Optimal. Leaf size=38 \[ \frac {b (c+d x)^5}{5 d^2}-\frac {(c+d x)^4 (b c-a d)}{4 d^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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 {b (c+d x)^5}{5 d^2}-\frac {(c+d x)^4 (b c-a d)}{4 d^2} \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 )^3}{(a+b x)^2} \, dx &=\int (a+b x) (c+d x)^3 \, dx\\ &=\int \left (\frac {(-b c+a d) (c+d x)^3}{d}+\frac {b (c+d x)^4}{d}\right ) \, dx\\ &=-\frac {(b c-a d) (c+d x)^4}{4 d^2}+\frac {b (c+d x)^5}{5 d^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 67, normalized size = 1.76 \begin {gather*} \frac {1}{2} c^2 x^2 (3 a d+b c)+\frac {1}{4} d^2 x^4 (a d+3 b c)+c d x^3 (a d+b c)+a c^3 x+\frac {1}{5} b d^3 x^5 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 31, normalized size = 0.82 \begin {gather*} \frac {(c+d x)^4 (5 a d+4 b (c+d x)-5 b c)}{20 d^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 69, normalized size = 1.82 \begin {gather*} \frac {1}{5} \, b d^{3} x^{5} + a c^{3} x + \frac {1}{4} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{4} + {\left (b c^{2} d + a c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 155, normalized size = 4.08 \begin {gather*} \frac {{\left (\frac {10 \, b^{3} c^{3}}{{\left (b x + a\right )}^{3}} + \frac {20 \, b^{2} c^{2} d}{{\left (b x + a\right )}^{2}} - \frac {30 \, a b^{2} c^{2} d}{{\left (b x + a\right )}^{3}} + \frac {15 \, b c d^{2}}{b x + a} - \frac {40 \, a b c d^{2}}{{\left (b x + a\right )}^{2}} + \frac {30 \, a^{2} b c d^{2}}{{\left (b x + a\right )}^{3}} - \frac {15 \, a d^{3}}{b x + a} + \frac {20 \, a^{2} d^{3}}{{\left (b x + a\right )}^{2}} - \frac {10 \, a^{3} d^{3}}{{\left (b x + a\right )}^{3}} + 4 \, d^{3}\right )} {\left (b x + a\right )}^{5}}{20 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 94, normalized size = 2.47 \begin {gather*} \frac {b \,d^{3} x^{5}}{5}+a \,c^{3} x +\frac {\left (2 b c \,d^{2}+\left (a d +b c \right ) d^{2}\right ) x^{4}}{4}+\frac {\left (a c \,d^{2}+b \,c^{2} d +2 \left (a d +b c \right ) c d \right ) x^{3}}{3}+\frac {\left (2 a \,c^{2} d +\left (a d +b c \right ) c^{2}\right ) x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.07, size = 69, normalized size = 1.82 \begin {gather*} \frac {1}{5} \, b d^{3} x^{5} + a c^{3} x + \frac {1}{4} \, {\left (3 \, b c d^{2} + a d^{3}\right )} x^{4} + {\left (b c^{2} d + a c d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{3} + 3 \, a c^{2} d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 65, normalized size = 1.71 \begin {gather*} x^2\,\left (\frac {b\,c^3}{2}+\frac {3\,a\,d\,c^2}{2}\right )+x^4\,\left (\frac {a\,d^3}{4}+\frac {3\,b\,c\,d^2}{4}\right )+\frac {b\,d^3\,x^5}{5}+a\,c^3\,x+c\,d\,x^3\,\left (a\,d+b\,c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 73, normalized size = 1.92 \begin {gather*} a c^{3} x + \frac {b d^{3} x^{5}}{5} + x^{4} \left (\frac {a d^{3}}{4} + \frac {3 b c d^{2}}{4}\right ) + x^{3} \left (a c d^{2} + b c^{2} d\right ) + x^{2} \left (\frac {3 a c^{2} d}{2} + \frac {b c^{3}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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