Optimal. Leaf size=103 \[ -\frac {b^3 x (3 b c-4 a d)}{d^4}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {b^4 x^2}{2 d^3} \]
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Rubi [A] time = 0.09, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \begin {gather*} -\frac {b^3 x (3 b c-4 a d)}{d^4}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {b^4 x^2}{2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac {(a+b x)^4}{(c+d x)^3} \, dx\\ &=\int \left (-\frac {b^3 (3 b c-4 a d)}{d^4}+\frac {b^4 x}{d^3}+\frac {(-b c+a d)^4}{d^4 (c+d x)^3}-\frac {4 b (b c-a d)^3}{d^4 (c+d x)^2}+\frac {6 b^2 (b c-a d)^2}{d^4 (c+d x)}\right ) \, dx\\ &=-\frac {b^3 (3 b c-4 a d) x}{d^4}+\frac {b^4 x^2}{2 d^3}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 167, normalized size = 1.62 \begin {gather*} \frac {-a^4 d^4-4 a^3 b d^3 (c+2 d x)+6 a^2 b^2 c d^2 (3 c+4 d x)+4 a b^3 d \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+12 b^2 (c+d x)^2 (b c-a d)^2 \log (c+d x)+b^4 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )}{2 d^5 (c+d x)^2} \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 {(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 291, normalized size = 2.83 \begin {gather*} \frac {b^{4} d^{4} x^{4} + 7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} - 4 \, {\left (b^{4} c d^{3} - 2 \, a b^{3} d^{4}\right )} x^{3} - {\left (11 \, b^{4} c^{2} d^{2} - 16 \, a b^{3} c d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 12 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 183, normalized size = 1.78 \begin {gather*} \frac {6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac {b^{4} d^{3} x^{2} - 6 \, b^{4} c d^{2} x + 8 \, a b^{3} d^{3} x}{2 \, d^{6}} + \frac {7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 245, normalized size = 2.38 \begin {gather*} -\frac {a^{4}}{2 \left (d x +c \right )^{2} d}+\frac {2 a^{3} b c}{\left (d x +c \right )^{2} d^{2}}-\frac {3 a^{2} b^{2} c^{2}}{\left (d x +c \right )^{2} d^{3}}+\frac {2 a \,b^{3} c^{3}}{\left (d x +c \right )^{2} d^{4}}-\frac {b^{4} c^{4}}{2 \left (d x +c \right )^{2} d^{5}}+\frac {b^{4} x^{2}}{2 d^{3}}-\frac {4 a^{3} b}{\left (d x +c \right ) d^{2}}+\frac {12 a^{2} b^{2} c}{\left (d x +c \right ) d^{3}}+\frac {6 a^{2} b^{2} \ln \left (d x +c \right )}{d^{3}}-\frac {12 a \,b^{3} c^{2}}{\left (d x +c \right ) d^{4}}-\frac {12 a \,b^{3} c \ln \left (d x +c \right )}{d^{4}}+\frac {4 a \,b^{3} x}{d^{3}}+\frac {4 b^{4} c^{3}}{\left (d x +c \right ) d^{5}}+\frac {6 b^{4} c^{2} \ln \left (d x +c \right )}{d^{5}}-\frac {3 b^{4} c x}{d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 191, normalized size = 1.85 \begin {gather*} \frac {7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} + \frac {b^{4} d x^{2} - 2 \, {\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} x}{2 \, d^{4}} + \frac {6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 196, normalized size = 1.90 \begin {gather*} x\,\left (\frac {4\,a\,b^3}{d^3}-\frac {3\,b^4\,c}{d^4}\right )-\frac {\frac {a^4\,d^4+4\,a^3\,b\,c\,d^3-18\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d-7\,b^4\,c^4}{2\,d}-x\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )}{c^2\,d^4+2\,c\,d^5\,x+d^6\,x^2}+\frac {b^4\,x^2}{2\,d^3}+\frac {\ln \left (c+d\,x\right )\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )}{d^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.37, size = 185, normalized size = 1.80 \begin {gather*} \frac {b^{4} x^{2}}{2 d^{3}} + \frac {6 b^{2} \left (a d - b c\right )^{2} \log {\left (c + d x \right )}}{d^{5}} + x \left (\frac {4 a b^{3}}{d^{3}} - \frac {3 b^{4} c}{d^{4}}\right ) + \frac {- a^{4} d^{4} - 4 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 20 a b^{3} c^{3} d + 7 b^{4} c^{4} + x \left (- 8 a^{3} b d^{4} + 24 a^{2} b^{2} c d^{3} - 24 a b^{3} c^{2} d^{2} + 8 b^{4} c^{3} d\right )}{2 c^{2} d^{5} + 4 c d^{6} x + 2 d^{7} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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