Optimal. Leaf size=145 \[ \frac {e^2 (a+b x)^2 (-4 a B e+A b e+3 b B d)}{2 b^5}-\frac {(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac {(b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{b^5}+\frac {3 e x (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac {B e^3 (a+b x)^3}{3 b^5} \]
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Rubi [A] time = 0.18, antiderivative size = 145, 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} \begin {gather*} \frac {e^2 (a+b x)^2 (-4 a B e+A b e+3 b B d)}{2 b^5}-\frac {(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac {3 e x (b d-a e) (-2 a B e+A b e+b B d)}{b^4}+\frac {(b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{b^5}+\frac {B e^3 (a+b x)^3}{3 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{(a+b x)^2} \, dx &=\int \left (\frac {3 e (b d-a e) (b B d+A b e-2 a B e)}{b^4}+\frac {(A b-a B) (b d-a e)^3}{b^4 (a+b x)^2}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e)}{b^4 (a+b x)}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)}{b^4}+\frac {B e^3 (a+b x)^2}{b^4}\right ) \, dx\\ &=\frac {3 e (b d-a e) (b B d+A b e-2 a B e) x}{b^4}-\frac {(A b-a B) (b d-a e)^3}{b^5 (a+b x)}+\frac {e^2 (3 b B d+A b e-4 a B e) (a+b x)^2}{2 b^5}+\frac {B e^3 (a+b x)^3}{3 b^5}+\frac {(b d-a e)^2 (b B d+3 A b e-4 a B e) \log (a+b x)}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 250, normalized size = 1.72 \begin {gather*} \frac {-3 A b \left (-2 a^3 e^3+2 a^2 b e^2 (3 d+2 e x)+3 a b^2 e \left (-2 d^2-2 d e x+e^2 x^2\right )+b^3 \left (2 d^3-6 d e^2 x^2-e^3 x^3\right )\right )+B \left (-6 a^4 e^3+18 a^3 b e^2 (d+e x)+6 a^2 b^2 e \left (-3 d^2-6 d e x+2 e^2 x^2\right )+a b^3 \left (6 d^3+18 d^2 e x-27 d e^2 x^2-4 e^3 x^3\right )+b^4 e x^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 (a+b x) (b d-a e)^2 \log (a+b x) (-4 a B e+3 A b e+b B d)}{6 b^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^3}{(a+b x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.91, size = 417, normalized size = 2.88 \begin {gather*} \frac {2 \, B b^{4} e^{3} x^{4} + 6 \, {\left (B a b^{3} - A b^{4}\right )} d^{3} - 18 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 18 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 6 \, {\left (B a^{4} - A a^{3} b\right )} e^{3} + {\left (9 \, B b^{4} d e^{2} - {\left (4 \, B a b^{3} - 3 \, A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B b^{4} d^{2} e - 3 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} d e^{2} + {\left (4 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 6 \, {\left (3 \, B a b^{3} d^{2} e - 3 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d e^{2} + {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} e^{3}\right )} x + 6 \, {\left (B a b^{3} d^{3} - 3 \, {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{2} - {\left (4 \, B a^{4} - 3 \, A a^{3} b\right )} e^{3} + {\left (B b^{4} d^{3} - 3 \, {\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{2} - {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{6} x + a b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 359, normalized size = 2.48 \begin {gather*} \frac {{\left (b x + a\right )}^{3} {\left (2 \, B e^{3} + \frac {3 \, {\left (3 \, B b^{2} d e^{2} - 4 \, B a b e^{3} + A b^{2} e^{3}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (B b^{4} d^{2} e - 3 \, B a b^{3} d e^{2} + A b^{4} d e^{2} + 2 \, B a^{2} b^{2} e^{3} - A a b^{3} e^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )}}{6 \, b^{5}} - \frac {{\left (B b^{3} d^{3} - 6 \, B a b^{2} d^{2} e + 3 \, A b^{3} d^{2} e + 9 \, B a^{2} b d e^{2} - 6 \, A a b^{2} d e^{2} - 4 \, B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5}} + \frac {\frac {B a b^{6} d^{3}}{b x + a} - \frac {A b^{7} d^{3}}{b x + a} - \frac {3 \, B a^{2} b^{5} d^{2} e}{b x + a} + \frac {3 \, A a