Optimal. Leaf size=196 \[ \frac {a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}-\frac {3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}-\frac {3 a (b c-a d) \left (5 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{b^7}+\frac {x (b c-a d) \left (10 a^2 d^2-8 a b c d+b^2 c^2\right )}{b^6}+\frac {3 d x^2 (b c-2 a d) (b c-a d)}{2 b^5}+\frac {d^2 x^3 (b c-a d)}{b^4}+\frac {d^3 x^4}{4 b^3} \]
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Rubi [A] time = 0.24, antiderivative size = 196, 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} \[ \frac {x (b c-a d) \left (10 a^2 d^2-8 a b c d+b^2 c^2\right )}{b^6}-\frac {3 a (b c-a d) \left (5 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{b^7}-\frac {3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}+\frac {a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}+\frac {d^2 x^3 (b c-a d)}{b^4}+\frac {3 d x^2 (b c-2 a d) (b c-a d)}{2 b^5}+\frac {d^3 x^4}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx &=\int \left (\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right )}{b^6}+\frac {3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac {3 d^2 (b c-a d) x^2}{b^4}+\frac {d^3 x^3}{b^3}+\frac {a^3 (-b c+a d)^3}{b^6 (a+b x)^3}-\frac {3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^6 (a+b x)^2}+\frac {3 a (b c-a d) \left (-b^2 c^2+5 a b c d-5 a^2 d^2\right )}{b^6 (a+b x)}\right ) \, dx\\ &=\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) x}{b^6}+\frac {3 d (b c-2 a d) (b c-a d) x^2}{2 b^5}+\frac {d^2 (b c-a d) x^3}{b^4}+\frac {d^3 x^4}{4 b^3}+\frac {a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}-\frac {3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}-\frac {3 a (b c-a d) \left (b^2 c^2-5 a b c d+5 a^2 d^2\right ) \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 207, normalized size = 1.06 \[ \frac {\frac {2 a^3 (b c-a d)^3}{(a+b x)^2}+6 b^2 d x^2 \left (2 a^2 d^2-3 a b c d+b^2 c^2\right )+\frac {12 a^2 (b c-a d)^2 (2 a d-b c)}{a+b x}+4 b x \left (-10 a^3 d^3+18 a^2 b c d^2-9 a b^2 c^2 d+b^3 c^3\right )+12 a \left (5 a^3 d^3-10 a^2 b c d^2+6 a b^2 c^2 d-b^3 c^3\right ) \log (a+b x)+4 b^3 d^2 x^3 (b c-a d)+b^4 d^3 x^4}{4 b^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 425, normalized size = 2.17 \[ \frac {b^{6} d^{3} x^{6} - 10 \, a^{3} b^{3} c^{3} + 42 \, a^{4} b^{2} c^{2} d - 54 \, a^{5} b c d^{2} + 22 \, a^{6} d^{3} + 2 \, {\left (2 \, b^{6} c d^{2} - a b^{5} d^{3}\right )} x^{5} + {\left (6 \, b^{6} c^{2} d - 10 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{4} + 4 \, {\left (b^{6} c^{3} - 6 \, a b^{5} c^{2} d + 10 \, a^{2} b^{4} c d^{2} - 5 \, a^{3} b^{3} d^{3}\right )} x^{3} + 2 \, {\left (4 \, a b^{5} c^{3} - 33 \, a^{2} b^{4} c^{2} d + 63 \, a^{3} b^{3} c d^{2} - 34 \, a^{4} b^{2} d^{3}\right )} x^{2} - 4 \, {\left (2 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 3 \, a^{4} b^{2} c d^{2} + 4 \, a^{5} b d^{3}\right )} x - 12 \, {\left (a^{3} b^{3} c^{3} - 6 \, a^{4} b^{2} c^{2} d + 10 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3} + {\left (a b^{5} c^{3} - 6 \, a^{2} b^{4} c^{2} d + 10 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a^{2} b^{4} c^{3} - 6 \, a^{3} b^{3} c^{2} d + 10 \, a^{4} b^{2} c d^{2} - 5 \, a^{5} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{4 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 277, normalized size = 1.41 \[ -\frac {3 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 10 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {5 \, a^{3} b^{3} c^{3} - 21 \, a^{4} b^{2} c^{2} d + 27 \, a^{5} b c d^{2} - 11 \, a^{6} d^{3} + 6 \, {\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{7}} + \frac {b^{9} d^{3} x^{4} + 4 \, b^{9} c d^{2} x^{3} - 4 \, a b^{8} d^{3} x^{3} + 6 \, b^{9} c^{2} d x^{2} - 18 \, a b^{8} c d^{2} x^{2} + 12 \, a^{2} b^{7} d^{3} x^{2} + 4 \, b^{9} c^{3} x - 36 \, a b^{8} c^{2} d x + 72 \, a^{2} b^{7} c d^{2} x - 40 \, a^{3} b^{6} d^{3} x}{4 \, b^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 335, normalized size = 1.71 \[ \frac {d^{3} x^{4}}{4 b^{3}}-\frac {a \,d^{3} x^{3}}{b^{4}}+\frac {c \,d^{2} x^{3}}{b^{3}}-\frac {a^{6} d^{3}}{2 \left (b x +a \right )^{2} b^{7}}+\frac {3 a^{5} c \,d^{2}}{2 \left (b x +a \right )^{2} b^{6}}-\frac {3 a^{4} c^{2} d}{2 \left (b x +a \right )^{2} b^{5}}+\frac {a^{3} c^{3}}{2 \left (b x +a \right )^{2} b^{4}}+\frac {3 a^{2} d^{3} x^{2}}{b^{5}}-\frac {9 a c \,d^{2} x^{2}}{2 b^{4}}+\frac {3 c^{2} d \,x^{2}}{2 b^{3}}+\frac {6 a^{5} d^{3}}{\left (b x +a \right ) b^{7}}-\frac {15 a^{4} c \,d^{2}}{\left (b x +a \right ) b^{6}}+\frac {15 a^{4} d^{3} \ln \left (b x +a \right )}{b^{7}}+\frac {12 a^{3} c^{2} d}{\left (b x +a \right ) b^{5}}-\frac {30 a^{3} c \,d^{2} \ln \left (b x +a \right )}{b^{6}}-\frac {10 a^{3} d^{3} x}{b^{6}}-\frac {3 a^{2} c^{3}}{\left (b x +a \right ) b^{4}}+\frac {18 a^{2} c^{2} d \ln \left (b x +a \right )}{b^{5}}+\frac {18 a^{2} c \,d^{2} x}{b^{5}}-\frac {3 a \,c^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {9 a \,c^{2} d x}{b^{4}}+\frac {c^{3} x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 277, normalized size = 1.41 \[ -\frac {5 \, a^{3} b^{3} c^{3} - 21 \, a^{4} b^{2} c^{2} d + 27 \, a^{5} b c d^{2} - 11 \, a^{6} d^{3} + 6 \, {\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x}{2 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac {b^{3} d^{3} x^{4} + 4 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} x}{4 \, b^{6}} - \frac {3 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 10 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 352, normalized size = 1.80 \[ \frac {x\,\left (6\,a^5\,d^3-15\,a^4\,b\,c\,d^2+12\,a^3\,b^2\,c^2\,d-3\,a^2\,b^3\,c^3\right )+\frac {11\,a^6\,d^3-27\,a^5\,b\,c\,d^2+21\,a^4\,b^2\,c^2\,d-5\,a^3\,b^3\,c^3}{2\,b}}{a^2\,b^6+2\,a\,b^7\,x+b^8\,x^2}-x^3\,\left (\frac {a\,d^3}{b^4}-\frac {c\,d^2}{b^3}\right )+x\,\left (\frac {c^3}{b^3}-\frac {3\,a\,\left (\frac {3\,c^2\,d}{b^3}+\frac {3\,a\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{b}-\frac {3\,a^2\,d^3}{b^5}\right )}{b}-\frac {a^3\,d^3}{b^6}+\frac {3\,a^2\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{b^2}\right )+x^2\,\left (\frac {3\,c^2\,d}{2\,b^3}+\frac {3\,a\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{2\,b}-\frac {3\,a^2\,d^3}{2\,b^5}\right )+\frac {d^3\,x^4}{4\,b^3}+\frac {\ln \left (a+b\,x\right )\,\left (15\,a^4\,d^3-30\,a^3\,b\,c\,d^2+18\,a^2\,b^2\,c^2\,d-3\,a\,b^3\,c^3\right )}{b^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 284, normalized size = 1.45 \[ \frac {3 a \left (a d - b c\right ) \left (5 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{3} \left (- \frac {a d^{3}}{b^{4}} + \frac {c d^{2}}{b^{3}}\right ) + x^{2} \left (\frac {3 a^{2} d^{3}}{b^{5}} - \frac {9 a c d^{2}}{2 b^{4}} + \frac {3 c^{2} d}{2 b^{3}}\right ) + x \left (- \frac {10 a^{3} d^{3}}{b^{6}} + \frac {18 a^{2} c d^{2}}{b^{5}} - \frac {9 a c^{2} d}{b^{4}} + \frac {c^{3}}{b^{3}}\right ) + \frac {11 a^{6} d^{3} - 27 a^{5} b c d^{2} + 21 a^{4} b^{2} c^{2} d - 5 a^{3} b^{3} c^{3} + x \left (12 a^{5} b d^{3} - 30 a^{4} b^{2} c d^{2} + 24 a^{3} b^{3} c^{2} d - 6 a^{2} b^{4} c^{3}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac {d^{3} x^{4}}{4 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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