Optimal. Leaf size=136 \[ \frac {a^3 (b c-a d)^2}{b^6 (a+b x)}+\frac {a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}-\frac {2 a x (b c-2 a d) (b c-a d)}{b^5}+\frac {x^2 (b c-3 a d) (b c-a d)}{2 b^4}+\frac {2 d x^3 (b c-a d)}{3 b^3}+\frac {d^2 x^4}{4 b^2} \]
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Rubi [A] time = 0.13, 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} \[ \frac {a^3 (b c-a d)^2}{b^6 (a+b x)}+\frac {a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}+\frac {2 d x^3 (b c-a d)}{3 b^3}+\frac {x^2 (b c-3 a d) (b c-a d)}{2 b^4}-\frac {2 a x (b c-2 a d) (b c-a d)}{b^5}+\frac {d^2 x^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^2}{(a+b x)^2} \, dx &=\int \left (\frac {2 a (b c-2 a d) (-b c+a d)}{b^5}+\frac {(b c-3 a d) (b c-a d) x}{b^4}+\frac {2 d (b c-a d) x^2}{b^3}+\frac {d^2 x^3}{b^2}-\frac {a^3 (-b c+a d)^2}{b^5 (a+b x)^2}+\frac {a^2 (3 b c-5 a d) (b c-a d)}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac {2 a (b c-2 a d) (b c-a d) x}{b^5}+\frac {(b c-3 a d) (b c-a d) x^2}{2 b^4}+\frac {2 d (b c-a d) x^3}{3 b^3}+\frac {d^2 x^4}{4 b^2}+\frac {a^3 (b c-a d)^2}{b^6 (a+b x)}+\frac {a^2 (3 b c-5 a d) (b c-a d) \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 149, normalized size = 1.10 \[ \frac {\frac {12 a^3 (b c-a d)^2}{a+b x}+6 b^2 x^2 \left (3 a^2 d^2-4 a b c d+b^2 c^2\right )-24 a b x \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+12 a^2 \left (5 a^2 d^2-8 a b c d+3 b^2 c^2\right ) \log (a+b x)+8 b^3 d x^3 (b c-a d)+3 b^4 d^2 x^4}{12 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 246, normalized size = 1.81 \[ \frac {3 \, b^{5} d^{2} x^{5} + 12 \, a^{3} b^{2} c^{2} - 24 \, a^{4} b c d + 12 \, a^{5} d^{2} + {\left (8 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{4} + 2 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{4} c d + 5 \, a^{2} b^{3} d^{2}\right )} x^{3} - 6 \, {\left (3 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 5 \, a^{3} b^{2} d^{2}\right )} x^{2} - 24 \, {\left (a^{2} b^{3} c^{2} - 3 \, a^{3} b^{2} c d + 2 \, a^{4} b d^{2}\right )} x + 12 \, {\left (3 \, a^{3} b^{2} c^{2} - 8 \, a^{4} b c d + 5 \, a^{5} d^{2} + {\left (3 \, a^{2} b^{3} c^{2} - 8 \, a^{3} b^{2} c d + 5 \, a^{4} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 236, normalized size = 1.74 \[ \frac {{\left (3 \, d^{2} + \frac {4 \, {\left (2 \, b^{2} c d - 5 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (b^{4} c^{2} - 8 \, a b^{3} c d + 10 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {12 \, {\left (3 \, a b^{5} c^{2} - 12 \, a^{2} b^{4} c d + 10 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{4}}{12 \, b^{6}} - \frac {{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} + \frac {\frac {a^{3} b^{6} c^{2}}{b x + a} - \frac {2 \, a^{4} b^{5} c d}{b x + a} + \frac {a^{5} b^{4} d^{2}}{b x + a}}{b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 205, normalized size = 1.51 \[ \frac {d^{2} x^{4}}{4 b^{2}}-\frac {2 a \,d^{2} x^{3}}{3 b^{3}}+\frac {2 c d \,x^{3}}{3 b^{2}}+\frac {3 a^{2} d^{2} x^{2}}{2 b^{4}}-\frac {2 a c d \,x^{2}}{b^{3}}+\frac {c^{2} x^{2}}{2 b^{2}}+\frac {a^{5} d^{2}}{\left (b x +a \right ) b^{6}}-\frac {2 a^{4} c d}{\left (b x +a \right ) b^{5}}+\frac {5 a^{4} d^{2} \ln \left (b x +a \right )}{b^{6}}+\frac {a^{3} c^{2}}{\left (b x +a \right ) b^{4}}-\frac {8 a^{3} c d \ln \left (b x +a \right )}{b^{5}}-\frac {4 a^{3} d^{2} x}{b^{5}}+\frac {3 a^{2} c^{2} \ln \left (b x +a \right )}{b^{4}}+\frac {6 a^{2} c d x}{b^{4}}-\frac {2 a \,c^{2} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 175, normalized size = 1.29 \[ \frac {a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} d^{2} x^{4} + 8 \, {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} - 4 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{2} - 24 \, {\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x}{12 \, b^{5}} + \frac {{\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} \log \left (b x + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 236, normalized size = 1.74 \[ x\,\left (\frac {a^2\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b}\right )+x^2\,\left (\frac {c^2}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{2\,b^4}\right )-x^3\,\left (\frac {2\,a\,d^2}{3\,b^3}-\frac {2\,c\,d}{3\,b^2}\right )+\frac {a^5\,d^2-2\,a^4\,b\,c\,d+a^3\,b^2\,c^2}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,d^2-8\,a^3\,b\,c\,d+3\,a^2\,b^2\,c^2\right )}{b^6}+\frac {d^2\,x^4}{4\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 175, normalized size = 1.29 \[ \frac {a^{2} \left (a d - b c\right ) \left (5 a d - 3 b c\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{3} \left (- \frac {2 a d^{2}}{3 b^{3}} + \frac {2 c d}{3 b^{2}}\right ) + x^{2} \left (\frac {3 a^{2} d^{2}}{2 b^{4}} - \frac {2 a c d}{b^{3}} + \frac {c^{2}}{2 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{2}}{b^{5}} + \frac {6 a^{2} c d}{b^{4}} - \frac {2 a c^{2}}{b^{3}}\right ) + \frac {a^{5} d^{2} - 2 a^{4} b c d + a^{3} b^{2} c^{2}}{a b^{6} + b^{7} x} + \frac {d^{2} x^{4}}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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