Optimal. Leaf size=187 \[ \frac {(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}+\frac {e x (A b-a B) (b d-a e)^4}{b^6}+\frac {(d+e x)^2 (A b-a B) (b d-a e)^3}{2 b^5}+\frac {(d+e x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {(d+e x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac {(d+e x)^5 (A b-a B)}{5 b^2}+\frac {B (d+e x)^6}{6 b e} \]
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Rubi [A] time = 0.13, 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} \[ \frac {(d+e x)^5 (A b-a B)}{5 b^2}+\frac {(d+e x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac {(d+e x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {(d+e x)^2 (A b-a B) (b d-a e)^3}{2 b^5}+\frac {e x (A b-a B) (b d-a e)^4}{b^6}+\frac {(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}+\frac {B (d+e x)^6}{6 b e} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^5}{a+b x} \, dx &=\int \left (\frac {(A b-a B) e (b d-a e)^4}{b^6}+\frac {(A b-a B) (b d-a e)^5}{b^6 (a+b x)}+\frac {(A b-a B) e (b d-a e)^3 (d+e x)}{b^5}+\frac {(A b-a B) e (b d-a e)^2 (d+e x)^2}{b^4}+\frac {(A b-a B) e (b d-a e) (d+e x)^3}{b^3}+\frac {(A b-a B) e (d+e x)^4}{b^2}+\frac {B (d+e x)^5}{b}\right ) \, dx\\ &=\frac {(A b-a B) e (b d-a e)^4 x}{b^6}+\frac {(A b-a B) (b d-a e)^3 (d+e x)^2}{2 b^5}+\frac {(A b-a B) (b d-a e)^2 (d+e x)^3}{3 b^4}+\frac {(A b-a B) (b d-a e) (d+e x)^4}{4 b^3}+\frac {(A b-a B) (d+e x)^5}{5 b^2}+\frac {B (d+e x)^6}{6 b e}+\frac {(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 368, normalized size = 1.97 \[ \frac {b x \left (-60 a^5 B e^5+30 a^4 b e^4 (2 A e+10 B d+B e x)-10 a^3 b^2 e^3 \left (3 A e (10 d+e x)+B \left (60 d^2+15 d e x+2 e^2 x^2\right )\right )+5 a^2 b^3 e^2 \left (2 A e \left (60 d^2+15 d e x+2 e^2 x^2\right )+B \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )\right )-a b^4 e \left (5 A e \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )+B \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )+10 B \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )\right )\right )+60 (A b-a B) (b d-a e)^5 \log (a+b x)}{60 b^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 566, normalized size = 3.03 \[ \frac {10 \, B b^{6} e^{5} x^{6} + 12 \, {\left (5 \, B b^{6} d e^{4} - {\left (B a b^{5} - A b^{6}\right )} e^{5}\right )} x^{5} + 15 \, {\left (10 \, B b^{6} d^{2} e^{3} - 5 \, {\left (B a b^{5} - A b^{6}\right )} d e^{4} + {\left (B a^{2} b^{4} - A a b^{5}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B b^{6} d^{3} e^{2} - 10 \, {\left (B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d e^{4} - {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B b^{6} d^{4} e - 10 \, {\left (B a b^{5} - A b^{6}\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} - 5 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d