b^{6} d^{2} e}{b x + a} + \frac {3 \, B a^{3} b^{4} d e^{2}}{b x + a} - \frac {3 \, A a^{2} b^{5} d e^{2}}{b x + a} - \frac {B a^{4} b^{3} e^{3}}{b x + a} + \frac {A a^{3} b^{4} e^{3}}{b x + a}}{b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 376, normalized size = 2.59 \begin {gather*} \frac {B \,e^{3} x^{3}}{3 b^{2}}+\frac {A \,e^{3} x^{2}}{2 b^{2}}-\frac {B a \,e^{3} x^{2}}{b^{3}}+\frac {3 B d \,e^{2} x^{2}}{2 b^{2}}+\frac {A \,a^{3} e^{3}}{\left (b x +a \right ) b^{4}}-\frac {3 A \,a^{2} d \,e^{2}}{\left (b x +a \right ) b^{3}}+\frac {3 A \,a^{2} e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 A a \,d^{2} e}{\left (b x +a \right ) b^{2}}-\frac {6 A a d \,e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {2 A a \,e^{3} x}{b^{3}}-\frac {A \,d^{3}}{\left (b x +a \right ) b}+\frac {3 A \,d^{2} e \ln \left (b x +a \right )}{b^{2}}+\frac {3 A d \,e^{2} x}{b^{2}}-\frac {B \,a^{4} e^{3}}{\left (b x +a \right ) b^{5}}+\frac {3 B \,a^{3} d \,e^{2}}{\left (b x +a \right ) b^{4}}-\frac {4 B \,a^{3} e^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {3 B \,a^{2} d^{2} e}{\left (b x +a \right ) b^{3}}+\frac {9 B \,a^{2} d \,e^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {3 B \,a^{2} e^{3} x}{b^{4}}+\frac {B a \,d^{3}}{\left (b x +a \right ) b^{2}}-\frac {6 B a \,d^{2} e \ln \left (b x +a \right )}{b^{3}}-\frac {6 B a d \,e^{2} x}{b^{3}}+\frac {B \,d^{3} \ln \left (b x +a \right )}{b^{2}}+\frac {3 B \,d^{2} e x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 273, normalized size = 1.88 \begin {gather*} \frac {{\left (B a b^{3} - A b^{4}\right )} d^{3} - 3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} d^{2} e + 3 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - {\left (B a^{4} - A a^{3} b\right )} e^{3}}{b^{6} x + a b^{5}} + \frac {2 \, B b^{2} e^{3} x^{3} + 3 \, {\left (3 \, B b^{2} d e^{2} - {\left (2 \, B a b - A b^{2}\right )} e^{3}\right )} x^{2} + 6 \, {\left (3 \, B b^{2} d^{2} e - 3 \, {\left (2 \, B a b - A b^{2}\right )} d e^{2} + {\left (3 \, B a^{2} - 2 \, A a b\right )} e^{3}\right )} x}{6 \, b^{4}} + \frac {{\left (B b^{3} d^{3} - 3 \, {\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{2} - {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} \log \left (b x + a\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 293, normalized size = 2.02 \begin {gather*} x^2\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{2\,b^2}-\frac {B\,a\,e^3}{b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {A\,e^3+3\,B\,d\,e^2}{b^2}-\frac {2\,B\,a\,e^3}{b^3}\right )}{b}-\frac {3\,d\,e\,\left (A\,e+B\,d\right )}{b^2}+\frac {B\,a^2\,e^3}{b^4}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-4\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3-6\,B\,a\,b^2\,d^2\,e-6\,A\,a\,b^2\,d\,e^2+B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{b^5}-\frac {B\,a^4\,e^3-3\,B\,a^3\,b\,d\,e^2-A\,a^3\,b\,e^3+3\,B\,a^2\,b^2\,d^2\,e+3\,A\,a^2\,b^2\,d\,e^2-B\,a\,b^3\,d^3-3\,A\,a\,b^3\,d^2\,e+A\,b^4\,d^3}{b\,\left (x\,b^5+a\,b^4\right )}+\frac {B\,e^3\,x^3}{3\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.53, size = 257, normalized size = 1.77 \begin {gather*} \frac {B e^{3} x^{3}}{3 b^{2}} + x^{2} \left (\frac {A e^{3}}{2 b^{2}} - \frac {B a e^{3}}{b^{3}} + \frac {3 B d e^{2}}{2 b^{2}}\right ) + x \left (- \frac {2 A a e^{3}}{b^{3}} + \frac {3 A d e^{2}}{b^{2}} + \frac {3 B a^{2} e^{3}}{b^{4}} - \frac {6 B a d e^{2}}{b^{3}} + \frac {3 B d^{2} e}{b^{2}}\right ) + \frac {A a^{3} b e^{3} - 3 A a^{2} b^{2} d e^{2} + 3 A a b^{3} d^{2} e - A b^{4} d^{3} - B a^{4} e^{3} + 3 B a^{3} b d e^{2} - 3 B a^{2} b^{2} d^{2} e + B a b^{3} d^{3}}{a b^{5} + b^{6} x} - \frac {\left (a e - b d\right )^{2} \left (- 3 A b e + 4 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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