Optimal. Leaf size=136 \[ -\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac {x (b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d x^2 (b c-a d)^2}{2 b^4}+\frac {d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac {d^3 x^4}{4 b^2} \]
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Rubi [A] time = 0.12, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ -\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}+\frac {d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac {3 d x^2 (b c-a d)^2}{2 b^4}+\frac {x (b c-4 a d) (b c-a d)^2}{b^5}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac {d^3 x^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (\frac {(b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d (b c-a d)^2 x}{b^4}+\frac {d^2 (3 b c-2 a d) x^2}{b^3}+\frac {d^3 x^3}{b^2}-\frac {a^2 (-b c+a d)^3}{b^5 (a+b x)^2}+\frac {a (-b c+a d)^2 (-2 b c+5 a d)}{b^5 (a+b x)}\right ) \, dx\\ &=\frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^3}{3 b^3}+\frac {d^3 x^4}{4 b^2}-\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 130, normalized size = 0.96 \[ \frac {\frac {12 a^2 (a d-b c)^3}{a+b x}+4 b^3 d^2 x^3 (3 b c-2 a d)+18 b^2 d x^2 (b c-a d)^2+12 b x (b c-4 a d) (b c-a d)^2+12 a (b c-a d)^2 (5 a d-2 b c) \log (a+b x)+3 b^4 d^3 x^4}{12 b^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 314, normalized size = 2.31 \[ \frac {3 \, b^{5} d^{3} x^{5} - 12 \, a^{2} b^{3} c^{3} + 36 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 12 \, a^{5} d^{3} + {\left (12 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \, {\left (9 \, b^{5} c^{2} d - 12 \, a b^{4} c d^{2} + 5 \, a^{2} b^{3} d^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 12 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{2} + 12 \, {\left (a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\right )} x - 12 \, {\left (2 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 12 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 286, normalized size = 2.10 \[ \frac {{\left (3 \, d^{3} + \frac {4 \, {\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (3 \, b^{4} c^{2} d - 12 \, a b^{3} c d^{2} + 10 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {12 \, {\left (b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 18 \, a^{2} b^{4} c d^{2} - 10 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{4}}{12 \, b^{6}} + \frac {{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} - \frac {\frac {a^{2} b^{7} c^{3}}{b x + a} - \frac {3 \, a^{3} b^{6} c^{2} d}{b x + a} + \frac {3 \, a^{4} b^{5} c d^{2}}{b x + a} - \frac {a^{5} b^{4} d^{3}}{b x + a}}{b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 260, normalized size = 1.91 \[ \frac {d^{3} x^{4}}{4 b^{2}}-\frac {2 a \,d^{3} x^{3}}{3 b^{3}}+\frac {c \,d^{2} x^{3}}{b^{2}}+\frac {3 a^{2} d^{3} x^{2}}{2 b^{4}}-\frac {3 a c \,d^{2} x^{2}}{b^{3}}+\frac {3 c^{2} d \,x^{2}}{2 b^{2}}+\frac {a^{5} d^{3}}{\left (b x +a \right ) b^{6}}-\frac {3 a^{4} c \,d^{2}}{\left (b x +a \right ) b^{5}}+\frac {5 a^{4} d^{3} \ln \left (b x +a \right )}{b^{6}}+\frac {3 a^{3} c^{2} d}{\left (b x +a \right ) b^{4}}-\frac {12 a^{3} c \,d^{2} \ln \left (b x +a \right )}{b^{5}}-\frac {4 a^{3} d^{3} x}{b^{5}}-\frac {a^{2} c^{3}}{\left (b x +a \right ) b^{3}}+\frac {9 a^{2} c^{2} d \ln \left (b x +a \right )}{b^{4}}+\frac {9 a^{2} c \,d^{2} x}{b^{4}}-\frac {2 a \,c^{3} \ln \left (b x +a \right )}{b^{3}}-\frac {6 a \,c^{2} d x}{b^{3}}+\frac {c^{3} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 220, normalized size = 1.62 \[ -\frac {a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} d^{3} x^{4} + 4 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{3} + 18 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x}{12 \, b^{5}} - \frac {{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 281, normalized size = 2.07 \[ x\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,d^3}{3\,b^3}-\frac {c\,d^2}{b^2}\right )+x^2\,\left (\frac {3\,c^2\,d}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{2\,b^4}\right )+\frac {a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {d^3\,x^4}{4\,b^2}+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,d^3-12\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3\right )}{b^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 204, normalized size = 1.50 \[ \frac {a \left (a d - b c\right )^{2} \left (5 a d - 2 b c\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{3} \left (- \frac {2 a d^{3}}{3 b^{3}} + \frac {c d^{2}}{b^{2}}\right ) + x^{2} \left (\frac {3 a^{2} d^{3}}{2 b^{4}} - \frac {3 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{2 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{3}}{b^{5}} + \frac {9 a^{2} c d^{2}}{b^{4}} - \frac {6 a c^{2} d}{b^{3}} + \frac {c^{3}}{b^{2}}\right ) + \frac {a^{5} d^{3} - 3 a^{4} b c d^{2} + 3 a^{3} b^{2} c^{2} d - a^{2} b^{3} c^{3}}{a b^{6} + b^{7} x} + \frac {d^{3} x^{4}}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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