e^{4} + {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 60 \, {\left (B b^{6} d^{5} - 5 \, {\left (B a b^{5} - A b^{6}\right )} d^{4} e + 10 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d e^{4} - {\left (B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x - 60 \, {\left ({\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} - {\left (B a^{6} - A a^{5} b\right )} e^{5}\right )} \log \left (b x + a\right )}{60 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 635, normalized size = 3.40 \[ \frac {10 \, B b^{5} x^{6} e^{5} + 60 \, B b^{5} d x^{5} e^{4} + 150 \, B b^{5} d^{2} x^{4} e^{3} + 200 \, B b^{5} d^{3} x^{3} e^{2} + 150 \, B b^{5} d^{4} x^{2} e + 60 \, B b^{5} d^{5} x - 12 \, B a b^{4} x^{5} e^{5} + 12 \, A b^{5} x^{5} e^{5} - 75 \, B a b^{4} d x^{4} e^{4} + 75 \, A b^{5} d x^{4} e^{4} - 200 \, B a b^{4} d^{2} x^{3} e^{3} + 200 \, A b^{5} d^{2} x^{3} e^{3} - 300 \, B a b^{4} d^{3} x^{2} e^{2} + 300 \, A b^{5} d^{3} x^{2} e^{2} - 300 \, B a b^{4} d^{4} x e + 300 \, A b^{5} d^{4} x e + 15 \, B a^{2} b^{3} x^{4} e^{5} - 15 \, A a b^{4} x^{4} e^{5} + 100 \, B a^{2} b^{3} d x^{3} e^{4} - 100 \, A a b^{4} d x^{3} e^{4} + 300 \, B a^{2} b^{3} d^{2} x^{2} e^{3} - 300 \, A a b^{4} d^{2} x^{2} e^{3} + 600 \, B a^{2} b^{3} d^{3} x e^{2} - 600 \, A a b^{4} d^{3} x e^{2} - 20 \, B a^{3} b^{2} x^{3} e^{5} + 20 \, A a^{2} b^{3} x^{3} e^{5} - 150 \, B a^{3} b^{2} d x^{2} e^{4} + 150 \, A a^{2} b^{3} d x^{2} e^{4} - 600 \, B a^{3} b^{2} d^{2} x e^{3} + 600 \, A a^{2} b^{3} d^{2} x e^{3} + 30 \, B a^{4} b x^{2} e^{5} - 30 \, A a^{3} b^{2} x^{2} e^{5} + 300 \, B a^{4} b d x e^{4} - 300 \, A a^{3} b^{2} d x e^{4} - 60 \, B a^{5} x e^{5} + 60 \, A a^{4} b x e^{5}}{60 \, b^{6}} - \frac {{\left (B a b^{5} d^{5} - A b^{6} d^{5} - 5 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 10 \, B a^{3} b^{3} d^{3} e^{2} - 10 \, A a^{2} b^{4} d^{3} e^{2} - 10 \, B a^{4} b^{2} d^{2} e^{3} + 10 \, A a^{3} b^{3} d^{2} e^{3} + 5 \, B a^{5} b d e^{4} - 5 \, A a^{4} b^{2} d e^{4} - B a^{6} e^{5} + A a^{5} b e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 737, normalized size = 3.94 \[ \frac {B \,e^{5} x^{6}}{6 b}+\frac {A \,e^{5} x^{5}}{5 b}-\frac {B a \,e^{5} x^{5}}{5 b^{2}}+\frac {B d \,e^{4} x^{5}}{b}-\frac {A a \,e^{5} x^{4}}{4 b^{2}}+\frac {5 A d \,e^{4} x^{4}}{4 b}+\frac {B \,a^{2} e^{5} x^{4}}{4 b^{3}}-\frac {5 B a d \,e^{4} x^{4}}{4 b^{2}}+\frac {5 B \,d^{2} e^{3} x^{4}}{2 b}+\frac {A \,a^{2} e^{5} x^{3}}{3 b^{3}}-\frac {5 A a d \,e^{4} x^{3}}{3 b^{2}}+\frac {10 A \,d^{2} e^{3} x^{3}}{3 b}-\frac {B \,a^{3} e^{5} x^{3}}{3 b^{4}}+\frac {5 B \,a^{2} d \,e^{4} x^{3}}{3 b^{3}}-\frac {10 B a \,d^{2} e^{3} x^{3}}{3 b^{2}}+\frac {10 B \,d^{3} e^{2} x^{3}}{3 b}-\frac {A \,a^{3} e^{5} x^{2}}{2 b^{4}}+\frac {5 A \,a^{2} d \,e^{4} x^{2}}{2 