Optimal. Leaf size=91 \[ \frac {(A b-a B) (b d-a e)^2 \log (a+b x)}{b^4}+\frac {e x (A b-a B) (b d-a e)}{b^3}+\frac {(d+e x)^2 (A b-a B)}{2 b^2}+\frac {B (d+e x)^3}{3 b e} \]
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Rubi [A] time = 0.06, antiderivative size = 91, 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} \[ \frac {(d+e x)^2 (A b-a B)}{2 b^2}+\frac {e x (A b-a B) (b d-a e)}{b^3}+\frac {(A b-a B) (b d-a e)^2 \log (a+b x)}{b^4}+\frac {B (d+e x)^3}{3 b e} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^2}{a+b x} \, dx &=\int \left (\frac {(A b-a B) e (b d-a e)}{b^3}+\frac {(A b-a B) (b d-a e)^2}{b^3 (a+b x)}+\frac {(A b-a B) e (d+e x)}{b^2}+\frac {B (d+e x)^2}{b}\right ) \, dx\\ &=\frac {(A b-a B) e (b d-a e) x}{b^3}+\frac {(A b-a B) (d+e x)^2}{2 b^2}+\frac {B (d+e x)^3}{3 b e}+\frac {(A b-a B) (b d-a e)^2 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 102, normalized size = 1.12 \[ \frac {b x \left (6 a^2 B e^2-3 a b e (2 A e+4 B d+B e x)+b^2 \left (3 A e (4 d+e x)+2 B \left (3 d^2+3 d e x+e^2 x^2\right )\right )\right )+6 (A b-a B) (b d-a e)^2 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 158, normalized size = 1.74 \[ \frac {2 \, B b^{3} e^{2} x^{3} + 3 \, {\left (2 \, B b^{3} d e - {\left (B a b^{2} - A b^{3}\right )} e^{2}\right )} x^{2} + 6 \, {\left (B b^{3} d^{2} - 2 \, {\left (B a b^{2} - A b^{3}\right )} d e + {\left (B a^{2} b - A a b^{2}\right )} e^{2}\right )} x - 6 \, {\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e + {\left (B a^{3} - A a^{2} b\right )} e^{2}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 164, normalized size = 1.80 \[ \frac {2 \, B b^{2} x^{3} e^{2} + 6 \, B b^{2} d x^{2} e + 6 \, B b^{2} d^{2} x - 3 \, B a b x^{2} e^{2} + 3 \, A b^{2} x^{2} e^{2} - 12 \, B a b d x e + 12 \, A b^{2} d x e + 6 \, B a^{2} x e^{2} - 6 \, A a b x e^{2}}{6 \, b^{3}} - \frac {{\left (B a b^{2} d^{2} - A b^{3} d^{2} - 2 \, B a^{2} b d e + 2 \, A a b^{2} d e + B a^{3} e^{2} - A a^{2} b e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 197, normalized size = 2.16 \[ \frac {B \,e^{2} x^{3}}{3 b}+\frac {A \,e^{2} x^{2}}{2 b}-\frac {B a \,e^{2} x^{2}}{2 b^{2}}+\frac {B d e \,x^{2}}{b}+\frac {A \,a^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {2 A a d e \ln \left (b x +a \right )}{b^{2}}-\frac {A a \,e^{2} x}{b^{2}}+\frac {A \,d^{2} \ln \left (b x +a \right )}{b}+\frac {2 A d e x}{b}-\frac {B \,a^{3} e^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {2 B \,a^{2} d e \ln \left (b x +a \right )}{b^{3}}+\frac {B \,a^{2} e^{2} x}{b^{3}}-\frac {B a \,d^{2} \ln \left (b x +a \right )}{b^{2}}-\frac {2 B a d e x}{b^{2}}+\frac {B \,d^{2} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 155, normalized size = 1.70 \[ \frac {2 \, B b^{2} e^{2} x^{3} + 3 \, {\left (2 \, B b^{2} d e - {\left (B a b - A b^{2}\right )} e^{2}\right )} x^{2} + 6 \, {\left (B b^{2} d^{2} - 2 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - A a b\right )} e^{2}\right )} x}{6 \, b^{3}} - \frac {{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} - 2 \, {\left (B a^{2} b - A a b^{2}\right )} d e + {\left (B a^{3} - A a^{2} b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 159, normalized size = 1.75 \[ x\,\left (\frac {B\,d^2+2\,A\,e\,d}{b}-\frac {a\,\left (\frac {A\,e^2+2\,B\,d\,e}{b}-\frac {B\,a\,e^2}{b^2}\right )}{b}\right )+x^2\,\left (\frac {A\,e^2+2\,B\,d\,e}{2\,b}-\frac {B\,a\,e^2}{2\,b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-B\,a^3\,e^2+2\,B\,a^2\,b\,d\,e+A\,a^2\,b\,e^2-B\,a\,b^2\,d^2-2\,A\,a\,b^2\,d\,e+A\,b^3\,d^2\right )}{b^4}+\frac {B\,e^2\,x^3}{3\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.49, size = 117, normalized size = 1.29 \[ \frac {B e^{2} x^{3}}{3 b} + x^{2} \left (\frac {A e^{2}}{2 b} - \frac {B a e^{2}}{2 b^{2}} + \frac {B d e}{b}\right ) + x \left (- \frac {A a e^{2}}{b^{2}} + \frac {2 A d e}{b} + \frac {B a^{2} e^{2}}{b^{3}} - \frac {2 B a d e}{b^{2}} + \frac {B d^{2}}{b}\right ) - \frac {\left (- A b + B a\right ) \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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