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 = 0.03, 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 = {43} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \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 \begin {gather*} -\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} \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 {(c+d x)^2}{(a+b x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.49, size = 98, normalized size = 1.51 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 96, normalized size = 1.48 \begin {gather*} -\frac {\frac {3 \, c^{2}}{{\left (b x + a\right )}^{4}} + \frac {8 \, c d}{{\left (b x + a\right )}^{3} b} - \frac {6 \, a c d}{{\left (b x + a\right )}^{4} b} + \frac {6 \, d^{2}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {8 \, a d^{2}}{{\left (b x + a\right )}^{3} b^{2}} + \frac {3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{4} b^{2}}}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 1.09 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 98, normalized size = 1.51 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 39, normalized size = 0.60 \begin {gather*} \frac {{\left (c+d\,x\right )}^3\,\left (4\,a\,d-3\,b\,c+b\,d\,x\right )}{12\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.76, size = 104, normalized size = 1.60 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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