Optimal. Leaf size=199 \[ -\frac {3 b^2 \log (x) (b c-a d)^2 (2 b c-a d)}{a^7}+\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}-\frac {b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac {b (5 b c-2 a d) (b c-a d)^2}{a^6 x}+\frac {(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac {c (b c-a d)^2}{a^4 x^3}+\frac {c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac {c^3}{5 a^2 x^5} \]
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Rubi [A] time = 0.18, antiderivative size = 199, 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} \[ -\frac {b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac {3 b^2 \log (x) (b c-a d)^2 (2 b c-a d)}{a^7}+\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}+\frac {c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac {c (b c-a d)^2}{a^4 x^3}+\frac {(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac {b (5 b c-2 a d) (b c-a d)^2}{a^6 x}-\frac {c^3}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx &=\int \left (\frac {c^3}{a^2 x^6}+\frac {c^2 (-2 b c+3 a d)}{a^3 x^5}+\frac {3 c (-b c+a d)^2}{a^4 x^4}+\frac {(-4 b c+a d) (-b c+a d)^2}{a^5 x^3}-\frac {b (-b c+a d)^2 (-5 b c+2 a d)}{a^6 x^2}+\frac {3 b^2 (-2 b c+a d) (-b c+a d)^2}{a^7 x}-\frac {b^3 (-b c+a d)^3}{a^6 (a+b x)^2}-\frac {3 b^3 (-2 b c+a d) (-b c+a d)^2}{a^7 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{5 a^2 x^5}+\frac {c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac {c (b c-a d)^2}{a^4 x^3}+\frac {(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac {b (5 b c-2 a d) (b c-a d)^2}{a^6 x}-\frac {b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (x)}{a^7}+\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 189, normalized size = 0.95 \[ -\frac {\frac {4 a^5 c^3}{x^5}+\frac {5 a^4 c^2 (3 a d-2 b c)}{x^4}+\frac {20 a^3 c (b c-a d)^2}{x^3}+\frac {10 a^2 (b c-a d)^2 (a d-4 b c)}{x^2}-\frac {20 a b^2 (a d-b c)^3}{a+b x}+60 b^2 \log (x) (b c-a d)^2 (2 b c-a d)-60 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)-\frac {20 a b (b c-a d)^2 (2 a d-5 b c)}{x}}{20 a^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 438, normalized size = 2.20 \[ -\frac {4 \, a^{6} c^{3} + 60 \, {\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5} + 30 \, {\left (2 \, a^{2} b^{4} c^{3} - 5 \, a^{3} b^{3} c^{2} d + 4 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{4} - 10 \, {\left (2 \, a^{3} b^{3} c^{3} - 5 \, a^{4} b^{2} c^{2} d + 4 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} x^{3} + 5 \, {\left (2 \, a^{4} b^{2} c^{3} - 5 \, a^{5} b c^{2} d + 4 \, a^{6} c d^{2}\right )} x^{2} - 3 \, {\left (2 \, a^{5} b c^{3} - 5 \, a^{6} c^{2} d\right )} x - 60 \, {\left ({\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + {\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + {\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5}\right )} \log \relax (x)}{20 \, {\left (a^{7} b x^{6} + a^{8} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.93, size = 436, normalized size = 2.19 \[ -\frac {3 \, {\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{7} b} - \frac {\frac {b^{11} c^{3}}{b x + a} - \frac {3 \, a b^{10} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{9} c d^{2}}{b x + a} - \frac {a^{3} b^{8} d^{3}}{b x + a}}{a^{6} b^{6}} + \frac {174 \, b^{5} c^{3} - 385 \, a b^{4} c^{2} d + 260 \, a^{2} b^{3} c d^{2} - 50 \, a^{3} b^{2} d^{3} - \frac {5 \, {\left (154 \, a b^{6} c^{3} - 337 \, a^{2} b^{5} c^{2} d + 224 \, a^{3} b^{4} c d^{2} - 42 \, a^{4} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {10 \, {\left (130 \, a^{2} b^{7} c^{3} - 280 \, a^{3} b^{6} c^{2} d + 182 \, a^{4} b^{5} c d^{2} - 33 \, a^{5} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {10 \, {\left (100 \, a^{3} b^{8} c^{3} - 210 \, a^{4} b^{7} c^{2} d + 132 \, a^{5} b^{6} c d^{2} - 23 \, a^{6} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {60 \, {\left (5 \, a^{4} b^{9} c^{3} - 10 \, a^{5} b^{8} c^{2} d + 6 \, a^{6} b^{7} c d^{2} - a^{7} b^{6} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}}}{20 \, a^{7} {\left (\frac {a}{b x + a} - 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 382, normalized size = 1.92 \[ \frac {b^{2} d^{3}}{\left (b x +a \right ) a^{3}}-\frac {3 b^{3} c \,d^{2}}{\left (b x +a \right ) a^{4}}+\frac {3 b^{2} d^{3} \ln \relax (x )}{a^{4}}-\frac {3 b^{2} d^{3} \ln \left (b x +a \right )}{a^{4}}+\frac {3 b^{4} c^{2} d}{\left (b x +a \right ) a^{5}}-\frac {12 b^{3} c \,d^{2} \ln \relax (x )}{a^{5}}+\frac {12 b^{3} c \,d^{2} \ln \left (b x +a \right )}{a^{5}}-\frac {b^{5} c^{3}}{\left (b x +a \right ) a^{6}}+\frac {15 b^{4} c^{2} d \ln \relax (x )}{a^{6}}-\frac {15 b^{4} c^{2} d \ln \left (b x +a \right )}{a^{6}}-\frac {6 b^{5} c^{3} \ln \relax (x )}{a^{7}}+\frac {6 b^{5} c^{3} \ln \left (b x +a \right )}{a^{7}}+\frac {2 b \,d^{3}}{a^{3} x}-\frac {9 b^{2} c \,d^{2}}{a^{4} x}+\frac {12 b^{3} c^{2} d}{a^{5} x}-\frac {5 b^{4} c^{3}}{a^{6} x}-\frac {d^{3}}{2 a^{2} x^{2}}+\frac {3 b c \,d^{2}}{a^{3} x^{2}}-\frac {9 b^{2} c^{2} d}{2 a^{4} x^{2}}+\frac {2 b^{3} c^{3}}{a^{5} x^{2}}-\frac {c \,d^{2}}{a^{2} x^{3}}+\frac {2 b \,c^{2} d}{a^{3} x^{3}}-\frac {b^{2} c^{3}}{a^{4} x^{3}}-\frac {3 c^{2} d}{4 a^{2} x^{4}}+\frac {b \,c^{3}}{2 a^{3} x^{4}}-\frac {c^{3}}{5 a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 332, normalized size = 1.67 \[ -\frac {4 \, a^{5} c^{3} + 60 \, {\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 30 \, {\left (2 \, a b^{4} c^{3} - 5 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 10 \, {\left (2 \, a^{2} b^{3} c^{3} - 5 \, a^{3} b^{2} c^{2} d + 4 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{2} c^{3} - 5 \, a^{4} b c^{2} d + 4 \, a^{5} c d^{2}\right )} x^{2} - 3 \, {\left (2 \, a^{4} b c^{3} - 5 \, a^{5} c^{2} d\right )} x}{20 \, {\left (a^{6} b x^{6} + a^{7} x^{5}\right )}} + \frac {3 \, {\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (b x + a\right )}{a^{7}} - \frac {3 \, {\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \relax (x)}{a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 314, normalized size = 1.58 \[ \frac {6\,b^2\,\mathrm {atanh}\left (\frac {3\,b^2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (-3\,a^3\,b^2\,d^3+12\,a^2\,b^3\,c\,d^2-15\,a\,b^4\,c^2\,d+6\,b^5\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )}{a^7}-\frac {\frac {c^3}{5\,a}+\frac {x^3\,\left (a^3\,d^3-4\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^4}+\frac {3\,c^2\,x\,\left (5\,a\,d-2\,b\,c\right )}{20\,a^2}-\frac {3\,b^2\,x^5\,\left (a^3\,d^3-4\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{a^6}+\frac {c\,x^2\,\left (4\,a^2\,d^2-5\,a\,b\,c\,d+2\,b^2\,c^2\right )}{4\,a^3}-\frac {3\,b\,x^4\,\left (a^3\,d^3-4\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^5}}{b\,x^6+a\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.69, size = 530, normalized size = 2.66 \[ \frac {- 4 a^{5} c^{3} + x^{5} \left (60 a^{3} b^{2} d^{3} - 240 a^{2} b^{3} c d^{2} + 300 a b^{4} c^{2} d - 120 b^{5} c^{3}\right ) + x^{4} \left (30 a^{4} b d^{3} - 120 a^{3} b^{2} c d^{2} + 150 a^{2} b^{3} c^{2} d - 60 a b^{4} c^{3}\right ) + x^{3} \left (- 10 a^{5} d^{3} + 40 a^{4} b c d^{2} - 50 a^{3} b^{2} c^{2} d + 20 a^{2} b^{3} c^{3}\right ) + x^{2} \left (- 20 a^{5} c d^{2} + 25 a^{4} b c^{2} d - 10 a^{3} b^{2} c^{3}\right ) + x \left (- 15 a^{5} c^{2} d + 6 a^{4} b c^{3}\right )}{20 a^{7} x^{5} + 20 a^{6} b x^{6}} + \frac {3 b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{4} b^{2} d^{3} - 12 a^{3} b^{3} c d^{2} + 15 a^{2} b^{4} c^{2} d - 6 a b^{5} c^{3} - 3 a b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2}}{6 a^{3} b^{3} d^{3} - 24 a^{2} b^{4} c d^{2} + 30 a b^{5} c^{2} d - 12 b^{6} c^{3}} \right )}}{a^{7}} - \frac {3 b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{4} b^{2} d^{3} - 12 a^{3} b^{3} c d^{2} + 15 a^{2} b^{4} c^{2} d - 6 a b^{5} c^{3} + 3 a b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2}}{6 a^{3} b^{3} d^{3} - 24 a^{2} b^{4} c d^{2} + 30 a b^{5} c^{2} d - 12 b^{6} c^{3}} \right )}}{a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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