Optimal. Leaf size=65 \[ -\frac {d (b c-a d)}{3 b^3 (a+b x)^6}-\frac {(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac {d^2}{5 b^3 (a+b x)^5} \]
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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} \begin {gather*} -\frac {d (b c-a d)}{3 b^3 (a+b x)^6}-\frac {(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac {d^2}{5 b^3 (a+b x)^5} \end {gather*}
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)^{10}} \, dx &=\int \frac {(c+d x)^2}{(a+b x)^8} \, dx\\ &=\int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^8}+\frac {2 d (b c-a d)}{b^2 (a+b x)^7}+\frac {d^2}{b^2 (a+b x)^6}\right ) \, dx\\ &=-\frac {(b c-a d)^2}{7 b^3 (a+b x)^7}-\frac {d (b c-a d)}{3 b^3 (a+b x)^6}-\frac {d^2}{5 b^3 (a+b x)^5}\\ \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 (5 c+7 d x)+b^2 \left (15 c^2+35 c d x+21 d^2 x^2\right )}{105 b^3 (a+b x)^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^{10}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 131, normalized size = 2.02 \begin {gather*} -\frac {21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \, {\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \, {\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 61, normalized size = 0.94 \begin {gather*} -\frac {21 \, b^{2} d^{2} x^{2} + 35 \, b^{2} c d x + 7 \, a b d^{2} x + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2}}{105 \, {\left (b x + a\right )}^{7} b^{3}} \end {gather*}
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 \begin {gather*} -\frac {d^{2}}{5 \left (b x +a \right )^{5} b^{3}}+\frac {\left (a d -b c \right ) d}{3 \left (b x +a \right )^{6} b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{7 \left (b x +a \right )^{7} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.22, size = 131, normalized size = 2.02 \begin {gather*} -\frac {21 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c^{2} + 5 \, a b c d + a^{2} d^{2} + 7 \, {\left (5 \, b^{2} c d + a b d^{2}\right )} x}{105 \, {\left (b^{10} x^{7} + 7 \, a b^{9} x^{6} + 21 \, a^{2} b^{8} x^{5} + 35 \, a^{3} b^{7} x^{4} + 35 \, a^{4} b^{6} x^{3} + 21 \, a^{5} b^{5} x^{2} + 7 \, a^{6} b^{4} x + a^{7} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 129, normalized size = 1.98 \begin {gather*} -\frac {\frac {a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2}{105\,b^3}+\frac {d^2\,x^2}{5\,b}+\frac {d\,x\,\left (a\,d+5\,b\,c\right )}{15\,b^2}}{a^7+7\,a^6\,b\,x+21\,a^5\,b^2\,x^2+35\,a^4\,b^3\,x^3+35\,a^3\,b^4\,x^4+21\,a^2\,b^5\,x^5+7\,a\,b^6\,x^6+b^7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.95, size = 139, normalized size = 2.14 \begin {gather*} \frac {- a^{2} d^{2} - 5 a b c d - 15 b^{2} c^{2} - 21 b^{2} d^{2} x^{2} + x \left (- 7 a b d^{2} - 35 b^{2} c d\right )}{105 a^{7} b^{3} + 735 a^{6} b^{4} x + 2205 a^{5} b^{5} x^{2} + 3675 a^{4} b^{6} x^{3} + 3675 a^{3} b^{7} x^{4} + 2205 a^{2} b^{8} x^{5} + 735 a b^{9} x^{6} + 105 b^{10} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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