Optimal. Leaf size=187 \[ \frac{7 d^6 (a+b x)^3 (b c-a d)}{3 b^8}+\frac{21 d^5 (a+b x)^2 (b c-a d)^2}{2 b^8}+\frac{35 d^3 (b c-a d)^4 \log (a+b x)}{b^8}-\frac{21 d^2 (b c-a d)^5}{b^8 (a+b x)}-\frac{7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac{(b c-a d)^7}{3 b^8 (a+b x)^3}+\frac{d^7 (a+b x)^4}{4 b^8}+\frac{35 d^4 x (b c-a d)^3}{b^7} \]
[Out]
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Rubi [A] time = 0.476113, antiderivative size = 187, 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 \[ \frac{7 d^6 (a+b x)^3 (b c-a d)}{3 b^8}+\frac{21 d^5 (a+b x)^2 (b c-a d)^2}{2 b^8}+\frac{35 d^3 (b c-a d)^4 \log (a+b x)}{b^8}-\frac{21 d^2 (b c-a d)^5}{b^8 (a+b x)}-\frac{7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac{(b c-a d)^7}{3 b^8 (a+b x)^3}+\frac{d^7 (a+b x)^4}{4 b^8}+\frac{35 d^4 x (b c-a d)^3}{b^7} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^7/(a + b*x)^4,x]
[Out]
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Rubi in Sympy [A] time = 71.1405, size = 172, normalized size = 0.92 \[ - \frac{35 d^{4} x \left (a d - b c\right )^{3}}{b^{7}} + \frac{d^{7} \left (a + b x\right )^{4}}{4 b^{8}} - \frac{7 d^{6} \left (a + b x\right )^{3} \left (a d - b c\right )}{3 b^{8}} + \frac{21 d^{5} \left (a + b x\right )^{2} \left (a d - b c\right )^{2}}{2 b^{8}} + \frac{35 d^{3} \left (a d - b c\right )^{4} \log{\left (a + b x \right )}}{b^{8}} + \frac{21 d^{2} \left (a d - b c\right )^{5}}{b^{8} \left (a + b x\right )} - \frac{7 d \left (a d - b c\right )^{6}}{2 b^{8} \left (a + b x\right )^{2}} + \frac{\left (a d - b c\right )^{7}}{3 b^{8} \left (a + b x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**7/(b*x+a)**4,x)
[Out]
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Mathematica [A] time = 0.1722, size = 199, normalized size = 1.06 \[ \frac{6 b^2 d^5 x^2 \left (10 a^2 d^2-28 a b c d+21 b^2 c^2\right )+12 b d^4 x \left (-20 a^3 d^3+70 a^2 b c d^2-84 a b^2 c^2 d+35 b^3 c^3\right )+4 b^3 d^6 x^3 (7 b c-4 a d)+420 d^3 (b c-a d)^4 \log (a+b x)+\frac{252 d^2 (a d-b c)^5}{a+b x}-\frac{42 d (b c-a d)^6}{(a+b x)^2}-\frac{4 (b c-a d)^7}{(a+b x)^3}+3 b^4 d^7 x^4}{12 b^8} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^7/(a + b*x)^4,x]
[Out]
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Maple [B] time = 0.018, size = 622, normalized size = 3.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^7/(b*x+a)^4,x)
[Out]
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Maxima [A] time = 1.37739, size = 653, normalized size = 3.49 \[ -\frac{2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \,{\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 21 \,{\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \,{\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} + \frac{3 \, b^{3} d^{7} x^{4} + 4 \,{\left (7 \, b^{3} c d^{6} - 4 \, a b^{2} d^{7}\right )} x^{3} + 6 \,{\left (21 \, b^{3} c^{2} d^{5} - 28 \, a b^{2} c d^{6} + 10 \, a^{2} b d^{7}\right )} x^{2} + 12 \,{\left (35 \, b^{3} c^{3} d^{4} - 84 \, a b^{2} c^{2} d^{5} + 70 \, a^{2} b c d^{6} - 20 \, a^{3} d^{7}\right )} x}{12 \, b^{7}} + \frac{35 \,{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^7/(b*x + a)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20226, size = 998, normalized size = 5.34 \[ \frac{3 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 84 \, a^{2} b^{5} c^{5} d^{2} + 770 \, a^{3} b^{4} c^{4} d^{3} - 1820 \, a^{4} b^{3} c^{3} d^{4} + 1974 \, a^{5} b^{2} c^{2} d^{5} - 1036 \, a^{6} b c d^{6} + 214 \, a^{7} d^{7} + 7 \,{\left (4 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 21 \,{\left (6 \, b^{7} c^{2} d^{5} - 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \,{\left (4 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 2 \,{\left (630 \, a b^{6} c^{3} d^{4} - 1323 \, a^{2} b^{5} c^{2} d^{5} + 1022 \, a^{3} b^{4} c d^{6} - 278 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \,{\left (42 \, b^{7} c^{5} d^{2} - 210 \, a b^{6} c^{4} d^{3} + 210 \, a^{2} b^{5} c^{3} d^{4} + 63 \, a^{3} b^{4} c^{2} d^{5} - 182 \, a^{4} b^{3} c d^{6} + 68 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \,{\left (7 \, b^{7} c^{6} d + 42 \, a b^{6} c^{5} d^{2} - 315 \, a^{2} b^{5} c^{4} d^{3} + 630 \, a^{3} b^{4} c^{3} d^{4} - 567 \, a^{4} b^{3} c^{2} d^{5} + 238 \, a^{5} b^{2} c d^{6} - 37 \, a^{6} b d^{7}\right )} x + 420 \,{\left (a^{3} b^{4} c^{4} d^{3} - 4 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} - 4 \, a^{6} b c d^{6} + a^{7} d^{7} +{\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 3 \,{\left (a b^{6} c^{4} d^{3} - 4 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} - 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} d^{3} - 4 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} - 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^7/(b*x + a)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.3712, size = 468, normalized size = 2.5 \[ \frac{107 a^{7} d^{7} - 518 a^{6} b c d^{6} + 987 a^{5} b^{2} c^{2} d^{5} - 910 a^{4} b^{3} c^{3} d^{4} + 385 a^{3} b^{4} c^{4} d^{3} - 42 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 2 b^{7} c^{7} + x^{2} \left (126 a^{5} b^{2} d^{7} - 630 a^{4} b^{3} c d^{6} + 1260 a^{3} b^{4} c^{2} d^{5} - 1260 a^{2} b^{5} c^{3} d^{4} + 630 a b^{6} c^{4} d^{3} - 126 b^{7} c^{5} d^{2}\right ) + x \left (231 a^{6} b d^{7} - 1134 a^{5} b^{2} c d^{6} + 2205 a^{4} b^{3} c^{2} d^{5} - 2100 a^{3} b^{4} c^{3} d^{4} + 945 a^{2} b^{5} c^{4} d^{3} - 126 a b^{6} c^{5} d^{2} - 21 b^{7} c^{6} d\right )}{6 a^{3} b^{8} + 18 a^{2} b^{9} x + 18 a b^{10} x^{2} + 6 b^{11} x^{3}} + \frac{d^{7} x^{4}}{4 b^{4}} - \frac{x^{3} \left (4 a d^{7} - 7 b c d^{6}\right )}{3 b^{5}} + \frac{x^{2} \left (10 a^{2} d^{7} - 28 a b c d^{6} + 21 b^{2} c^{2} d^{5}\right )}{2 b^{6}} - \frac{x \left (20 a^{3} d^{7} - 70 a^{2} b c d^{6} + 84 a b^{2} c^{2} d^{5} - 35 b^{3} c^{3} d^{4}\right )}{b^{7}} + \frac{35 d^{3} \left (a d - b c\right )^{4} \log{\left (a + b x \right )}}{b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**7/(b*x+a)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.227247, size = 635, normalized size = 3.4 \[ \frac{35 \,{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{8}} - \frac{2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \,{\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 21 \,{\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \,{\left (b x + a\right )}^{3} b^{8}} + \frac{3 \, b^{12} d^{7} x^{4} + 28 \, b^{12} c d^{6} x^{3} - 16 \, a b^{11} d^{7} x^{3} + 126 \, b^{12} c^{2} d^{5} x^{2} - 168 \, a b^{11} c d^{6} x^{2} + 60 \, a^{2} b^{10} d^{7} x^{2} + 420 \, b^{12} c^{3} d^{4} x - 1008 \, a b^{11} c^{2} d^{5} x + 840 \, a^{2} b^{10} c d^{6} x - 240 \, a^{3} b^{9} d^{7} x}{12 \, b^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^7/(b*x + a)^4,x, algorithm="giac")
[Out]