Optimal. Leaf size=187 \[ \frac {7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {21 d^5 x (b c-a d)^2}{b^7} \]
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Rubi [A] time = 0.20, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac {21 d^5 x (b c-a d)^2}{b^7}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac {d^7 (a+b x)^3}{3 b^8} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^5} \, dx &=\int \left (\frac {21 d^5 (b c-a d)^2}{b^7}+\frac {(b c-a d)^7}{b^7 (a+b x)^5}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^4}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^3}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^2}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)}{b^7}+\frac {d^7 (a+b x)^2}{b^7}\right ) \, dx\\ &=\frac {21 d^5 (b c-a d)^2 x}{b^7}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)^2}{2 b^8}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 173, normalized size = 0.93 \[ \frac {12 b d^5 x \left (15 a^2 d^2-35 a b c d+21 b^2 c^2\right )+6 b^2 d^6 x^2 (7 b c-5 a d)+420 d^4 (b c-a d)^3 \log (a+b x)-\frac {420 d^3 (b c-a d)^4}{a+b x}+\frac {126 d^2 (a d-b c)^5}{(a+b x)^2}-\frac {28 d (b c-a d)^6}{(a+b x)^3}-\frac {3 (b c-a d)^7}{(a+b x)^4}+4 b^3 d^7 x^3}{12 b^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 754, normalized size = 4.03 \[ \frac {4 \, b^{7} d^{7} x^{7} - 3 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 105 \, a^{3} b^{4} c^{4} d^{3} + 875 \, a^{4} b^{3} c^{3} d^{4} - 1617 \, a^{5} b^{2} c^{2} d^{5} + 1197 \, a^{6} b c d^{6} - 319 \, a^{7} d^{7} + 14 \, {\left (3 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (3 \, b^{7} c^{2} d^{5} - 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 4 \, {\left (252 \, a b^{6} c^{2} d^{5} - 357 \, a^{2} b^{5} c d^{6} + 139 \, a^{3} b^{4} d^{7}\right )} x^{4} - 4 \, {\left (105 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 252 \, a^{2} b^{5} c^{2} d^{5} + 168 \, a^{3} b^{4} c d^{6} - 136 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \, {\left (21 \, b^{7} c^{5} d^{2} + 105 \, a b^{6} c^{4} d^{3} - 630 \, a^{2} b^{5} c^{3} d^{4} + 882 \, a^{3} b^{4} c^{2} d^{5} - 462 \, a^{4} b^{3} c d^{6} + 74 \, a^{5} b^{2} d^{7}\right )} x^{2} - 4 \, {\left (7 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} - 770 \, a^{3} b^{4} c^{3} d^{4} + 1302 \, a^{4} b^{3} c^{2} d^{5} - 882 \, a^{5} b^{2} c d^{6} + 214 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 4 \, {\left (a b^{6} c^{3} d^{4} - 3 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 6 \, {\left (a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 4 \, {\left (a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 660, normalized size = 3.53 \[ \frac {{\left (2 \, d^{7} + \frac {21 \, {\left (b^{2} c d^{6} - a b d^{7}\right )}}{{\left (b x + a\right )} b} + \frac {126 \, {\left (b^{4} c^{2} d^{5} - 2 \, a b^{3} c d^{6} + a^{2} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{3}}{6 \, b^{8}} - \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {3 \, b^{43} c^{7}}{{\left (b x + a\right )}^{4}} + \frac {28 \, b^{42} c^{6} d}{{\left (b x + a\right )}^{3}} - \frac {21 \, a b^{42} c^{6} d}{{\left (b x + a\right )}^{4}} + \frac {126 \, b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{2}} - \frac {168 \, a b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{3}} + \frac {63 \, a^{2} b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{4}} + \frac {420 \, b^{40} c^{4} d^{3}}{b x + a} - \frac {630 \, a b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{2}} + \frac {420 \, a^{2} b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{3}} - \frac {105 \, a^{3} b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{4}} - \frac {1680 \, a b^{39} c^{3} d^{4}}{b x + a} + \frac {1260 \, a^{2} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{2}} - \frac {560 \, a^{3} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{3}} + \frac {105 \, a^{4} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{4}} + \frac {2520 \, a^{2} b^{38} c^{2} d^{5}}{b x + a} - \frac {1260 \, a^{3} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{2}} + \frac {420 \, a^{4} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{3}} - \frac {63 \, a^{5} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{4}} - \frac {1680 \, a^{3} b^{37} c d^{6}}{b x + a} + \frac {630 \, a^{4} b^{37} c d^{6}}{{\left (b x + a\right )}^{2}} - \frac {168 \, a^{5} b^{37} c d^{6}}{{\left (b x + a\right )}^{3}} + \frac {21 \, a^{6} b^{37} c d^{6}}{{\left (b x + a\right )}^{4}} + \frac {420 \, a^{4} b^{36} d^{7}}{b x + a} - \frac {126 \, a^{5} b^{36} d^{7}}{{\left (b x + a\right )}^{2}} + \frac {28 \, a^{6} b^{36} d^{7}}{{\left (b x + a\right )}^{3}} - \frac {3 \, a^{7} b^{36} d^{7}}{{\left (b x + a\right )}^{4}}}{12 \, b^{44}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 641, normalized size = 3.43 \[ \frac {a^{7} d^{7}}{4 \left (b x +a \right )^{4} b^{8}}-\frac {7 a^{6} c \,d^{6}}{4 \left (b x +a \right )^{4} b^{7}}+\frac {21 a^{5} c^{2} d^{5}}{4 \left (b x +a \right )^{4} b^{6}}-\frac {35 a^{4} c^{3} d^{4}}{4 \left (b x +a \right )^{4} b^{5}}+\frac {35 a^{3} c^{4} d^{3}}{4 \left (b x +a \right )^{4} b^{4}}-\frac {21 a^{2} c^{5} d^{2}}{4 \left (b x +a \right )^{4} b^{3}}+\frac {7 a \,c^{6} d}{4 \left (b x +a \right )^{4} b^{2}}-\frac {c^{7}}{4 \left (b x +a \right )^{4} b}-\frac {7 a^{6} d^{7}}{3 \left (b x +a \right )^{3} b^{8}}+\frac {14 a^{5} c \,d^{6}}{\left (b x +a \right )^{3} b^{7}}-\frac {35 a^{4} c^{2} d^{5}}{\left (b x +a \right )^{3} b^{6}}+\frac {140 a^{3} c^{3} d^{4}}{3 \left (b x +a \right )^{3} b^{5}}-\frac {35 a^{2} c^{4} d^{3}}{\left (b x +a \right )^{3} b^{4}}+\frac {14 a \,c^{5} d^{2}}{\left (b x +a \right )^{3} b^{3}}-\frac {7 c^{6} d}{3 \left (b x +a \right )^{3} b^{2}}+\frac {d^{7} x^{3}}{3 b^{5}}+\frac {21 a^{5} d^{7}}{2 \left (b x +a \right )^{2} b^{8}}-\frac {105 a^{4} c \,d^{6}}{2 \left (b x +a \right )^{2} b^{7}}+\frac {105 a^{3} c^{2} d^{5}}{\left (b x +a \right )^{2} b^{6}}-\frac {105 a^{2} c^{3} d^{4}}{\left (b x +a \right )^{2} b^{5}}+\frac {105 a \,c^{4} d^{3}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {5 a \,d^{7} x^{2}}{2 b^{6}}-\frac {21 c^{5} d^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {7 c \,d^{6} x^{2}}{2 b^{5}}-\frac {35 a^{4} d^{7}}{\left (b x +a \right ) b^{8}}+\frac {140 a^{3} c \,d^{6}}{\left (b x +a \right ) b^{7}}-\frac {35 a^{3} d^{7} \ln \left (b x +a \right )}{b^{8}}-\frac {210 a^{2} c^{2} d^{5}}{\left (b x +a \right ) b^{6}}+\frac {105 a^{2} c \,d^{6} \ln \left (b x +a \right )}{b^{7}}+\frac {15 a^{2} d^{7} x}{b^{7}}+\frac {140 a \,c^{3} d^{4}}{\left (b x +a \right ) b^{5}}-\frac {105 a \,c^{2} d^{5} \ln \left (b x +a \right )}{b^{6}}-\frac {35 a c \,d^{6} x}{b^{6}}-\frac {35 c^{4} d^{3}}{\left (b x +a \right ) b^{4}}+\frac {35 c^{3} d^{4} \ln \left (b x +a \right )}{b^{5}}+\frac {21 c^{2} d^{5} x}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.73, size = 494, normalized size = 2.64 \[ -\frac {3 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} - 875 \, a^{4} b^{3} c^{3} d^{4} + 1617 \, a^{5} b^{2} c^{2} d^{5} - 1197 \, a^{6} b c d^{6} + 319 \, a^{7} d^{7} + 420 \, {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 126 \, {\left (b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} + 50 \, a^{3} b^{4} c^{2} d^{5} - 35 \, a^{4} b^{3} c d^{6} + 9 \, a^{5} b^{2} d^{7}\right )} x^{2} + 28 \, {\left (b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 110 \, a^{3} b^{4} c^{3} d^{4} + 195 \, a^{4} b^{3} c^{2} d^{5} - 141 \, a^{5} b^{2} c d^{6} + 37 \, a^{6} b d^{7}\right )} x}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} + \frac {2 \, b^{2} d^{7} x^{3} + 3 \, {\left (7 \, b^{2} c d^{6} - 5 \, a b d^{7}\right )} x^{2} + 6 \, {\left (21 \, b^{2} c^{2} d^{5} - 35 \, a b c d^{6} + 15 \, a^{2} d^{7}\right )} x}{6 \, b^{7}} + \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.77, size = 512, normalized size = 2.74 \[ x\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^7}{b^6}-\frac {7\,c\,d^6}{b^5}\right )}{b}-\frac {10\,a^2\,d^7}{b^7}+\frac {21\,c^2\,d^5}{b^5}\right )-x^2\,\left (\frac {5\,a\,d^7}{2\,b^6}-\frac {7\,c\,d^6}{2\,b^5}\right )-\frac {\frac {319\,a^7\,d^7-1197\,a^6\,b\,c\,d^6+1617\,a^5\,b^2\,c^2\,d^5-875\,a^4\,b^3\,c^3\,d^4+105\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+3\,b^7\,c^7}{12\,b}+x\,\left (\frac {259\,a^6\,d^7}{3}-329\,a^5\,b\,c\,d^6+455\,a^4\,b^2\,c^2\,d^5-\frac {770\,a^3\,b^3\,c^3\,d^4}{3}+35\,a^2\,b^4\,c^4\,d^3+7\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{3}\right )+x^3\,\left (35\,a^4\,b^2\,d^7-140\,a^3\,b^3\,c\,d^6+210\,a^2\,b^4\,c^2\,d^5-140\,a\,b^5\,c^3\,d^4+35\,b^6\,c^4\,d^3\right )+x^2\,\left (\frac {189\,a^5\,b\,d^7}{2}-\frac {735\,a^4\,b^2\,c\,d^6}{2}+525\,a^3\,b^3\,c^2\,d^5-315\,a^2\,b^4\,c^3\,d^4+\frac {105\,a\,b^5\,c^4\,d^3}{2}+\frac {21\,b^6\,c^5\,d^2}{2}\right )}{a^4\,b^7+4\,a^3\,b^8\,x+6\,a^2\,b^9\,x^2+4\,a\,b^{10}\,x^3+b^{11}\,x^4}-\frac {\ln \left (a+b\,x\right )\,\left (35\,a^3\,d^7-105\,a^2\,b\,c\,d^6+105\,a\,b^2\,c^2\,d^5-35\,b^3\,c^3\,d^4\right )}{b^8}+\frac {d^7\,x^3}{3\,b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 22.44, size = 500, normalized size = 2.67 \[ x^{2} \left (- \frac {5 a d^{7}}{2 b^{6}} + \frac {7 c d^{6}}{2 b^{5}}\right ) + x \left (\frac {15 a^{2} d^{7}}{b^{7}} - \frac {35 a c d^{6}}{b^{6}} + \frac {21 c^{2} d^{5}}{b^{5}}\right ) + \frac {- 319 a^{7} d^{7} + 1197 a^{6} b c d^{6} - 1617 a^{5} b^{2} c^{2} d^{5} + 875 a^{4} b^{3} c^{3} d^{4} - 105 a^{3} b^{4} c^{4} d^{3} - 21 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 3 b^{7} c^{7} + x^{3} \left (- 420 a^{4} b^{3} d^{7} + 1680 a^{3} b^{4} c d^{6} - 2520 a^{2} b^{5} c^{2} d^{5} + 1680 a b^{6} c^{3} d^{4} - 420 b^{7} c^{4} d^{3}\right ) + x^{2} \left (- 1134 a^{5} b^{2} d^{7} + 4410 a^{4} b^{3} c d^{6} - 6300 a^{3} b^{4} c^{2} d^{5} + 3780 a^{2} b^{5} c^{3} d^{4} - 630 a b^{6} c^{4} d^{3} - 126 b^{7} c^{5} d^{2}\right ) + x \left (- 1036 a^{6} b d^{7} + 3948 a^{5} b^{2} c d^{6} - 5460 a^{4} b^{3} c^{2} d^{5} + 3080 a^{3} b^{4} c^{3} d^{4} - 420 a^{2} b^{5} c^{4} d^{3} - 84 a b^{6} c^{5} d^{2} - 28 b^{7} c^{6} d\right )}{12 a^{4} b^{8} + 48 a^{3} b^{9} x + 72 a^{2} b^{10} x^{2} + 48 a b^{11} x^{3} + 12 b^{12} x^{4}} + \frac {d^{7} x^{3}}{3 b^{5}} - \frac {35 d^{4} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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