Optimal. Leaf size=191 \[ \frac {e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac {2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac {2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac {B e^4 (a+b x)^3}{3 b^6} \]
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Rubi [A] time = 0.25, antiderivative size = 191, 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} \[ \frac {e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}+\frac {2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac {2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac {B e^4 (a+b x)^3}{3 b^6} \]
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)^3} \, dx &=\int \left (\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e)}{b^5}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)^3}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^5 (a+b x)^2}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^5 (a+b x)}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)}{b^5}+\frac {B e^4 (a+b x)^2}{b^5}\right ) \, dx\\ &=\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) x}{b^5}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^6 (a+b x)}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^2}{2 b^6}+\frac {B e^4 (a+b x)^3}{3 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 187, normalized size = 0.98 \[ \frac {6 b e^2 x \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+3 b^2 e^3 x^2 (-3 a B e+A b e+4 b B d)-\frac {6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{a+b x}-\frac {3 (A b-a B) (b d-a e)^4}{(a+b x)^2}+12 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)+2 b^3 B e^4 x^3}{6 b^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 668, normalized size = 3.50 \[ \frac {2 \, B b^{5} e^{4} x^{5} - 3 \, {\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \, {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \, {\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + {\left (12 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \, {\left (9 \, B b^{5} d^{2} e^{2} - 6 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (24 \, B a b^{4} d^{2} e^{2} - 4 \, {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} + {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \, {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} - {\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (2 \, B a^{2} b^{3} d^{3} e - 3 \, {\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} - {\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} + {\left (2 \, B b^{5} d^{3} e - 3 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \, {\left (2 \, B a b^{4} d^{3} e - 3 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 420, normalized size = 2.20 \[ \frac {2 \, {\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \, {\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, B b^{6} x^{3} e^{4} + 12 \, B b^{6} d x^{2} e^{3} + 36 \, B b^{6} d^{2} x e^{2} - 9 \, B a b^{5} x^{2} e^{4} + 3 \, A b^{6} x^{2} e^{4} - 72 \, B a b^{5} d x e^{3} + 24 \, A b^{6} d x e^{3} + 36 \, B a^{2} b^{4} x e^{4} - 18 \, A a b^{5} x e^{4}}{6 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 601, normalized size = 3.15 \[ \frac {B \,e^{4} x^{3}}{3 b^{3}}-\frac {A \,a^{4} e^{4}}{2 \left (b x +a \right )^{2} b^{5}}+\frac {2 A \,a^{3} d \,e^{3}}{\left (b x +a \right )^{2} b^{4}}-\frac {3 A \,a^{2} d^{2} e^{2}}{\left (b x +a \right )^{2} b^{3}}+\frac {2 A a \,d^{3} e}{\left (b x +a \right )^{2} b^{2}}-\frac {A \,d^{4}}{2 \left (b x +a \right )^{2} b}+\frac {A \,e^{4} x^{2}}{2 b^{3}}+\frac {B \,a^{5} e^{4}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {2 B \,a^{4} d \,e^{3}}{\left (b x +a \right )^{2} b^{5}}+\frac {3 B \,a^{3} d^{2} e^{2}}{\left (b x +a \right )^{2} b^{4}}-\frac {2 B \,a^{2} d^{3} e}{\left (b x +a \right )^{2} b^{3}}+\frac {B a \,d^{4}}{2 \left (b x +a \right )^{2} b^{2}}-\frac {3 B a \,e^{4} x^{2}}{2 b^{4}}+\frac {2 B d \,e^{3} x^{2}}{b^{3}}+\frac {4 A \,a^{3} e^{4}}{\left (b x +a \right ) b^{5}}-\frac {12 A \,a^{2} d \,e^{3}}{\left (b x +a \right ) b^{4}}+\frac {6 A \,a^{2} e^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {12 A a \,d^{2} e^{2}}{\left (b x +a \right ) b^{3}}-\frac {12 A a d \,e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {3 A a \,e^{4} x}{b^{4}}-\frac {4 A \,d^{3} e}{\left (b x +a \right ) b^{2}}+\frac {6 A \,d^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {4 A d \,e^{3} x}{b^{3}}-\frac {5 B \,a^{4} e^{4}}{\left (b x +a \right ) b^{6}}+\frac {16 B \,a^{3} d \,e^{3}}{\left (b x +a \right ) b^{5}}-\frac {10 B \,a^{3} e^{4} \ln \left (b x +a \right )}{b^{6}}-\frac {18 B \,a^{2} d^{2} e^{2}}{\left (b x +a \right ) b^{4}}+\frac {24 B \,a^{2} d \,e^{3} \ln \left (b x +a \right )}{b^{5}}+\frac {6 B \,a^{2} e^{4} x}{b^{5}}+\frac {8 B a \,d^{3} e}{\left (b x +a \right ) b^{3}}-\frac {18 B a \,d^{2} e^{2} \ln \left (b x +a \right )}{b^{4}}-\frac {12 B a d \,e^{3} x}{b^{4}}-\frac {B \,d^{4}}{\left (b x +a \right ) b^{2}}+\frac {4 B \,d^{3} e \ln \left (b x +a \right )}{b^{3}}+\frac {6 B \,d^{2} e^{2} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 424, normalized size = 2.22 \[ -\frac {{\left (B a b^{4} + A b^{5}\right )} d^{4} - 4 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} + {\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + 2 \, {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, B b^{2} e^{4} x^{3} + 3 \, {\left (4 \, B b^{2} d e^{3} - {\left (3 \, B a b - A b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (6 \, B b^{2} d^{2} e^{2} - 4 \, {\left (3 \, B a b - A b^{2}\right )} d e^{3} + 3 \, {\left (2 \, B a^{2} - A a b\right )} e^{4}\right )} x}{6 \, b^{5}} + \frac {2 \, {\left (2 \, B b^{3} d^{3} e - 3 \, {\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{3} - {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 451, normalized size = 2.36 \[ x^2\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{2\,b^3}-\frac {3\,B\,a\,e^4}{2\,b^4}\right )-\frac {\frac {9\,B\,a^5\,e^4-28\,B\,a^4\,b\,d\,e^3-7\,A\,a^4\,b\,e^4+30\,B\,a^3\,b^2\,d^2\,e^2+20\,A\,a^3\,b^2\,d\,e^3-12\,B\,a^2\,b^3\,d^3\,e-18\,A\,a^2\,b^3\,d^2\,e^2+B\,a\,b^4\,d^4+4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}{2\,b}+x\,\left (5\,B\,a^4\,e^4-16\,B\,a^3\,b\,d\,e^3-4\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3-8\,B\,a\,b^3\,d^3\,e-12\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}-x\,\left (\frac {3\,a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^3}-\frac {3\,B\,a\,e^4}{b^4}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^3}+\frac {3\,B\,a^2\,e^4}{b^5}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-10\,B\,a^3\,e^4+24\,B\,a^2\,b\,d\,e^3+6\,A\,a^2\,b\,e^4-18\,B\,a\,b^2\,d^2\,e^2-12\,A\,a\,b^2\,d\,e^3+4\,B\,b^3\,d^3\,e+6\,A\,b^3\,d^2\,e^2\right )}{b^6}+\frac {B\,e^4\,x^3}{3\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.10, size = 444, normalized size = 2.32 \[ \frac {B e^{4} x^{3}}{3 b^{3}} + x^{2} \left (\frac {A e^{4}}{2 b^{3}} - \frac {3 B a e^{4}}{2 b^{4}} + \frac {2 B d e^{3}}{b^{3}}\right ) + x \left (- \frac {3 A a e^{4}}{b^{4}} + \frac {4 A d e^{3}}{b^{3}} + \frac {6 B a^{2} e^{4}}{b^{5}} - \frac {12 B a d e^{3}}{b^{4}} + \frac {6 B d^{2} e^{2}}{b^{3}}\right ) + \frac {7 A a^{4} b e^{4} - 20 A a^{3} b^{2} d e^{3} + 18 A a^{2} b^{3} d^{2} e^{2} - 4 A a b^{4} d^{3} e - A b^{5} d^{4} - 9 B a^{5} e^{4} + 28 B a^{4} b d e^{3} - 30 B a^{3} b^{2} d^{2} e^{2} + 12 B a^{2} b^{3} d^{3} e - B a b^{4} d^{4} + x \left (8 A a^{3} b^{2} e^{4} - 24 A a^{2} b^{3} d e^{3} + 24 A a b^{4} d^{2} e^{2} - 8 A b^{5} d^{3} e - 10 B a^{4} b e^{4} + 32 B a^{3} b^{2} d e^{3} - 36 B a^{2} b^{3} d^{2} e^{2} + 16 B a b^{4} d^{3} e - 2 B b^{5} d^{4}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac {2 e \left (a e - b d\right )^{2} \left (- 3 A b e + 5 B a e - 2 B b d\right ) \log {\left (a + b x \right )}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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