Optimal. Leaf size=112 \[ -\frac {c^3}{a^3 x}-\frac {(b c-a d)^3}{2 a^2 b^2 (a+b x)^2}-\frac {(b c-a d)^2 (2 b c+a d)}{a^3 b^2 (a+b x)}-\frac {3 c^2 (b c-a d) \log (x)}{a^4}+\frac {3 c^2 (b c-a d) \log (a+b x)}{a^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90}
\begin {gather*} -\frac {3 c^2 \log (x) (b c-a d)}{a^4}+\frac {3 c^2 (b c-a d) \log (a+b x)}{a^4}-\frac {(b c-a d)^2 (a d+2 b c)}{a^3 b^2 (a+b x)}-\frac {c^3}{a^3 x}-\frac {(b c-a d)^3}{2 a^2 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{x^2 (a+b x)^3} \, dx &=\int \left (\frac {c^3}{a^3 x^2}+\frac {3 c^2 (-b c+a d)}{a^4 x}-\frac {(-b c+a d)^3}{a^2 b (a+b x)^3}+\frac {(-b c+a d)^2 (2 b c+a d)}{a^3 b (a+b x)^2}-\frac {3 b c^2 (-b c+a d)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{a^3 x}-\frac {(b c-a d)^3}{2 a^2 b^2 (a+b x)^2}-\frac {(b c-a d)^2 (2 b c+a d)}{a^3 b^2 (a+b x)}-\frac {3 c^2 (b c-a d) \log (x)}{a^4}+\frac {3 c^2 (b c-a d) \log (a+b x)}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 106, normalized size = 0.95 \begin {gather*} \frac {-\frac {2 a c^3}{x}+\frac {a^2 (-b c+a d)^3}{b^2 (a+b x)^2}-\frac {2 a (b c-a d)^2 (2 b c+a d)}{b^2 (a+b x)}+6 c^2 (-b c+a d) \log (x)+6 c^2 (b c-a d) \log (a+b x)}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 145, normalized size = 1.29
method | result | size |
norman | \(\frac {\frac {\left (3 a^{2} d^{2} c -6 a b \,c^{2} d +6 b^{2} c^{3}\right ) x^{2}}{a^{3}}-\frac {c^{3}}{a}+\frac {\left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +9 b^{3} c^{3}\right ) x^{3}}{2 a^{4}}}{x \left (b x +a \right )^{2}}+\frac {3 c^{2} \left (a d -b c \right ) \ln \left (x \right )}{a^{4}}-\frac {3 c^{2} \left (a d -b c \right ) \ln \left (b x +a \right )}{a^{4}}\) | \(139\) |
default | \(-\frac {a^{3} d^{3}-3 a \,b^{2} c^{2} d +2 b^{3} c^{3}}{a^{3} b^{2} \left (b x +a \right )}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 b^{2} a^{2} \left (b x +a \right )^{2}}-\frac {3 c^{2} \left (a d -b c \right ) \ln \left (b x +a \right )}{a^{4}}-\frac {c^{3}}{a^{3} x}+\frac {3 c^{2} \left (a d -b c \right ) \ln \left (x \right )}{a^{4}}\) | \(145\) |
risch | \(\frac {-\frac {\left (a^{3} d^{3}-3 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) x^{2}}{a^{3} b}-\frac {\left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +9 b^{3} c^{3}\right ) x}{2 a^{2} b^{2}}-\frac {c^{3}}{a}}{x \left (b x +a \right )^{2}}+\frac {3 c^{2} \ln \left (-x \right ) d}{a^{3}}-\frac {3 c^{3} \ln \left (-x \right ) b}{a^{4}}-\frac {3 c^{2} \ln \left (b x +a \right ) d}{a^{3}}+\frac {3 c^{3} \ln \left (b x +a \right ) b}{a^{4}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 164, normalized size = 1.46 \begin {gather*} -\frac {2 \, a^{2} b^{2} c^{3} + 2 \, {\left (3 \, b^{4} c^{3} - 3 \, a b^{3} c^{2} d + a^{3} b d^{3}\right )} x^{2} + {\left (9 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x}{2 \, {\left (a^{3} b^{4} x^{3} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2} x\right )}} + \frac {3 \, {\left (b c^{3} - a c^{2} d\right )} \log \left (b x + a\right )}{a^{4}} - \frac {3 \, {\left (b c^{3} - a c^{2} d\right )} \log \left (x\right )}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs.
\(2 (110) = 220\).
time = 0.73, size = 284, normalized size = 2.54 \begin {gather*} -\frac {2 \, a^{3} b^{2} c^{3} + 2 \, {\left (3 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{4} b d^{3}\right )} x^{2} + {\left (9 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x - 6 \, {\left ({\left (b^{5} c^{3} - a b^{4} c^{2} d\right )} x^{3} + 2 \, {\left (a b^{4} c^{3} - a^{2} b^{3} c^{2} d\right )} x^{2} + {\left (a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (b^{5} c^{3} - a b^{4} c^{2} d\right )} x^{3} + 2 \, {\left (a b^{4} c^{3} - a^{2} b^{3} c^{2} d\right )} x^{2} + {\left (a^{2} b^{3} c^{3} - a^{3} b^{2} c^{2} d\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{4} b^{4} x^{3} + 2 \, a^{5} b^{3} x^{2} + a^{6} b^{2} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs.
\(2 (100) = 200\).
time = 0.86, size = 262, normalized size = 2.34 \begin {gather*} \frac {- 2 a^{2} b^{2} c^{3} + x^{2} \left (- 2 a^{3} b d^{3} + 6 a b^{3} c^{2} d - 6 b^{4} c^{3}\right ) + x \left (- a^{4} d^{3} - 3 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - 9 a b^{3} c^{3}\right )}{2 a^{5} b^{2} x + 4 a^{4} b^{3} x^{2} + 2 a^{3} b^{4} x^{3}} + \frac {3 c^{2} \left (a d - b c\right ) \log {\left (x + \frac {3 a^{2} c^{2} d - 3 a b c^{3} - 3 a c^{2} \left (a d - b c\right )}{6 a b c^{2} d - 6 b^{2} c^{3}} \right )}}{a^{4}} - \frac {3 c^{2} \left (a d - b c\right ) \log {\left (x + \frac {3 a^{2} c^{2} d - 3 a b c^{3} + 3 a c^{2} \left (a d - b c\right )}{6 a b c^{2} d - 6 b^{2} c^{3}} \right )}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 161, normalized size = 1.44 \begin {gather*} -\frac {3 \, {\left (b c^{3} - a c^{2} d\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {3 \, {\left (b^{2} c^{3} - a b c^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} b^{2} c^{3} + 2 \, {\left (3 \, a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + a^{4} b d^{3}\right )} x^{2} + {\left (9 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} + a^{5} d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} a^{4} b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 169, normalized size = 1.51 \begin {gather*} \frac {6\,c^2\,\mathrm {atanh}\left (\frac {3\,c^2\,\left (a\,d-b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (3\,b\,c^3-3\,a\,c^2\,d\right )}\right )\,\left (a\,d-b\,c\right )}{a^4}-\frac {\frac {c^3}{a}+\frac {x^2\,\left (a^3\,d^3-3\,a\,b^2\,c^2\,d+3\,b^3\,c^3\right )}{a^3\,b}+\frac {x\,\left (a^3\,d^3+3\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+9\,b^3\,c^3\right )}{2\,a^2\,b^2}}{a^2\,x+2\,a\,b\,x^2+b^2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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