Optimal. Leaf size=198 \[ -\frac{d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac{21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac{35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac{7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac{21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac{7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac{(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac{d^7}{6 b^8 (a+b x)^6} \]
[Out]
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Rubi [A] time = 0.411383, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac{21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac{35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac{7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac{21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac{7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac{(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac{d^7}{6 b^8 (a+b x)^6} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^7/(a + b*x)^14,x]
[Out]
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Rubi in Sympy [A] time = 81.8883, size = 180, normalized size = 0.91 \[ - \frac{d^{7}}{6 b^{8} \left (a + b x\right )^{6}} + \frac{d^{6} \left (a d - b c\right )}{b^{8} \left (a + b x\right )^{7}} - \frac{21 d^{5} \left (a d - b c\right )^{2}}{8 b^{8} \left (a + b x\right )^{8}} + \frac{35 d^{4} \left (a d - b c\right )^{3}}{9 b^{8} \left (a + b x\right )^{9}} - \frac{7 d^{3} \left (a d - b c\right )^{4}}{2 b^{8} \left (a + b x\right )^{10}} + \frac{21 d^{2} \left (a d - b c\right )^{5}}{11 b^{8} \left (a + b x\right )^{11}} - \frac{7 d \left (a d - b c\right )^{6}}{12 b^{8} \left (a + b x\right )^{12}} + \frac{\left (a d - b c\right )^{7}}{13 b^{8} \left (a + b x\right )^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**7/(b*x+a)**14,x)
[Out]
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Mathematica [A] time = 0.245387, size = 369, normalized size = 1.86 \[ -\frac{a^7 d^7+a^6 b d^6 (6 c+13 d x)+3 a^5 b^2 d^5 \left (7 c^2+26 c d x+26 d^2 x^2\right )+a^4 b^3 d^4 \left (56 c^3+273 c^2 d x+468 c d^2 x^2+286 d^3 x^3\right )+a^3 b^4 d^3 \left (126 c^4+728 c^3 d x+1638 c^2 d^2 x^2+1716 c d^3 x^3+715 d^4 x^4\right )+3 a^2 b^5 d^2 \left (84 c^5+546 c^4 d x+1456 c^3 d^2 x^2+2002 c^2 d^3 x^3+1430 c d^4 x^4+429 d^5 x^5\right )+a b^6 d \left (462 c^6+3276 c^5 d x+9828 c^4 d^2 x^2+16016 c^3 d^3 x^3+15015 c^2 d^4 x^4+7722 c d^5 x^5+1716 d^6 x^6\right )+b^7 \left (792 c^7+6006 c^6 d x+19656 c^5 d^2 x^2+36036 c^4 d^3 x^3+40040 c^3 d^4 x^4+27027 c^2 d^5 x^5+10296 c d^6 x^6+1716 d^7 x^7\right )}{10296 b^8 (a+b x)^{13}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^7/(a + b*x)^14,x]
[Out]
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Maple [B] time = 0.011, size = 463, normalized size = 2.3 \[ -{\frac{{d}^{7}}{6\,{b}^{8} \left ( bx+a \right ) ^{6}}}-{\frac{7\,d \left ({a}^{6}{d}^{6}-6\,{a}^{5}bc{d}^{5}+15\,{a}^{4}{b}^{2}{c}^{2}{d}^{4}-20\,{a}^{3}{b}^{3}{c}^{3}{d}^{3}+15\,{a}^{2}{b}^{4}{c}^{4}{d}^{2}-6\,a{b}^{5}{c}^{5}d+{b}^{6}{c}^{6} \right ) }{12\,{b}^{8} \left ( bx+a \right ) ^{12}}}+{\frac{{d}^{6} \left ( ad-bc \right ) }{{b}^{8} \left ( bx+a \right ) ^{7}}}-{\frac{21\,{d}^{5} \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ) }{8\,{b}^{8} \left ( bx+a \right ) ^{8}}}-{\frac{7\,{d}^{3} \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) }{2\,{b}^{8} \left ( bx+a \right ) ^{10}}}+{\frac{21\,{d}^{2} \left ({a}^{5}{d}^{5}-5\,{a}^{4}bc{d}^{4}+10\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-10\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+5\,a{b}^{4}{c}^{4}d-{b}^{5}{c}^{5} \right ) }{11\,{b}^{8} \left ( bx+a \right ) ^{11}}}-{\frac{-{a}^{7}{d}^{7}+7\,c{d}^{6}{a}^{6}b-21\,{a}^{5}{c}^{2}{d}^{5}{b}^{2}+35\,{a}^{4}{b}^{3}{c}^{3}{d}^{4}-35\,{a}^{3}{b}^{4}{c}^{4}{d}^{3}+21\,{a}^{2}{c}^{5}{d}^{2}{b}^{5}-7\,a{b}^{6}{c}^{6}d+{c}^{7}{b}^{7}}{13\,{b}^{8} \left ( bx+a \right ) ^{13}}}+{\frac{35\,{d}^{4} \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ) }{9\,{b}^{8} \left ( bx+a \right ) ^{9}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^7/(b*x+a)^14,x)
[Out]
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Maxima [A] time = 1.39982, size = 799, normalized size = 4.04 \[ -\frac{1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \,{\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \,{\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \,{\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \,{\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \,{\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \,{\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \,{\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^7/(b*x + a)^14,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229904, size = 799, normalized size = 4.04 \[ -\frac{1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \,{\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \,{\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \,{\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \,{\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \,{\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \,{\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \,{\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^7/(b*x + a)^14,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**7/(b*x+a)**14,x)
[Out]
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GIAC/XCAS [A] time = 0.225605, size = 670, normalized size = 3.38 \[ -\frac{1716 \, b^{7} d^{7} x^{7} + 10296 \, b^{7} c d^{6} x^{6} + 1716 \, a b^{6} d^{7} x^{6} + 27027 \, b^{7} c^{2} d^{5} x^{5} + 7722 \, a b^{6} c d^{6} x^{5} + 1287 \, a^{2} b^{5} d^{7} x^{5} + 40040 \, b^{7} c^{3} d^{4} x^{4} + 15015 \, a b^{6} c^{2} d^{5} x^{4} + 4290 \, a^{2} b^{5} c d^{6} x^{4} + 715 \, a^{3} b^{4} d^{7} x^{4} + 36036 \, b^{7} c^{4} d^{3} x^{3} + 16016 \, a b^{6} c^{3} d^{4} x^{3} + 6006 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1716 \, a^{3} b^{4} c d^{6} x^{3} + 286 \, a^{4} b^{3} d^{7} x^{3} + 19656 \, b^{7} c^{5} d^{2} x^{2} + 9828 \, a b^{6} c^{4} d^{3} x^{2} + 4368 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 1638 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 468 \, a^{4} b^{3} c d^{6} x^{2} + 78 \, a^{5} b^{2} d^{7} x^{2} + 6006 \, b^{7} c^{6} d x + 3276 \, a b^{6} c^{5} d^{2} x + 1638 \, a^{2} b^{5} c^{4} d^{3} x + 728 \, a^{3} b^{4} c^{3} d^{4} x + 273 \, a^{4} b^{3} c^{2} d^{5} x + 78 \, a^{5} b^{2} c d^{6} x + 13 \, a^{6} b d^{7} x + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7}}{10296 \,{\left (b x + a\right )}^{13} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^7/(b*x + a)^14,x, algorithm="giac")
[Out]