b^{3}}-\frac {5 A a \,d^{2} e^{3} x^{2}}{b^{2}}+\frac {5 A \,d^{3} e^{2} x^{2}}{b}+\frac {B \,a^{4} e^{5} x^{2}}{2 b^{5}}-\frac {5 B \,a^{3} d \,e^{4} x^{2}}{2 b^{4}}+\frac {5 B \,a^{2} d^{2} e^{3} x^{2}}{b^{3}}-\frac {5 B a \,d^{3} e^{2} x^{2}}{b^{2}}+\frac {5 B \,d^{4} e \,x^{2}}{2 b}-\frac {A \,a^{5} e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {5 A \,a^{4} d \,e^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {A \,a^{4} e^{5} x}{b^{5}}-\frac {10 A \,a^{3} d^{2} e^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {5 A \,a^{3} d \,e^{4} x}{b^{4}}+\frac {10 A \,a^{2} d^{3} e^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {10 A \,a^{2} d^{2} e^{3} x}{b^{3}}-\frac {5 A a \,d^{4} e \ln \left (b x +a \right )}{b^{2}}-\frac {10 A a \,d^{3} e^{2} x}{b^{2}}+\frac {A \,d^{5} \ln \left (b x +a \right )}{b}+\frac {5 A \,d^{4} e x}{b}+\frac {B \,a^{6} e^{5} \ln \left (b x +a \right )}{b^{7}}-\frac {5 B \,a^{5} d \,e^{4} \ln \left (b x +a \right )}{b^{6}}-\frac {B \,a^{5} e^{5} x}{b^{6}}+\frac {10 B \,a^{4} d^{2} e^{3} \ln \left (b x +a \right )}{b^{5}}+\frac {5 B \,a^{4} d \,e^{4} x}{b^{5}}-\frac {10 B \,a^{3} d^{3} e^{2} \ln \left (b x +a \right )}{b^{4}}-\frac {10 B \,a^{3} d^{2} e^{3} x}{b^{4}}+\frac {5 B \,a^{2} d^{4} e \ln \left (b x +a \right )}{b^{3}}+\frac {10 B \,a^{2} d^{3} e^{2} x}{b^{3}}-\frac {B a \,d^{5} \ln \left (b x +a \right )}{b^{2}}-\frac {5 B a \,d^{4} e x}{b^{2}}+\frac {B \,d^{5} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 563, normalized size = 3.01 \[ \frac {10 \, B b^{5} e^{5} x^{6} + 12 \, {\left (5 \, B b^{5} d e^{4} - {\left (B a b^{4} - A b^{5}\right )} e^{5}\right )} x^{5} + 15 \, {\left (10 \, B b^{5} d^{2} e^{3} - 5 \, {\left (B a b^{4} - A b^{5}\right )} d e^{4} + {\left (B a^{2} b^{3} - A a b^{4}\right )} e^{5}\right )} x^{4} + 20 \, {\left (10 \, B b^{5} d^{3} e^{2} - 10 \, {\left (B a b^{4} - A b^{5}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{4} - {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{5}\right )} x^{3} + 30 \, {\left (5 \, B b^{5} d^{4} e - 10 \, {\left (B a b^{4} - A b^{5}\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{3} - 5 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{4} + {\left (B a^{4} b - A a^{3} b^{2}\right )} e^{5}\right )} x^{2} + 60 \, {\left (B b^{5} d^{5} - 5 \, {\left (B a b^{4} - A b^{5}\right )} d^{4} e + 10 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{4} - {\left (B a^{5} - A a^{4} b\right )} e^{5}\right )} x}{60 \, b^{6}} - \frac {{\left ({\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \, {\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \, {\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} - {\left (B a^{6} - A a^{5} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 565, normalized size = 3.02 \[ x\,\left (\frac {B\,d^5+5\,A\,e\,d^4}{b}+\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b}\right )}{b}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b}\right )}{b}-\frac {5\,d^3\,e\,\left (2\,A\,e+B\,d\right )}{b}\right )}{b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b}\right )}{3\,b}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{3\,b}\right )-x^4\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{4\,b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{4\,b}\right )+x^5\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{5\,b}-\frac {B\,a\,e^5}{5\,b^2}\right )-x^2\,\left (\frac {a\,\left (\frac {a\,\left (\frac {a\,\left (\frac {A\,e^5+5\,B\,d\,e^4}{b}-\frac {B\,a\,e^5}{b^2}\right )}{b}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{b}\right )}{b}+\frac {10\,d^2\,e^2\,\left (A\,e+B\,d\right )}{b}\right )}{2\,b}-\frac {5\,d^3\,e\,\left (2\,A\,e+B\,d\right )}{2\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^6\,e^5-5\,B\,a^5\,b\,d\,e^4-A\,a^5\,b\,e^5+10\,B\,a^4\,b^2\,d^2\,e^3+5\,A\,a^4\,b^2\,d\,e^4-10\,B\,a^3\,b^3\,d^3\,e^2-10\,A\,a^3\,b^3\,d^2\,e^3+5\,B\,a^2\,b^4\,d^4\,e+10\,A\,a^2\,b^4\,d^3\,e^2-B\,a\,b^5\,d^5-5\,A\,a\,b^5\,d^4\,e+A\,b^6\,d^5\right )}{b^7}+\frac {B\,e^5\,x^6}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.37, size = 534, normalized size = 2.86 \[ \frac {B e^{5} x^{6}}{6 b} + x^{5} \left (\frac {A e^{5}}{5 b} - \frac {B a e^{5}}{5 b^{2}} + \frac {B d e^{4}}{b}\right ) + x^{4} \left (- \frac {A a e^{5}}{4 b^{2}} + \frac {5 A d e^{4}}{4 b} + \frac {B a^{2} e^{5}}{4 b^{3}} - \frac {5 B a d e^{4}}{4 b^{2}} + \frac {5 B d^{2} e^{3}}{2 b}\right ) + x^{3} \left (\frac {A a^{2} e^{5}}{3 b^{3}} - \frac {5 A a d e^{4}}{3 b^{2}} + \frac {10 A d^{2} e^{3}}{3 b} - \frac {B a^{3} e^{5}}{3 b^{4}} + \frac {5 B a^{2} d e^{4}}{3 b^{3}} - \frac {10 B a d^{2} e^{3}}{3 b^{2}} + \frac {10 B d^{3} e^{2}}{3 b}\right ) + x^{2} \left (- \frac {A a^{3} e^{5}}{2 b^{4}} + \frac {5 A a^{2} d e^{4}}{2 b^{3}} - \frac {5 A a d^{2} e^{3}}{b^{2}} + \frac {5 A d^{3} e^{2}}{b} + \frac {B a^{4} e^{5}}{2 b^{5}} - \frac {5 B a^{3} d e^{4}}{2 b^{4}} + \frac {5 B a^{2} d^{2} e^{3}}{b^{3}} - \frac {5 B a d^{3} e^{2}}{b^{2}} + \frac {5 B d^{4} e}{2 b}\right ) + x \left (\frac {A a^{4} e^{5}}{b^{5}} - \frac {5 A a^{3} d e^{4}}{b^{4}} + \frac {10 A a^{2} d^{2} e^{3}}{b^{3}} - \frac {10 A a d^{3} e^{2}}{b^{2}} + \frac {5 A d^{4} e}{b} - \frac {B a^{5} e^{5}}{b^{6}} + \frac {5 B a^{4} d e^{4}}{b^{5}} - \frac {10 B a^{3} d^{2} e^{3}}{b^{4}} + \frac {10 B a^{2} d^{3} e^{2}}{b^{3}} - \frac {5 B a d^{4} e}{b^{2}} + \frac {B d^{5}}{b}\right ) + \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{5} \log {\left (a + b x \right )}}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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