Optimal. Leaf size=58 \[ -\frac {(c+d x)^4}{5 (b c-a d) (a+b x)^5}+\frac {d (c+d x)^4}{20 (b c-a d)^2 (a+b x)^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37}
\begin {gather*} \frac {d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac {(c+d x)^4}{5 (a+b x)^5 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+b x)^6} \, dx &=-\frac {(c+d x)^4}{5 (b c-a d) (a+b x)^5}-\frac {d \int \frac {(c+d x)^3}{(a+b x)^5} \, dx}{5 (b c-a d)}\\ &=-\frac {(c+d x)^4}{5 (b c-a d) (a+b x)^5}+\frac {d (c+d x)^4}{20 (b c-a d)^2 (a+b x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 97, normalized size = 1.67 \begin {gather*} -\frac {a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )}{20 b^4 (a+b x)^5} \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. \(120\) vs.
\(2(54)=108\).
time = 0.14, size = 121, normalized size = 2.09
method | result | size |
risch | \(\frac {-\frac {d^{3} x^{3}}{2 b}-\frac {d^{2} \left (a d +2 b c \right ) x^{2}}{2 b^{2}}-\frac {d \left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) x}{4 b^{3}}-\frac {a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}}{20 b^{4}}}{\left (b x +a \right )^{5}}\) | \(110\) |
gosper | \(-\frac {10 d^{3} x^{3} b^{3}+10 a \,b^{2} d^{3} x^{2}+20 b^{3} c \,d^{2} x^{2}+5 a^{2} b \,d^{3} x +10 a \,b^{2} c \,d^{2} x +15 b^{3} c^{2} d x +a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}}{20 b^{4} \left (b x +a \right )^{5}}\) | \(115\) |
default | \(-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 b^{4} \left (b x +a \right )^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{5 b^{4} \left (b x +a \right )^{5}}-\frac {d^{3}}{2 b^{4} \left (b x +a \right )^{2}}+\frac {d^{2} \left (a d -b c \right )}{b^{4} \left (b x +a \right )^{3}}\) | \(121\) |
norman | \(\frac {-\frac {d^{3} x^{3}}{2 b}+\frac {\left (-a b \,d^{3}-2 b^{2} c \,d^{2}\right ) x^{2}}{2 b^{3}}+\frac {\left (-a^{2} b \,d^{3}-2 a \,b^{2} c \,d^{2}-3 b^{3} c^{2} d \right ) x}{4 b^{4}}+\frac {-a^{3} b \,d^{3}-2 b^{2} a^{2} c \,d^{2}-3 a \,b^{3} c^{2} d -4 b^{4} c^{3}}{20 b^{5}}}{\left (b x +a \right )^{5}}\) | \(126\) |
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. 160 vs.
\(2 (54) = 108\).
time = 0.30, size = 160, normalized size = 2.76 \begin {gather*} -\frac {10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \, {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} 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. 160 vs.
\(2 (54) = 108\).
time = 0.54, size = 160, normalized size = 2.76 \begin {gather*} -\frac {10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \, {\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \, {\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \, {\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} 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. 172 vs.
\(2 (46) = 92\).
time = 1.13, size = 172, normalized size = 2.97 \begin {gather*} \frac {- a^{3} d^{3} - 2 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 4 b^{3} c^{3} - 10 b^{3} d^{3} x^{3} + x^{2} \left (- 10 a b^{2} d^{3} - 20 b^{3} c d^{2}\right ) + x \left (- 5 a^{2} b d^{3} - 10 a b^{2} c d^{2} - 15 b^{3} c^{2} d\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \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. 114 vs.
\(2 (54) = 108\).
time = 1.11, size = 114, normalized size = 1.97 \begin {gather*} -\frac {10 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c d^{2} x^{2} + 10 \, a b^{2} d^{3} x^{2} + 15 \, b^{3} c^{2} d x + 10 \, a b^{2} c d^{2} x + 5 \, a^{2} b d^{3} x + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}}{20 \, {\left (b x + a\right )}^{5} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 39, normalized size = 0.67 \begin {gather*} \frac {{\left (c+d\,x\right )}^4\,\left (5\,a\,d-4\,b\,c+b\,d\,x\right )}{20\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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