Optimal. Leaf size=151 \[ -\frac {d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac {d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac {d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \]
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Rubi [A] time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \begin {gather*} -\frac {d^4 (c+d x)^8}{3960 (a+b x)^8 (b c-a d)^5}+\frac {d^3 (c+d x)^8}{495 (a+b x)^9 (b c-a d)^4}-\frac {d^2 (c+d x)^8}{110 (a+b x)^{10} (b c-a d)^3}+\frac {d (c+d x)^8}{33 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}-\frac {d \int \frac {(c+d x)^7}{(a+b x)^{12}} \, dx}{3 (b c-a d)}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}+\frac {d^2 \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx}{11 (b c-a d)^2}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}-\frac {d^3 \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx}{55 (b c-a d)^3}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac {d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}+\frac {d^4 \int \frac {(c+d x)^7}{(a+b x)^9} \, dx}{495 (b c-a d)^4}\\ &=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac {d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}-\frac {d^4 (c+d x)^8}{3960 (b c-a d)^5 (a+b x)^8}\\ \end {align*}
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Mathematica [B] time = 0.13, size = 371, normalized size = 2.46 \begin {gather*} -\frac {a^7 d^7+a^6 b d^6 (5 c+12 d x)+3 a^5 b^2 d^5 \left (5 c^2+20 c d x+22 d^2 x^2\right )+5 a^4 b^3 d^4 \left (7 c^3+36 c^2 d x+66 c d^2 x^2+44 d^3 x^3\right )+5 a^3 b^4 d^3 \left (14 c^4+84 c^3 d x+198 c^2 d^2 x^2+220 c d^3 x^3+99 d^4 x^4\right )+3 a^2 b^5 d^2 \left (42 c^5+280 c^4 d x+770 c^3 d^2 x^2+1100 c^2 d^3 x^3+825 c d^4 x^4+264 d^5 x^5\right )+a b^6 d \left (210 c^6+1512 c^5 d x+4620 c^4 d^2 x^2+7700 c^3 d^3 x^3+7425 c^2 d^4 x^4+3960 c d^5 x^5+924 d^6 x^6\right )+b^7 \left (330 c^7+2520 c^6 d x+8316 c^5 d^2 x^2+15400 c^4 d^3 x^3+17325 c^3 d^4 x^4+11880 c^2 d^5 x^5+4620 c d^6 x^6+792 d^7 x^7\right )}{3960 b^8 (a+b x)^{12}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.42, size = 581, normalized size = 3.85 \begin {gather*} -\frac {792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \, {\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \, {\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \, {\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \, {\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \, {\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \, {\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \, {\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.26, size = 496, normalized size = 3.28 \begin {gather*} -\frac {792 \, b^{7} d^{7} x^{7} + 4620 \, b^{7} c d^{6} x^{6} + 924 \, a b^{6} d^{7} x^{6} + 11880 \, b^{7} c^{2} d^{5} x^{5} + 3960 \, a b^{6} c d^{6} x^{5} + 792 \, a^{2} b^{5} d^{7} x^{5} + 17325 \, b^{7} c^{3} d^{4} x^{4} + 7425 \, a b^{6} c^{2} d^{5} x^{4} + 2475 \, a^{2} b^{5} c d^{6} x^{4} + 495 \, a^{3} b^{4} d^{7} x^{4} + 15400 \, b^{7} c^{4} d^{3} x^{3} + 7700 \, a b^{6} c^{3} d^{4} x^{3} + 3300 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1100 \, a^{3} b^{4} c d^{6} x^{3} + 220 \, a^{4} b^{3} d^{7} x^{3} + 8316 \, b^{7} c^{5} d^{2} x^{2} + 4620 \, a b^{6} c^{4} d^{3} x^{2} + 2310 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 990 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 330 \, a^{4} b^{3} c d^{6} x^{2} + 66 \, a^{5} b^{2} d^{7} x^{2} + 2520 \, b^{7} c^{6} d x + 1512 \, a b^{6} c^{5} d^{2} x + 840 \, a^{2} b^{5} c^{4} d^{3} x + 420 \, a^{3} b^{4} c^{3} d^{4} x + 180 \, a^{4} b^{3} c^{2} d^{5} x + 60 \, a^{5} b^{2} c d^{6} x + 12 \, a^{6} b d^{7} x + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7}}{3960 \, {\left (b x + a\right )}^{12} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 464, normalized size = 3.07 \begin {gather*} -\frac {d^{7}}{5 \left (b x +a \right )^{5} b^{8}}+\frac {7 \left (a d -b c \right ) d^{6}}{6 \left (b x +a \right )^{6} b^{8}}-\frac {3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d^{5}}{\left (b x +a \right )^{7} b^{8}}+\frac {35 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d^{4}}{8 \left (b x +a \right )^{8} b^{8}}-\frac {35 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) d^{3}}{9 \left (b x +a \right )^{9} b^{8}}+\frac {21 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) d^{2}}{10 \left (b x +a \right )^{10} b^{8}}-\frac {7 \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) d}{11 \left (b x +a \right )^{11} b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} c^{3} d^{4} b^{3}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} c^{5} d^{2} b^{5}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{12 \left (b x +a \right )^{12} b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.76, size = 581, normalized size = 3.85 \begin {gather*} -\frac {792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \, {\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \, {\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \, {\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \, {\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \, {\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \, {\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \, {\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 559, normalized size = 3.70 \begin {gather*} -\frac {\frac {a^7\,d^7+5\,a^6\,b\,c\,d^6+15\,a^5\,b^2\,c^2\,d^5+35\,a^4\,b^3\,c^3\,d^4+70\,a^3\,b^4\,c^4\,d^3+126\,a^2\,b^5\,c^5\,d^2+210\,a\,b^6\,c^6\,d+330\,b^7\,c^7}{3960\,b^8}+\frac {d^7\,x^7}{5\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+5\,a^4\,b\,c\,d^4+15\,a^3\,b^2\,c^2\,d^3+35\,a^2\,b^3\,c^3\,d^2+70\,a\,b^4\,c^4\,d+126\,b^5\,c^5\right )}{60\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+5\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+35\,b^3\,c^3\right )}{8\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+5\,b\,c\right )}{30\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+5\,a^3\,b\,c\,d^3+15\,a^2\,b^2\,c^2\,d^2+35\,a\,b^3\,c^3\,d+70\,b^4\,c^4\right )}{18\,b^5}+\frac {d\,x\,\left (a^6\,d^6+5\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4+35\,a^3\,b^3\,c^3\,d^3+70\,a^2\,b^4\,c^4\,d^2+126\,a\,b^5\,c^5\,d+210\,b^6\,c^6\right )}{330\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2\right )}{5\,b^3}}{a^{12}+12\,a^{11}\,b\,x+66\,a^{10}\,b^2\,x^2+220\,a^9\,b^3\,x^3+495\,a^8\,b^4\,x^4+792\,a^7\,b^5\,x^5+924\,a^6\,b^6\,x^6+792\,a^5\,b^7\,x^7+495\,a^4\,b^8\,x^8+220\,a^3\,b^9\,x^9+66\,a^2\,b^{10}\,x^{10}+12\,a\,b^{11}\,x^{11}+b^{12}\,x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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