Optimal. Leaf size=75 \[ -\frac {(b c-a d)^3}{b^4 (a+b x)}+\frac {3 d (b c-a d)^2 \log (a+b x)}{b^4}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {d^3 x^2}{2 b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {d^2 x (3 b c-2 a d)}{b^3}-\frac {(b c-a d)^3}{b^4 (a+b x)}+\frac {3 d (b c-a d)^2 \log (a+b x)}{b^4}+\frac {d^3 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+b x)^2} \, dx &=\int \left (\frac {d^2 (3 b c-2 a d)}{b^3}+\frac {d^3 x}{b^2}+\frac {(b c-a d)^3}{b^3 (a+b x)^2}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)}\right ) \, dx\\ &=\frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^2}{2 b^2}-\frac {(b c-a d)^3}{b^4 (a+b x)}+\frac {3 d (b c-a d)^2 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 72, normalized size = 0.96 \[ \frac {2 b d^2 x (3 b c-2 a d)-\frac {2 (b c-a d)^3}{a+b x}+6 d (b c-a d)^2 \log (a+b x)+b^2 d^3 x^2}{2 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 173, normalized size = 2.31 \[ \frac {b^{3} d^{3} x^{3} - 2 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + 3 \, {\left (2 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x + 6 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.01, size = 167, normalized size = 2.23 \[ \frac {{\left (d^{3} + \frac {6 \, {\left (b^{2} c d^{2} - a b d^{3}\right )}}{{\left (b x + a\right )} b}\right )} {\left (b x + a\right )}^{2}}{2 \, b^{4}} - \frac {3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} - \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 149, normalized size = 1.99 \[ \frac {d^{3} x^{2}}{2 b^{2}}+\frac {a^{3} d^{3}}{\left (b x +a \right ) b^{4}}-\frac {3 a^{2} c \,d^{2}}{\left (b x +a \right ) b^{3}}+\frac {3 a^{2} d^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 a \,c^{2} d}{\left (b x +a \right ) b^{2}}-\frac {6 a c \,d^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {2 a \,d^{3} x}{b^{3}}-\frac {c^{3}}{\left (b x +a \right ) b}+\frac {3 c^{2} d \ln \left (b x +a \right )}{b^{2}}+\frac {3 c \,d^{2} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 118, normalized size = 1.57 \[ -\frac {b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{b^{5} x + a b^{4}} + \frac {b d^{3} x^{2} + 2 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{2 \, b^{3}} + \frac {3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 123, normalized size = 1.64 \[ \frac {\ln \left (a+b\,x\right )\,\left (3\,a^2\,d^3-6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{b^4}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )+\frac {a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}{b\,\left (x\,b^4+a\,b^3\right )}+\frac {d^3\,x^2}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 102, normalized size = 1.36 \[ x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{a b^{4} + b^{5} x} + \frac {d^{3} x^{2}}{2 b^{2}} + \frac {3 d \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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