Optimal. Leaf size=28 \[ -\frac {(c+d x)^4}{4 (b c-a d) (a+b x)^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 37}
\begin {gather*} -\frac {(c+d x)^4}{4 (a+b x)^4 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^8} \, dx &=\int \frac {(c+d x)^3}{(a+b x)^5} \, dx\\ &=-\frac {(c+d x)^4}{4 (b c-a d) (a+b x)^4}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(28)=56\).
time = 0.02, size = 91, normalized size = 3.25 \begin {gather*} -\frac {a^3 d^3+a^2 b d^2 (c+4 d x)+a b^2 d \left (c^2+4 c d x+6 d^2 x^2\right )+b^3 \left (c^3+4 c^2 d x+6 c d^2 x^2+4 d^3 x^3\right )}{4 b^4 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(121\) vs.
\(2(26)=52\).
time = 0.66, size = 122, normalized size = 4.36
method | result | size |
risch | \(\frac {-\frac {d^{3} x^{3}}{b}-\frac {3 d^{2} \left (a d +b c \right ) x^{2}}{2 b^{2}}-\frac {d \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x}{b^{3}}-\frac {a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4}}}{\left (b x +a \right )^{4}}\) | \(104\) |
gosper | \(-\frac {4 d^{3} x^{3} b^{3}+6 a \,b^{2} d^{3} x^{2}+6 b^{3} c \,d^{2} x^{2}+4 a^{2} b \,d^{3} x +4 b^{2} d^{2} x a c +4 b^{3} c^{2} d x +a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}\) | \(112\) |
default | \(-\frac {d^{3}}{b^{4} \left (b x +a \right )}-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{4} \left (b x +a \right )^{3}}+\frac {3 d^{2} \left (a d -b c \right )}{2 b^{4} \left (b x +a \right )^{2}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{4 b^{4} \left (b x +a \right )^{4}}\) | \(122\) |
norman | \(\frac {-b^{2} d^{3} x^{6}+\frac {a^{3} \left (-a^{3} b^{3} d^{3}-a^{2} b^{4} c \,d^{2}-c^{2} d a \,b^{5}-c^{3} b^{6}\right )}{4 b^{7}}+\frac {3 \left (-3 a \,b^{3} d^{3}-b^{4} c \,d^{2}\right ) x^{5}}{2 b^{2}}+\frac {\left (-17 a^{2} b^{3} d^{3}-11 a \,b^{4} c \,d^{2}-2 c^{2} d \,b^{5}\right ) x^{4}}{2 b^{3}}+\frac {\left (-35 a^{3} b^{3} d^{3}-31 a^{2} b^{4} c \,d^{2}-13 c^{2} d a \,b^{5}-c^{3} b^{6}\right ) x^{3}}{4 b^{4}}+\frac {3 a \left (-7 a^{3} b^{3} d^{3}-7 a^{2} b^{4} c \,d^{2}-5 c^{2} d a \,b^{5}-c^{3} b^{6}\right ) x^{2}}{4 b^{5}}+\frac {a^{2} \left (-7 a^{3} b^{3} d^{3}-7 a^{2} b^{4} c \,d^{2}-7 c^{2} d a \,b^{5}-3 c^{3} b^{6}\right ) x}{4 b^{6}}}{\left (b x +a \right )^{7}}\) | \(289\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (26) = 52\).
time = 0.28, size = 143, normalized size = 5.11 \begin {gather*} -\frac {4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \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. 143 vs.
\(2 (26) = 52\).
time = 2.56, size = 143, normalized size = 5.11 \begin {gather*} -\frac {4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \, {\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs.
\(2 (22) = 44\).
time = 2.59, size = 155, normalized size = 5.54 \begin {gather*} \frac {- a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d - b^{3} c^{3} - 4 b^{3} d^{3} x^{3} + x^{2} \left (- 6 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (- 4 a^{2} b d^{3} - 4 a b^{2} c d^{2} - 4 b^{3} c^{2} d\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs.
\(2 (26) = 52\).
time = 1.07, size = 111, normalized size = 3.96 \begin {gather*} -\frac {4 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 4 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}}{4 \, {\left (b x + a\right )}^{4} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 135, normalized size = 4.82 \begin {gather*} -\frac {\frac {a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3}{4\,b^4}+\frac {d^3\,x^3}{b}+\frac {d\,x\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{b^3}+\frac {3\,d^2\,x^2\,\left (a\,d+b\,c\right )}{2\,b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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