Optimal. Leaf size=187 \[ \frac{(d+e x)^5 (A b-a B)}{5 b^2}+\frac{(d+e x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{(d+e x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac{(d+e x)^2 (A b-a B) (b d-a e)^3}{2 b^5}+\frac{e x (A b-a B) (b d-a e)^4}{b^6}+\frac{(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}+\frac{B (d+e x)^6}{6 b e} \]
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Rubi [A] time = 0.132561, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}
Mathematica [A] time = 0.187491, size = 368, normalized size = 1.97 \[ \frac{b x \left (5 a^2 b^3 e^2 \left (2 A e \left (60 d^2+15 d e x+2 e^2 x^2\right )+B \left (60 d^2 e x+120 d^3+20 d e^2 x^2+3 e^3 x^3\right )\right )-10 a^3 b^2 e^3 \left (3 A e (10 d+e x)+B \left (60 d^2+15 d e x+2 e^2 x^2\right )\right )+30 a^4 b e^4 (2 A e+10 B d+B e x)-60 a^5 B e^5-a b^4 e \left (5 A e \left (60 d^2 e x+120 d^3+20 d e^2 x^2+3 e^3 x^3\right )+B \left (200 d^2 e^2 x^2+300 d^3 e x+300 d^4+75 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (200 d^2 e^2 x^2+300 d^3 e x+300 d^4+75 d e^3 x^3+12 e^4 x^4\right )+10 B \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )\right )\right )+60 (A b-a B) (b d-a e)^5 \log (a+b x)}{60 b^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 737, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.26704, size = 760, normalized size = 4.06 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.56136, size = 1111, normalized size = 5.94 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.59197, size = 510, normalized size = 2.73 \begin{align*} \frac{B e^{5} x^{6}}{6 b} - \frac{x^{5} \left (- A b e^{5} + B a e^{5} - 5 B b d e^{4}\right )}{5 b^{2}} + \frac{x^{4} \left (- A a b e^{5} + 5 A b^{2} d e^{4} + B a^{2} e^{5} - 5 B a b d e^{4} + 10 B b^{2} d^{2} e^{3}\right )}{4 b^{3}} - \frac{x^{3} \left (- A a^{2} b e^{5} + 5 A a b^{2} d e^{4} - 10 A b^{3} d^{2} e^{3} + B a^{3} e^{5} - 5 B a^{2} b d e^{4} + 10 B a b^{2} d^{2} e^{3} - 10 B b^{3} d^{3} e^{2}\right )}{3 b^{4}} + \frac{x^{2} \left (- A a^{3} b e^{5} + 5 A a^{2} b^{2} d e^{4} - 10 A a b^{3} d^{2} e^{3} + 10 A b^{4} d^{3} e^{2} + B a^{4} e^{5} - 5 B a^{3} b d e^{4} + 10 B a^{2} b^{2} d^{2} e^{3} - 10 B a b^{3} d^{3} e^{2} + 5 B b^{4} d^{4} e\right )}{2 b^{5}} - \frac{x \left (- A a^{4} b e^{5} + 5 A a^{3} b^{2} d e^{4} - 10 A a^{2} b^{3} d^{2} e^{3} + 10 A a b^{4} d^{3} e^{2} - 5 A b^{5} d^{4} e + B a^{5} e^{5} - 5 B a^{4} b d e^{4} + 10 B a^{3} b^{2} d^{2} e^{3} - 10 B a^{2} b^{3} d^{3} e^{2} + 5 B a b^{4} d^{4} e - B b^{5} d^{5}\right )}{b^{6}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right )^{5} \log{\left (a + b x \right )}}{b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.57823, size = 857, normalized size = 4.58 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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