Optimal. Leaf size=86 \[ -\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {3 d^2 (b c-a d)}{b^4 (a+b x)}+\frac {d^3 \log (a+b x)}{b^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 86, 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 {3 d^2 (b c-a d)}{b^4 (a+b x)}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}+\frac {d^3 \log (a+b x)}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+b x)^4} \, dx &=\int \left (\frac {(b c-a d)^3}{b^3 (a+b x)^4}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^3}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^2}+\frac {d^3}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {3 d^2 (b c-a d)}{b^4 (a+b x)}+\frac {d^3 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 80, normalized size = 0.93 \begin {gather*} \frac {-\frac {(b c-a d) \left (11 a^2 d^2+a b d (5 c+27 d x)+b^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )\right )}{(a+b x)^3}+6 d^3 \log (a+b x)}{6 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 120, normalized size = 1.40
method | result | size |
risch | \(\frac {\frac {3 d^{2} \left (a d -b c \right ) x^{2}}{b^{2}}+\frac {3 d \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x}{2 b^{3}}+\frac {11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(115\) |
norman | \(\frac {\frac {11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4}}+\frac {3 \left (a \,d^{3}-b c \,d^{2}\right ) x^{2}}{b^{2}}+\frac {3 \left (3 a^{2} d^{3}-2 a b c \,d^{2}-b^{2} c^{2} d \right ) x}{2 b^{3}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(119\) |
default | \(\frac {3 d^{2} \left (a d -b c \right )}{b^{4} \left (b x +a \right )}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{4} \left (b x +a \right )^{2}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{3 b^{4} \left (b x +a \right )^{3}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 142, normalized size = 1.65 \begin {gather*} -\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {d^{3} \log \left (b x + a\right )}{b^{4}} \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. 176 vs.
\(2 (82) = 164\).
time = 0.74, size = 176, normalized size = 2.05 \begin {gather*} -\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.65, size = 148, normalized size = 1.72 \begin {gather*} \frac {11 a^{3} d^{3} - 6 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 2 b^{3} c^{3} + x^{2} \cdot \left (18 a b^{2} d^{3} - 18 b^{3} c d^{2}\right ) + x \left (27 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 9 b^{3} c^{2} d\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {d^{3} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 118, normalized size = 1.37 \begin {gather*} \frac {d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {18 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 9 \, {\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x + \frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 138, normalized size = 1.60 \begin {gather*} \frac {d^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {\frac {-11\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+2\,b^3\,c^3}{6\,b^4}+\frac {3\,x\,\left (-3\,a^2\,d^3+2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{2\,b^3}-\frac {3\,d^2\,x^2\,\left (a\,d-b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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