Optimal. Leaf size=78 \[ \frac {b^3 x}{d^3}+\frac {(b c-a d)^3}{2 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2}{d^4 (c+d x)}-\frac {3 b^2 (b c-a d) \log (c+d x)}{d^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 78, 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 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac {3 b (b c-a d)^2}{d^4 (c+d x)}+\frac {(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac {b^3 x}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{(c+d x)^3} \, dx &=\int \left (\frac {b^3}{d^3}+\frac {(-b c+a d)^3}{d^3 (c+d x)^3}+\frac {3 b (b c-a d)^2}{d^3 (c+d x)^2}-\frac {3 b^2 (b c-a d)}{d^3 (c+d x)}\right ) \, dx\\ &=\frac {b^3 x}{d^3}+\frac {(b c-a d)^3}{2 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2}{d^4 (c+d x)}-\frac {3 b^2 (b c-a d) \log (c+d x)}{d^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 114, normalized size = 1.46 \begin {gather*} \frac {-a^3 d^3-3 a^2 b d^2 (c+2 d x)+3 a b^2 c d (3 c+4 d x)+b^3 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )-6 b^2 (b c-a d) (c+d x)^2 \log (c+d x)}{2 d^4 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 114, normalized size = 1.46
method | result | size |
default | \(\frac {b^{3} x}{d^{3}}-\frac {3 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{4} \left (d x +c \right )}+\frac {3 b^{2} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 d^{4} \left (d x +c \right )^{2}}\) | \(114\) |
norman | \(\frac {\frac {b^{3} x^{3}}{d}-\frac {a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +9 b^{3} c^{3}}{2 d^{4}}-\frac {\left (3 a^{2} b \,d^{2}-6 a \,b^{2} c d +6 b^{3} c^{2}\right ) x}{d^{3}}}{\left (d x +c \right )^{2}}+\frac {3 b^{2} \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{4}}\) | \(116\) |
risch | \(\frac {b^{3} x}{d^{3}}+\frac {\left (-3 a^{2} b \,d^{2}+6 a \,b^{2} c d -3 b^{3} c^{2}\right ) x -\frac {a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}}{2 d}}{d^{3} \left (d x +c \right )^{2}}+\frac {3 b^{2} \ln \left (d x +c \right ) a}{d^{3}}-\frac {3 b^{3} \ln \left (d x +c \right ) c}{d^{4}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 125, normalized size = 1.60 \begin {gather*} \frac {b^{3} x}{d^{3}} - \frac {5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} - \frac {3 \, {\left (b^{3} c - a b^{2} d\right )} \log \left (d x + c\right )}{d^{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. 188 vs.
\(2 (76) = 152\).
time = 0.65, size = 188, normalized size = 2.41 \begin {gather*} \frac {2 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c d^{2} x^{2} - 5 \, b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - a^{3} d^{3} - 2 \, {\left (2 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{2} d - a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 128, normalized size = 1.64 \begin {gather*} \frac {b^{3} x}{d^{3}} + \frac {3 b^{2} \left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{4}} + \frac {- a^{3} d^{3} - 3 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - 5 b^{3} c^{3} + x \left (- 6 a^{2} b d^{3} + 12 a b^{2} c d^{2} - 6 b^{3} c^{2} d\right )}{2 c^{2} d^{4} + 4 c d^{5} x + 2 d^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 112, normalized size = 1.44 \begin {gather*} \frac {b^{3} x}{d^{3}} - \frac {3 \, {\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4}} - \frac {5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 130, normalized size = 1.67 \begin {gather*} \frac {b^3\,x}{d^3}-\frac {\ln \left (c+d\,x\right )\,\left (3\,b^3\,c-3\,a\,b^2\,d\right )}{d^4}-\frac {\frac {a^3\,d^3+3\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+5\,b^3\,c^3}{2\,d}+x\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}{c^2\,d^3+2\,c\,d^4\,x+d^5\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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