Optimal. Leaf size=245 \[ \frac {a^7}{2 b^5 (a+b x)^2 (b c-a d)^3}-\frac {a^6 (7 b c-4 a d)}{b^5 (a+b x) (b c-a d)^4}+\frac {3 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^5}-\frac {3 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac {3 x (a d+b c)}{b^4 d^4}-\frac {c^7}{2 d^5 (c+d x)^2 (b c-a d)^3}+\frac {c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}+\frac {x^2}{2 b^3 d^3} \]
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Rubi [A] time = 0.39, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \begin {gather*} -\frac {3 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}+\frac {3 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^5}+\frac {a^7}{2 b^5 (a+b x)^2 (b c-a d)^3}-\frac {a^6 (7 b c-4 a d)}{b^5 (a+b x) (b c-a d)^4}-\frac {3 x (a d+b c)}{b^4 d^4}+\frac {c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}-\frac {c^7}{2 d^5 (c+d x)^2 (b c-a d)^3}+\frac {x^2}{2 b^3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {x^7}{(a+b x)^3 (c+d x)^3} \, dx &=\int \left (-\frac {3 (b c+a d)}{b^4 d^4}+\frac {x}{b^3 d^3}-\frac {a^7}{b^4 (b c-a d)^3 (a+b x)^3}-\frac {a^6 (-7 b c+4 a d)}{b^4 (b c-a d)^4 (a+b x)^2}-\frac {3 a^5 \left (7 b^2 c^2-7 a b c d+2 a^2 d^2\right )}{b^4 (b c-a d)^5 (a+b x)}-\frac {c^7}{d^4 (-b c+a d)^3 (c+d x)^3}-\frac {c^6 (4 b c-7 a d)}{d^4 (-b c+a d)^4 (c+d x)^2}-\frac {3 c^5 \left (2 b^2 c^2-7 a b c d+7 a^2 d^2\right )}{d^4 (-b c+a d)^5 (c+d x)}\right ) \, dx\\ &=-\frac {3 (b c+a d) x}{b^4 d^4}+\frac {x^2}{2 b^3 d^3}+\frac {a^7}{2 b^5 (b c-a d)^3 (a+b x)^2}-\frac {a^6 (7 b c-4 a d)}{b^5 (b c-a d)^4 (a+b x)}-\frac {c^7}{2 d^5 (b c-a d)^3 (c+d x)^2}+\frac {c^6 (4 b c-7 a d)}{d^5 (b c-a d)^4 (c+d x)}-\frac {3 a^5 \left (7 b^2 c^2-7 a b c d+2 a^2 d^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}+\frac {3 c^5 \left (2 b^2 c^2-7 a b c d+7 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^5}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 241, normalized size = 0.98 \begin {gather*} \frac {1}{2} \left (\frac {a^7}{b^5 (a+b x)^2 (b c-a d)^3}+\frac {2 a^6 (4 a d-7 b c)}{b^5 (a+b x) (b c-a d)^4}-\frac {6 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (a d-b c)^5}-\frac {6 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac {6 x (a d+b c)}{b^4 d^4}+\frac {c^7}{d^5 (c+d x)^2 (a d-b c)^3}+\frac {2 c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}+\frac {x^2}{b^3 d^3}\right ) \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 {x^7}{(a+b x)^3 (c+d x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 2.18, size = 1567, normalized size = 6.40
