Optimal. Leaf size=65 \[ -\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {d^2}{3 b^3 (a+b x)^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 65, 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 {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d^2}{3 b^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+b x)^6} \, dx &=\int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^6}+\frac {2 d (b c-a d)}{b^2 (a+b x)^5}+\frac {d^2}{b^2 (a+b x)^4}\right ) \, dx\\ &=-\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {d^2}{3 b^3 (a+b x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 57, normalized size = 0.88 \begin {gather*} -\frac {a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )}{30 b^3 (a+b x)^5} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.89, size = 103, normalized size = 1.58 \begin {gather*} \frac {-a^2 d^2-3 a b c d-6 b^2 c^2-5 b d x \left (a d+3 b c\right )-10 b^2 d^2 x^2}{30 b^3 \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 71, normalized size = 1.09
method | result | size |
gosper | \(-\frac {10 d^{2} x^{2} b^{2}+5 a b \,d^{2} x +15 b^{2} c d x +a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}}{30 b^{3} \left (b x +a \right )^{5}}\) | \(62\) |
risch | \(\frac {-\frac {d^{2} x^{2}}{3 b}-\frac {d \left (a d +3 b c \right ) x}{6 b^{2}}-\frac {a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}}{30 b^{3}}}{\left (b x +a \right )^{5}}\) | \(63\) |
default | \(\frac {d \left (a d -b c \right )}{2 b^{3} \left (b x +a \right )^{4}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{5 b^{3} \left (b x +a \right )^{5}}-\frac {d^{2}}{3 b^{3} \left (b x +a \right )^{3}}\) | \(71\) |
norman | \(\frac {-\frac {d^{2} x^{2}}{3 b}+\frac {\left (-a \,b^{2} d^{2}-3 b^{3} c d \right ) x}{6 b^{4}}+\frac {-b^{2} a^{2} d^{2}-3 a \,b^{3} c d -6 b^{4} c^{2}}{30 b^{5}}}{\left (b x +a \right )^{5}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 109, normalized size = 1.68 \begin {gather*} -\frac {10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.28, size = 109, normalized size = 1.68 \begin {gather*} -\frac {10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\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. 116 vs.
\(2 (56) = 112\).
time = 0.55, size = 116, normalized size = 1.78 \begin {gather*} \frac {- a^{2} d^{2} - 3 a b c d - 6 b^{2} c^{2} - 10 b^{2} d^{2} x^{2} + x \left (- 5 a b d^{2} - 15 b^{2} c d\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 69, normalized size = 1.06 \begin {gather*} \frac {-10 x^{2} d^{2} b^{2}-5 x d^{2} b a-15 x d c b^{2}-d^{2} a^{2}-3 d c b a-6 c^{2} b^{2}}{30 b^{3} \left (x b+a\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 107, normalized size = 1.65 \begin {gather*} -\frac {\frac {a^2\,d^2+3\,a\,b\,c\,d+6\,b^2\,c^2}{30\,b^3}+\frac {d^2\,x^2}{3\,b}+\frac {d\,x\,\left (a\,d+3\,b\,c\right )}{6\,b^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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