Optimal. Leaf size=103 \[ -\frac {b^3 (3 b c-4 a d) x}{d^4}+\frac {b^4 x^2}{2 d^3}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 103, 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 {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 45
Rubi steps
\begin {align*} \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 = 0.04, 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 )+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 )+12 b^2 (b c-a d)^2 (c+d x)^2 \log (c+d x)}{2 d^5 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(103)=206\).
time = 3.92, size = 210, normalized size = 2.04 \begin {gather*} \frac {-a^4 d^4-4 a^3 b c d^3+12 b^2 \text {Log}\left [c+d x\right ] \left (c^2+2 c d x+d^2 x^2\right ) \left (a d-b c\right )^2+18 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+7 b^4 c^4-8 b d x \left (a^3 d^3-3 a^2 b c d^2+3 a b^2 c^2 d-b^3 c^3\right )+2 b^3 d x \left (4 a d-3 b c\right ) \left (c^2+2 c d x+d^2 x^2\right )+b^4 d^2 x^2 \left (c^2+2 c d x+d^2 x^2\right )}{2 d^5 \left (c^2+2 c d x+d^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 172, normalized size = 1.67
method | result | size |
default | \(\frac {b^{3} \left (\frac {1}{2} b d \,x^{2}+4 a d x -3 b c x \right )}{d^{4}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{5} \left (d x +c \right )}+\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5}}-\frac {a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}{2 d^{5} \left (d x +c \right )^{2}}\) | \(172\) |
norman | \(\frac {-\frac {a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+36 a \,b^{3} c^{3} d -18 b^{4} c^{4}}{2 d^{5}}+\frac {b^{4} x^{4}}{2 d}-\frac {2 \left (2 a^{3} b \,d^{3}-6 b^{2} a^{2} c \,d^{2}+12 a \,b^{3} c^{2} d -6 b^{4} c^{3}\right ) x}{d^{4}}+\frac {2 b^{3} \left (2 a d -b c \right ) x^{3}}{d^{2}}}{\left (d x +c \right )^{2}}+\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5}}\) | \(178\) |
risch | \(\frac {b^{4} x^{2}}{2 d^{3}}+\frac {4 a \,b^{3} x}{d^{3}}-\frac {3 b^{4} c x}{d^{4}}+\frac {\left (-4 a^{3} b \,d^{3}+12 b^{2} a^{2} c \,d^{2}-12 a \,b^{3} c^{2} d +4 b^{4} c^{3}\right ) x -\frac {a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d -7 b^{4} c^{4}}{2 d}}{d^{4} \left (d x +c \right )^{2}}+\frac {6 b^{2} \ln \left (d x +c \right ) a^{2}}{d^{3}}-\frac {12 b^{3} \ln \left (d x +c \right ) a c}{d^{4}}+\frac {6 b^{4} \ln \left (d x +c \right ) c^{2}}{d^{5}}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs.
\(2 (99) = 198\).
time = 0.30, 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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Sympy [A]
time = 0.72, 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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Giac [A]
time = 0.00, size = 198, normalized size = 1.92 \begin {gather*} \frac {\frac {1}{2} x^{2} b^{4} d^{3}-3 x b^{4} d^{2} c+4 x b^{3} a d^{3}}{d^{6}}+\frac {\frac {1}{2} \left (7 b^{4} c^{4}-20 b^{3} d c^{3} a+18 b^{2} d^{2} c^{2} a^{2}-4 b d^{3} c a^{3}-d^{4} a^{4}+\left (8 b^{4} d c^{3}-24 b^{3} d^{2} c^{2} a+24 b^{2} d^{3} c a^{2}-8 b d^{4} a^{3}\right ) x\right )}{d^{5} \left (x d+c\right )^{2}}+\frac {\left (6 b^{4} c^{2}-12 b^{3} a d c+6 b^{2} a^{2} d^{2}\right ) \ln \left |x d+c\right |}{d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, 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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