Optimal. Leaf size=65 \[ -\frac {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {d^2}{2 b^3 (a+b x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {d^2}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^7} \, dx &=\int \frac {(c+d x)^2}{(a+b x)^5} \, dx\\ &=\int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^5}+\frac {2 d (b c-a d)}{b^2 (a+b x)^4}+\frac {d^2}{b^2 (a+b x)^3}\right ) \, dx\\ &=-\frac {(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac {2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac {d^2}{2 b^3 (a+b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 0.86 \[ -\frac {a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )}{12 b^3 (a+b x)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 98, normalized size = 1.51 \[ -\frac {6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 61, normalized size = 0.94 \[ -\frac {6 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c d x + 4 \, a b d^{2} x + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}}{12 \, {\left (b x + a\right )}^{4} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 71, normalized size = 1.09 \[ -\frac {d^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {2 \left (a d -b c \right ) d}{3 \left (b x +a \right )^{3} b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{4 \left (b x +a \right )^{4} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 98, normalized size = 1.51 \[ -\frac {6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \, {\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 96, normalized size = 1.48 \[ -\frac {\frac {a^2\,d^2+2\,a\,b\,c\,d+3\,b^2\,c^2}{12\,b^3}+\frac {d^2\,x^2}{2\,b}+\frac {d\,x\,\left (a\,d+2\,b\,c\right )}{3\,b^2}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 104, normalized size = 1.60 \[ \frac {- a^{2} d^{2} - 2 a b c d - 3 b^{2} c^{2} - 6 b^{2} d^{2} x^{2} + x \left (- 4 a b d^{2} - 8 b^{2} c d\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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