \frac {12 B a^{2} d e^{3}}{b^{4}} - \frac {12 B a d^{2} e^{2}}{b^{3}} + \frac {4 B d^{3} e}{b^{2}}\right ) + \frac {- A a^{4} b e^{4} + 4 A a^{3} b^{2} d e^{3} - 6 A a^{2} b^{3} d^{2} e^{2} + 4 A a b^{4} d^{3} e - A b^{5} d^{4} + B a^{5} e^{4} - 4 B a^{4} b d e^{3} + 6 B a^{3} b^{2} d^{2} e^{2} - 4 B a^{2} b^{3} d^{3} e + B a b^{4} d^{4}}{a b^{6} + b^{7} x} + \frac {\left (a e - b d\right )^{3} \left (- 4 A b e + 5 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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