Optimal. Leaf size=26 \[ \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1607, 1598, 45}
\begin {gather*} \frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rule 1607
Rubi steps
\begin {align*} \int \frac {a x^2+b x^3}{c x^2+d x^3} \, dx &=\int \frac {x^2 (a+b x)}{c x^2+d x^3} \, dx\\ &=\int \frac {a+b x}{c+d x} \, dx\\ &=\int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx\\ &=\frac {b x}{d}-\frac {(b c-a d) \log (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.96 \begin {gather*} \frac {b x}{d}+\frac {(-b c+a d) \log (c+d x)}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 26, normalized size = 1.00
method | result | size |
default | \(\frac {x b}{d}+\frac {\left (a d -b c \right ) \ln \left (d x +c \right )}{d^{2}}\) | \(26\) |
norman | \(\frac {x b}{d}+\frac {\left (a d -b c \right ) \ln \left (d x +c \right )}{d^{2}}\) | \(26\) |
risch | \(\frac {x b}{d}+\frac {\ln \left (d x +c \right ) a}{d}-\frac {\ln \left (d x +c \right ) b c}{d^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 26, normalized size = 1.00 \begin {gather*} \frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 25, normalized size = 0.96 \begin {gather*} \frac {b d x - {\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.77 \begin {gather*} \frac {b x}{d} + \frac {\left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.62, size = 27, normalized size = 1.04 \begin {gather*} \frac {b x}{d} - \frac {{\left (b c - a d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.23, size = 25, normalized size = 0.96 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (a\,d-b\,c\right )}{d^2}+\frac {b\,x}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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