Optimal. Leaf size=200 \[ -\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac {3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac {a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac {x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}+\frac {3 d x^4 (b c-a d)^2}{4 b^4}+\frac {d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac {d^3 x^6}{6 b^2} \]
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Rubi [A] time = 0.27, antiderivative size = 200, 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} \[ \frac {3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac {d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac {3 d x^4 (b c-a d)^2}{4 b^4}+\frac {x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}-\frac {a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac {d^3 x^6}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx &=\int \left (-\frac {3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^7}+\frac {a (-b c+a d)^2 (-2 b c+5 a d) x}{b^6}+\frac {(b c-4 a d) (b c-a d)^2 x^2}{b^5}+\frac {3 d (b c-a d)^2 x^3}{b^4}+\frac {d^2 (3 b c-2 a d) x^4}{b^3}+\frac {d^3 x^5}{b^2}-\frac {a^4 (-b c+a d)^3}{b^7 (a+b x)^2}+\frac {a^3 (-b c+a d)^2 (-4 b c+7 a d)}{b^7 (a+b x)}\right ) \, dx\\ &=\frac {3 a^2 (b c-2 a d) (b c-a d)^2 x}{b^7}-\frac {a (2 b c-5 a d) (b c-a d)^2 x^2}{2 b^6}+\frac {(b c-4 a d) (b c-a d)^2 x^3}{3 b^5}+\frac {3 d (b c-a d)^2 x^4}{4 b^4}+\frac {d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac {d^3 x^6}{6 b^2}-\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 190, normalized size = 0.95 \[ \frac {\frac {60 a^4 (a d-b c)^3}{a+b x}+60 a^3 (b c-a d)^2 (7 a d-4 b c) \log (a+b x)-180 a^2 b x (b c-a d)^2 (2 a d-b c)+12 b^5 d^2 x^5 (3 b c-2 a d)+45 b^4 d x^4 (b c-a d)^2+20 b^3 x^3 (b c-4 a d) (b c-a d)^2+30 a b^2 x^2 (b c-a d)^2 (5 a d-2 b c)+10 b^6 d^3 x^6}{60 b^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 421, normalized size = 2.10 \[ \frac {10 \, b^{7} d^{3} x^{7} - 60 \, a^{4} b^{3} c^{3} + 180 \, a^{5} b^{2} c^{2} d - 180 \, a^{6} b c d^{2} + 60 \, a^{7} d^{3} + 2 \, {\left (18 \, b^{7} c d^{2} - 7 \, a b^{6} d^{3}\right )} x^{6} + 3 \, {\left (15 \, b^{7} c^{2} d - 18 \, a b^{6} c d^{2} + 7 \, a^{2} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (4 \, b^{7} c^{3} - 15 \, a b^{6} c^{2} d + 18 \, a^{2} b^{5} c d^{2} - 7 \, a^{3} b^{4} d^{3}\right )} x^{4} - 10 \, {\left (4 \, a b^{6} c^{3} - 15 \, a^{2} b^{5} c^{2} d + 18 \, a^{3} b^{4} c d^{2} - 7 \, a^{4} b^{3} d^{3}\right )} x^{3} + 30 \, {\left (4 \, a^{2} b^{5} c^{3} - 15 \, a^{3} b^{4} c^{2} d + 18 \, a^{4} b^{3} c d^{2} - 7 \, a^{5} b^{2} d^{3}\right )} x^{2} + 180 \, {\left (a^{3} b^{4} c^{3} - 4 \, a^{4} b^{3} c^{2} d + 5 \, a^{5} b^{2} c d^{2} - 2 \, a^{6} b d^{3}\right )} x - 60 \, {\left (4 \, a^{4} b^{3} c^{3} - 15 \, a^{5} b^{2} c^{2} d + 18 \, a^{6} b c d^{2} - 7 \, a^{7} d^{3} + {\left (4 \, a^{3} b^{4} c^{3} - 15 \, a^{4} b^{3} c^{2} d + 18 \, a^{5} b^{2} c d^{2} - 7 \, a^{6} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{9} x + a b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.02, size = 403, normalized size = 2.02 \[ \frac {{\left (10 \, d^{3} + \frac {12 \, {\left (3 \, b^{2} c d^{2} - 7 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {45 \, {\left (b^{4} c^{2} d - 6 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {20 \, {\left (b^{6} c^{3} - 15 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 35 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} - \frac {30 \, {\left (4 \, a b^{7} c^{3} - 30 \, a^{2} b^{6} c^{2} d + 60 \, a^{3} b^{5} c d^{2} - 35 \, a^{4} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {180 \, {\left (2 \, a^{2} b^{8} c^{3} - 10 \, a^{3} b^{7} c^{2} d + 15 \, a^{4} b^{6} c d^{2} - 7 \, a^{5} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )} {\left (b x + a\right )}^{6}}{60 \, b^{8}} + \frac {{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {a^{4} b^{9} c^{3}}{b x + a} - \frac {3 \, a^{5} b^{8} c^{2} d}{b x + a} + \frac {3 \, a^{6} b^{7} c d^{2}}{b x + a} - \frac {a^{7} b^{6} d^{3}}{b x + a}}{b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 378, normalized size = 1.89 \[ \frac {d^{3} x^{6}}{6 b^{2}}-\frac {2 a \,d^{3} x^{5}}{5 b^{3}}+\frac {3 c \,d^{2} x^{5}}{5 b^{2}}+\frac {3 a^{2} d^{3} x^{4}}{4 b^{4}}-\frac {3 a c \,d^{2} x^{4}}{2 b^{3}}+\frac {3 c^{2} d \,x^{4}}{4 b^{2}}-\frac {4 a^{3} d^{3} x^{3}}{3 b^{5}}+\frac {3 a^{2} c \,d^{2} x^{3}}{b^{4}}-\frac {2 a \,c^{2} d \,x^{3}}{b^{3}}+\frac {c^{3} x^{3}}{3 b^{2}}+\frac {5 a^{4} d^{3} x^{2}}{2 b^{6}}-\frac {6 a^{3} c \,d^{2} x^{2}}{b^{5}}+\frac {9 a^{2} c^{2} d \,x^{2}}{2 b^{4}}-\frac {a \,c^{3} x^{2}}{b^{3}}+\frac {a^{7} d^{3}}{\left (b x +a \right ) b^{8}}-\frac {3 a^{6} c \,d^{2}}{\left (b x +a \right ) b^{7}}+\frac {7 a^{6} d^{3} \ln \left (b x +a \right )}{b^{8}}+\frac {3 a^{5} c^{2} d}{\left (b x +a \right ) b^{6}}-\frac {18 a^{5} c \,d^{2} \ln \left (b x +a \right )}{b^{7}}-\frac {6 a^{5} d^{3} x}{b^{7}}-\frac {a^{4} c^{3}}{\left (b x +a \right ) b^{5}}+\frac {15 a^{4} c^{2} d \ln \left (b x +a \right )}{b^{6}}+\frac {15 a^{4} c \,d^{2} x}{b^{6}}-\frac {4 a^{3} c^{3} \ln \left (b x +a \right )}{b^{5}}-\frac {12 a^{3} c^{2} d x}{b^{5}}+\frac {3 a^{2} c^{3} x}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 323, normalized size = 1.62 \[ -\frac {a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}}{b^{9} x + a b^{8}} + \frac {10 \, b^{5} d^{3} x^{6} + 12 \, {\left (3 \, b^{5} c d^{2} - 2 \, a b^{4} d^{3}\right )} x^{5} + 45 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} - 6 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 4 \, a^{3} b^{2} d^{3}\right )} x^{3} - 30 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2} + 180 \, {\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} x}{60 \, b^{7}} - \frac {{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 688, normalized size = 3.44 \[ x^3\,\left (\frac {c^3}{3\,b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{3\,b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{3\,b^2}\right )-x^5\,\left (\frac {2\,a\,d^3}{5\,b^3}-\frac {3\,c\,d^2}{5\,b^2}\right )-x\,\left (\frac {a^2\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b^2}+\frac {2\,a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )}{b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{2\,b}-\frac {a^2\,d^3}{4\,b^4}\right )-x^2\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{2\,b^2}+\frac {a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (7\,a^6\,d^3-18\,a^5\,b\,c\,d^2+15\,a^4\,b^2\,c^2\,d-4\,a^3\,b^3\,c^3\right )}{b^8}+\frac {a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d-a^4\,b^3\,c^3}{b\,\left (x\,b^8+a\,b^7\right )}+\frac {d^3\,x^6}{6\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 323, normalized size = 1.62 \[ \frac {a^{3} \left (a d - b c\right )^{2} \left (7 a d - 4 b c\right ) \log {\left (a + b x \right )}}{b^{8}} + x^{5} \left (- \frac {2 a d^{3}}{5 b^{3}} + \frac {3 c d^{2}}{5 b^{2}}\right ) + x^{4} \left (\frac {3 a^{2} d^{3}}{4 b^{4}} - \frac {3 a c d^{2}}{2 b^{3}} + \frac {3 c^{2} d}{4 b^{2}}\right ) + x^{3} \left (- \frac {4 a^{3} d^{3}}{3 b^{5}} + \frac {3 a^{2} c d^{2}}{b^{4}} - \frac {2 a c^{2} d}{b^{3}} + \frac {c^{3}}{3 b^{2}}\right ) + x^{2} \left (\frac {5 a^{4} d^{3}}{2 b^{6}} - \frac {6 a^{3} c d^{2}}{b^{5}} + \frac {9 a^{2} c^{2} d}{2 b^{4}} - \frac {a c^{3}}{b^{3}}\right ) + x \left (- \frac {6 a^{5} d^{3}}{b^{7}} + \frac {15 a^{4} c d^{2}}{b^{6}} - \frac {12 a^{3} c^{2} d}{b^{5}} + \frac {3 a^{2} c^{3}}{b^{4}}\right ) + \frac {a^{7} d^{3} - 3 a^{6} b c d^{2} + 3 a^{5} b^{2} c^{2} d - a^{4} b^{3} c^{3}}{a b^{8} + b^{9} x} + \frac {d^{3} x^{6}}{6 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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