Optimal. Leaf size=187 \[ \frac {35 d^4 (b c-a d)^3 x}{b^7}-\frac {(b c-a d)^7}{3 b^8 (a+b x)^3}-\frac {7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac {21 d^2 (b c-a d)^5}{b^8 (a+b x)}+\frac {21 d^5 (b c-a d)^2 (a+b x)^2}{2 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^3}{3 b^8}+\frac {d^7 (a+b x)^4}{4 b^8}+\frac {35 d^3 (b c-a d)^4 \log (a+b x)}{b^8} \]
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Rubi [A]
time = 0.15, 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 = {45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^7}{(a+b x)^4} \, dx &=\int \left (\frac {35 d^4 (b c-a d)^3}{b^7}+\frac {(b c-a d)^7}{b^7 (a+b x)^4}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^3}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^2}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)}+\frac {21 d^5 (b c-a d)^2 (a+b x)}{b^7}+\frac {7 d^6 (b c-a d) (a+b x)^2}{b^7}+\frac {d^7 (a+b x)^3}{b^7}\right ) \, dx\\ &=\frac {35 d^4 (b c-a d)^3 x}{b^7}-\frac {(b c-a d)^7}{3 b^8 (a+b x)^3}-\frac {7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac {21 d^2 (b c-a d)^5}{b^8 (a+b x)}+\frac {21 d^5 (b c-a d)^2 (a+b x)^2}{2 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^3}{3 b^8}+\frac {d^7 (a+b x)^4}{4 b^8}+\frac {35 d^3 (b c-a d)^4 \log (a+b x)}{b^8}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 199, normalized size = 1.06 \begin {gather*} \frac {12 b d^4 \left (35 b^3 c^3-84 a b^2 c^2 d+70 a^2 b c d^2-20 a^3 d^3\right ) x+6 b^2 d^5 \left (21 b^2 c^2-28 a b c d+10 a^2 d^2\right ) x^2+4 b^3 d^6 (7 b c-4 a d) x^3+3 b^4 d^7 x^4-\frac {4 (b c-a d)^7}{(a+b x)^3}-\frac {42 d (b c-a d)^6}{(a+b x)^2}+\frac {252 d^2 (-b c+a d)^5}{a+b x}+420 d^3 (b c-a d)^4 \log (a+b x)}{12 b^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs.
\(2(177)=354\).
time = 0.14, size = 459, normalized size = 2.45
method | result | size |
norman | \(\frac {\frac {385 a^{7} d^{7}-1540 a^{6} b c \,d^{6}+2310 a^{5} b^{2} c^{2} d^{5}-1540 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}}{6 b^{8}}+\frac {d^{7} x^{7}}{4 b}+\frac {3 \left (35 a^{5} d^{7}-140 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}-140 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}-7 b^{5} c^{5} d^{2}\right ) x^{2}}{b^{6}}+\frac {\left (315 a^{6} d^{7}-1260 a^{5} b c \,d^{6}+1890 a^{4} b^{2} c^{2} d^{5}-1260 a^{3} b^{3} c^{3} d^{4}+315 a^{2} b^{4} c^{4} d^{3}-42 a \,b^{5} c^{5} d^{2}-7 b^{6} c^{6} d \right ) x}{2 b^{7}}-\frac {35 d^{4} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}+\frac {7 d^{5} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{5}}{4 b^{3}}-\frac {7 d^{6} \left (a d -4 b c \right ) x^{6}}{12 b^{2}}}{\left (b x +a \right )^{3}}+\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(448\) |
default | \(-\frac {d^{4} \left (-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {4}{3} a \,b^{2} d^{3} x^{3}-\frac {7}{3} b^{3} c \,d^{2} x^{3}-5 a^{2} b \,d^{3} x^{2}+14 a \,b^{2} c \,d^{2} x^{2}-\frac {21}{2} b^{3} c^{2} d \,x^{2}+20 a^{3} d^{3} x -70 a^{2} b c \,d^{2} x +84 a \,b^{2} c^{2} d x -35 b^{3} c^{3} x \right )}{b^{7}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{8} \left (b x +a \right )}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{3 b^{8} \left (b x +a \right )^{3}}\) | \(459\) |
