Optimal. Leaf size=94 \[ \frac {\left (a d^2+b c^2\right )^2 \log (c+d x)}{d^5}-\frac {b c x \left (2 a d^2+b c^2\right )}{d^4}+\frac {b x^2 \left (2 a d^2+b c^2\right )}{2 d^3}-\frac {b^2 c x^3}{3 d^2}+\frac {b^2 x^4}{4 d} \]
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Rubi [A] time = 0.13, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {1586, 28, 697} \[ \frac {b x^2 \left (2 a d^2+b c^2\right )}{2 d^3}-\frac {b c x \left (2 a d^2+b c^2\right )}{d^4}+\frac {\left (a d^2+b c^2\right )^2 \log (c+d x)}{d^5}-\frac {b^2 c x^3}{3 d^2}+\frac {b^2 x^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 28
Rule 697
Rule 1586
Rubi steps
\begin {align*} \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{(c+d x)^2} \, dx &=\int \frac {a^2+2 a b x^2+b^2 x^4}{c+d x} \, dx\\ &=\frac {\int \frac {\left (a b+b^2 x^2\right )^2}{c+d x} \, dx}{b^2}\\ &=\frac {\int \left (-\frac {b^3 c \left (b c^2+2 a d^2\right )}{d^4}+\frac {b^3 \left (b c^2+2 a d^2\right ) x}{d^3}-\frac {b^4 c x^2}{d^2}+\frac {b^4 x^3}{d}+\frac {b^2 \left (b c^2+a d^2\right )^2}{d^4 (c+d x)}\right ) \, dx}{b^2}\\ &=-\frac {b c \left (b c^2+2 a d^2\right ) x}{d^4}+\frac {b \left (b c^2+2 a d^2\right ) x^2}{2 d^3}-\frac {b^2 c x^3}{3 d^2}+\frac {b^2 x^4}{4 d}+\frac {\left (b c^2+a d^2\right )^2 \log (c+d x)}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 79, normalized size = 0.84 \[ \frac {12 \left (a d^2+b c^2\right )^2 \log (c+d x)+b d x \left (12 a d^2 (d x-2 c)+b \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )}{12 d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 105, normalized size = 1.12 \[ \frac {3 \, b^{2} d^{4} x^{4} - 4 \, b^{2} c d^{3} x^{3} + 6 \, {\left (b^{2} c^{2} d^{2} + 2 \, a b d^{4}\right )} x^{2} - 12 \, {\left (b^{2} c^{3} d + 2 \, a b c d^{3}\right )} x + 12 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{12 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 365, normalized size = 3.88 \[ -\frac {1}{12} \, b^{2} d {\left (\frac {{\left (d x + c\right )}^{4} {\left (\frac {20 \, c}{d x + c} - \frac {60 \, c^{2}}{{\left (d x + c\right )}^{2}} + \frac {120 \, c^{3}}{{\left (d x + c\right )}^{3}} - 3\right )}}{d^{6}} + \frac {60 \, c^{4} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{6}} - \frac {12 \, c^{5}}{{\left (d x + c\right )} d^{6}}\right )} - \frac {1}{3} \, b^{2} c {\left (\frac {{\left (d x + c\right )}^{3} {\left (\frac {6 \, c}{d x + c} - \frac {18 \, c^{2}}{{\left (d x + c\right )}^{2}} - 1\right )}}{d^{5}} - \frac {12 \, c^{3} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{5}} + \frac {3 \, c^{4}}{{\left (d x + c\right )} d^{5}}\right )} - a b d {\left (\frac {{\left (d x + c\right )}^{2} {\left (\frac {6 \, c}{d x + c} - 1\right )}}{d^{4}} + \frac {6 \, c^{2} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{4}} - \frac {2 \, c^{3}}{{\left (d x + c\right )} d^{4}}\right )} + 2 \, a b c {\left (\frac {2 \, c \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{3}} + \frac {d x + c}{d^{3}} - \frac {c^{2}}{{\left (d x + c\right )} d^{3}}\right )} - a^{2} {\left (\frac {\log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d} - \frac {c}{{\left (d x + c\right )} d}\right )} - \frac {a^{2} c}{{\left (d x + c\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 114, normalized size = 1.21 \[ \frac {b^{2} x^{4}}{4 d}-\frac {b^{2} c \,x^{3}}{3 d^{2}}+\frac {a b \,x^{2}}{d}+\frac {b^{2} c^{2} x^{2}}{2 d^{3}}+\frac {a^{2} \ln \left (d x +c \right )}{d}+\frac {2 a b \,c^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {2 a b c x}{d^{2}}+\frac {b^{2} c^{4} \ln \left (d x +c \right )}{d^{5}}-\frac {b^{2} c^{3} x}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 105, normalized size = 1.12 \[ \frac {3 \, b^{2} d^{3} x^{4} - 4 \, b^{2} c d^{2} x^{3} + 6 \, {\left (b^{2} c^{2} d + 2 \, a b d^{3}\right )} x^{2} - 12 \, {\left (b^{2} c^{3} + 2 \, a b c d^{2}\right )} x}{12 \, d^{4}} + \frac {{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \log \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 106, normalized size = 1.13 \[ x^2\,\left (\frac {b^2\,c^2}{2\,d^3}+\frac {a\,b}{d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (a^2\,d^4+2\,a\,b\,c^2\,d^2+b^2\,c^4\right )}{d^5}+\frac {b^2\,x^4}{4\,d}-\frac {b^2\,c\,x^3}{3\,d^2}-\frac {c\,x\,\left (\frac {b^2\,c^2}{d^3}+\frac {2\,a\,b}{d}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 88, normalized size = 0.94 \[ - \frac {b^{2} c x^{3}}{3 d^{2}} + \frac {b^{2} x^{4}}{4 d} + x^{2} \left (\frac {a b}{d} + \frac {b^{2} c^{2}}{2 d^{3}}\right ) + x \left (- \frac {2 a b c}{d^{2}} - \frac {b^{2} c^{3}}{d^{4}}\right ) + \frac {\left (a d^{2} + b c^{2}\right )^{2} \log {\left (c + d x \right )}}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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