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.62, size = 526, normalized size = 2.15 \begin {gather*} -\frac {3 \, {\left (7 \, a^{5} b^{2} c^{2} - 7 \, a^{6} b c d + 2 \, a^{7} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}} + \frac {3 \, {\left (2 \, b^{2} c^{7} - 7 \, a b c^{6} d + 7 \, a^{2} c^{5} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}} + \frac {b^{3} d^{3} x^{2} - 6 \, b^{3} c d^{2} x - 6 \, a b^{2} d^{3} x}{2 \, b^{6} d^{6}} + \frac {7 \, a^{2} b^{6} c^{8} - 13 \, a^{3} b^{5} c^{7} d - 13 \, a^{7} b c^{3} d^{5} + 7 \, a^{8} c^{2} d^{6} + 2 \, {\left (4 \, b^{8} c^{7} d - 7 \, a b^{7} c^{6} d^{2} - 7 \, a^{6} b^{2} c d^{7} + 4 \, a^{7} b d^{8}\right )} x^{3} + {\left (7 \, b^{8} c^{8} + 3 \, a b^{7} c^{7} d - 28 \, a^{2} b^{6} c^{6} d^{2} - 28 \, a^{6} b^{2} c^{2} d^{6} + 3 \, a^{7} b c d^{7} + 7 \, a^{8} d^{8}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} c^{8} - 9 \, a^{2} b^{6} c^{7} d - 7 \, a^{3} b^{5} c^{6} d^{2} - 7 \, a^{6} b^{2} c^{3} d^{5} - 9 \, a^{7} b c^{2} d^{6} + 7 \, a^{8} c d^{7}\right )} x}{2 \, {\left (b c - a d\right )}^{4} {\left (b x + a\right )}^{2} {\left (d x + c\right )}^{2} b^{5} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 347, normalized size = 1.42 \begin {gather*} \frac {6 a^{7} d^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{5}}-\frac {21 a^{6} c d \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{4}}+\frac {21 a^{5} c^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5} b^{3}}-\frac {21 a^{2} c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{3}}+\frac {21 a b \,c^{6} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{4}}-\frac {6 b^{2} c^{7} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5} d^{5}}+\frac {4 a^{7} d}{\left (a d -b c \right )^{4} \left (b x +a \right ) b^{5}}-\frac {7 a^{6} c}{\left (a d -b c \right )^{4} \left (b x +a \right ) b^{4}}-\frac {7 a \,c^{6}}{\left (a d -b c \right )^{4} \left (d x +c \right ) d^{4}}+\frac {4 b \,c^{7}}{\left (a d -b c \right )^{4} \left (d x +c \right ) d^{5}}-\frac {a^{7}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2} b^{5}}+\frac {c^{7}}{2 \left (a d -b c \right )^{3} \left (d x +c \right )^{2} d^{5}}+\frac {x^{2}}{2 b^{3} d^{3}}-\frac {3 a x}{b^{4} d^{3}}-\frac {3 c x}{b^{3} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.24, size = 841, normalized size = 3.43 \begin {gather*} -\frac {3 \, {\left (7 \, a^{5} b^{2} c^{2} - 7 \, a^{6} b c d + 2 \, a^{7} d^{2}\right )} \log \left (b x + a\right )}{b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}} + \frac {3 \, {\left (2 \, b^{2} c^{7} - 7 \, a b c^{6} d + 7 \, a^{2} c^{5} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} d^{5} - 5 \, a b^{4} c^{4} d^{6} + 10 \, a^{2} b^{3} c^{3} d^{7} - 10 \, a^{3} b^{2} c^{2} d^{8} + 5 \, a^{4} b c d^{9} - a^{5} d^{10}} + \frac {7 \, a^{2} b^{6} c^{8} - 13 \, a^{3} b^{5} c^{7} d - 13 \, a^{7} b c^{3} d^{5} + 7 \, a^{8} c^{2} d^{6} + 2 \, {\left (4 \, b^{8} c^{7} d - 7 \, a b^{7} c^{6} d^{2} - 7 \, a^{6} b^{2} c d^{7} + 4 \, a^{7} b d^{8}\right )} x^{3} + {\left (7 \, b^{8} c^{8} + 3 \, a b^{7} c^{7} d - 28 \, a^{2} b^{6} c^{6} d^{2} - 28 \, a^{6} b^{2} c^{2} d^{6} + 3 \, a^{7} b c d^{7} + 7 \, a^{8} d^{8}\right )} x^{2} + 2 \, {\left (7 \, a b^{7} c^{8} - 9 \, a^{2} b^{6} c^{7} d - 7 \, a^{3} b^{5} c^{6} d^{2} - 7 \, a^{6} b^{2} c^{3} d^{5} - 9 \, a^{7} b c^{2} d^{6} + 7 \, a^{8} c d^{7}\right )} x}{2 \, {\left (a^{2} b^{9} c^{6} d^{5} - 4 \, a^{3} b^{8} c^{5} d^{6} + 6 \, a^{4} b^{7} c^{4} d^{7} - 4 \, a^{5} b^{6} c^{3} d^{8} + a^{6} b^{5} c^{2} d^{9} + {\left (b^{11} c^{4} d^{7} - 4 \, a b^{10} c^{3} d^{8} + 6 \, a^{2} b^{9} c^{2} d^{9} - 4 \, a^{3} b^{8} c d^{10} + a^{4} b^{7} d^{11}\right )} x^{4} + 2 \, {\left (b^{11} c^{5} d^{6} - 3 \, a b^{10} c^{4} d^{7} + 2 \, a^{2} b^{9} c^{3} d^{8} + 2 \, a^{3} b^{8} c^{2} d^{9} - 3 \, a^{4} b^{7} c d^{10} + a^{5} b^{6} d^{11}\right )} x^{3} + {\left (b^{11} c^{6} d^{5} - 9 \, a^{2} b^{9} c^{4} d^{7} + 16 \, a^{3} b^{8} c^{3} d^{8} - 9 \, a^{4} b^{7} c^{2} d^{9} + a^{6} b^{5} d^{11}\right )} x^{2} + 2 \, {\left (a b^{10} c^{6} d^{5} - 3 \, a^{2} b^{9} c^{5} d^{6} + 2 \, a^{3} b^{8} c^{4} d^{7} + 2 \, a^{4} b^{7} c^{3} d^{8} - 3 \, a^{5} b^{6} c^{2} d^{9} + a^{6} b^{5} c d^{10}\right )} x\right )}} + \frac {b d x^{2} - 6 \, {\left (b c + a d\right )} x}{2 \, b^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 880, normalized size = 3.59 \begin {gather*} \frac {\frac {x^3\,\left (a\,d+b\,c\right )\,\left (4\,a^6\,d^6-11\,a^5\,b\,c\,d^5+11\,a^4\,b^2\,c^2\,d^4-11\,a^3\,b^3\,c^3\,d^3+11\,a^2\,b^4\,c^4\,d^2-11\,a\,b^5\,c^5\,d+4\,b^6\,c^6\right )}{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}+\frac {7\,a^8\,c^2\,d^6-13\,a^7\,b\,c^3\,d^5-13\,a^3\,b^5\,c^7\,d+7\,a^2\,b^6\,c^8}{2\,b\,d\,\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 )}+\frac {x^2\,\left (7\,a^8\,d^8+3\,a^7\,b\,c\,d^7-28\,a^6\,b^2\,c^2\,d^6-28\,a^2\,b^6\,c^6\,d^2+3\,a\,b^7\,c^7\,d+7\,b^8\,c^8\right )}{2\,b\,d\,\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 )}+\frac {x\,\left (a\,d+b\,c\right )\,\left (7\,a^7\,c\,d^6-16\,a^6\,b\,c^2\,d^5+9\,a^5\,b^2\,c^3\,d^4-9\,a^4\,b^3\,c^4\,d^3+9\,a^3\,b^4\,c^5\,d^2-16\,a^2\,b^5\,c^6\,d+7\,a\,b^6\,c^7\right )}{b\,d\,\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 )}}{x^3\,\left (2\,c\,b^6\,d^5+2\,a\,b^5\,d^6\right )+x\,\left (2\,a^2\,b^4\,c\,d^5+2\,a\,b^5\,c^2\,d^4\right )+x^2\,\left (a^2\,b^4\,d^6+4\,a\,b^5\,c\,d^5+b^6\,c^2\,d^4\right )+b^6\,d^6\,x^4+a^2\,b^4\,c^2\,d^4}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a^7\,d^2-21\,a^6\,b\,c\,d+21\,a^5\,b^2\,c^2\right )}{-a^5\,b^5\,d^5+5\,a^4\,b^6\,c\,d^4-10\,a^3\,b^7\,c^2\,d^3+10\,a^2\,b^8\,c^3\,d^2-5\,a\,b^9\,c^4\,d+b^{10}\,c^5}+\frac {x^2}{2\,b^3\,d^3}-\frac {\ln \left (c+d\,x\right )\,\left (21\,a^2\,c^5\,d^2-21\,a\,b\,c^6\,d+6\,b^2\,c^7\right )}{a^5\,d^{10}-5\,a^4\,b\,c\,d^9+10\,a^3\,b^2\,c^2\,d^8-10\,a^2\,b^3\,c^3\,d^7+5\,a\,b^4\,c^4\,d^6-b^5\,c^5\,d^5}-\frac {3\,x\,\left (a\,d+b\,c\right )}{b^4\,d^4} \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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