risch | \(\frac {d^{7} x^{4}}{4 b^{4}}-\frac {4 d^{7} a \,x^{3}}{3 b^{5}}+\frac {7 d^{6} c \,x^{3}}{3 b^{4}}+\frac {5 d^{7} a^{2} x^{2}}{b^{6}}-\frac {14 d^{6} a c \,x^{2}}{b^{5}}+\frac {21 d^{5} c^{2} x^{2}}{2 b^{4}}-\frac {20 d^{7} a^{3} x}{b^{7}}+\frac {70 d^{6} a^{2} c x}{b^{6}}-\frac {84 d^{5} a \,c^{2} x}{b^{5}}+\frac {35 d^{4} c^{3} x}{b^{4}}+\frac {\left (21 a^{5} b \,d^{7}-105 a^{4} b^{2} c \,d^{6}+210 a^{3} b^{3} c^{2} d^{5}-210 a^{2} b^{4} c^{3} d^{4}+105 a \,b^{5} c^{4} d^{3}-21 b^{6} c^{5} d^{2}\right ) x^{2}+\frac {7 d \left (11 a^{6} d^{6}-54 a^{5} b c \,d^{5}+105 a^{4} b^{2} c^{2} d^{4}-100 a^{3} b^{3} c^{3} d^{3}+45 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d -b^{6} c^{6}\right ) x}{2}+\frac {107 a^{7} d^{7}-518 a^{6} b c \,d^{6}+987 a^{5} b^{2} c^{2} d^{5}-910 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}}{6 b}}{b^{7} \left (b x +a \right )^{3}}+\frac {35 d^{7} \ln \left (b x +a \right ) a^{4}}{b^{8}}-\frac {140 d^{6} \ln \left (b x +a \right ) a^{3} c}{b^{7}}+\frac {210 d^{5} \ln \left (b x +a \right ) a^{2} c^{2}}{b^{6}}-\frac {140 d^{4} \ln \left (b x +a \right ) a \,c^{3}}{b^{5}}+\frac {35 d^{3} \ln \left (b x +a \right ) c^{4}}{b^{4}}\) | \(488\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 484 vs.
\(2 (177) = 354\).
time = 0.30, size = 484, normalized size = 2.59 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 739 vs.
\(2 (177) = 354\).
time = 0.81, size = 739, normalized size = 3.95 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs.
\(2 (172) = 344\).
time = 22.92, size = 474, normalized size = 2.53 \begin {gather*} x^{3} \left (- \frac {4 a d^{7}}{3 b^{5}} + \frac {7 c d^{6}}{3 b^{4}}\right ) + x^{2} \cdot \left (\frac {5 a^{2} d^{7}}{b^{6}} - \frac {14 a c d^{6}}{b^{5}} + \frac {21 c^{2} d^{5}}{2 b^{4}}\right ) + x \left (- \frac {20 a^{3} d^{7}}{b^{7}} + \frac {70 a^{2} c d^{6}}{b^{6}} - \frac {84 a c^{2} d^{5}}{b^{5}} + \frac {35 c^{3} d^{4}}{b^{4}}\right ) + \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} \cdot \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 {35 d^{3} \left (a d - b c\right )^{4} \log {\left (a + b x \right )}}{b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 470 vs.
\(2 (177) = 354\).
time = 0.74, size = 470, normalized size = 2.51 \begin {gather*} \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 ({\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 559, normalized size = 2.99 \begin {gather*} x^2\,\left (\frac {2\,a\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b}-\frac {3\,a^2\,d^7}{b^6}+\frac {21\,c^2\,d^5}{2\,b^4}\right )-x^3\,\left (\frac {4\,a\,d^7}{3\,b^5}-\frac {7\,c\,d^6}{3\,b^4}\right )-\frac {\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}{6\,b}+x\,\left (-\frac {77\,a^6\,d^7}{2}+189\,a^5\,b\,c\,d^6-\frac {735\,a^4\,b^2\,c^2\,d^5}{2}+350\,a^3\,b^3\,c^3\,d^4-\frac {315\,a^2\,b^4\,c^4\,d^3}{2}+21\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{2}\right )-x^2\,\left (21\,a^5\,b\,d^7-105\,a^4\,b^2\,c\,d^6+210\,a^3\,b^3\,c^2\,d^5-210\,a^2\,b^4\,c^3\,d^4+105\,a\,b^5\,c^4\,d^3-21\,b^6\,c^5\,d^2\right )}{a^3\,b^7+3\,a^2\,b^8\,x+3\,a\,b^9\,x^2+b^{10}\,x^3}-x\,\left (\frac {4\,a\,\left (\frac {4\,a\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b}-\frac {6\,a^2\,d^7}{b^6}+\frac {21\,c^2\,d^5}{b^4}\right )}{b}+\frac {4\,a^3\,d^7}{b^7}-\frac {35\,c^3\,d^4}{b^4}-\frac {6\,a^2\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (35\,a^4\,d^7-140\,a^3\,b\,c\,d^6+210\,a^2\,b^2\,c^2\,d^5-140\,a\,b^3\,c^3\,d^4+35\,b^4\,c^4\,d^3\right )}{b^8}+\frac {d^7\,x^4}{4\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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