Optimal. Leaf size=73 \[ \frac {d (b c-a d)^2 x}{b^3}+\frac {(b c-a d) (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{3 b}+\frac {(b c-a d)^3 \log (a+b x)}{b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, 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 = {640, 45}
\begin {gather*} \frac {(b c-a d)^3 \log (a+b x)}{b^4}+\frac {d x (b c-a d)^2}{b^3}+\frac {(c+d x)^2 (b c-a d)}{2 b^2}+\frac {(c+d x)^3}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^4} \, dx &=\int \frac {(c+d x)^3}{a+b x} \, dx\\ &=\int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx\\ &=\frac {d (b c-a d)^2 x}{b^3}+\frac {(b c-a d) (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{3 b}+\frac {(b c-a d)^3 \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 74, normalized size = 1.01 \begin {gather*} \frac {b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 109, normalized size = 1.49
method | result | size |
default | \(\frac {d \left (\frac {1}{3} b^{2} d^{2} x^{3}-\frac {1}{2} a b \,d^{2} x^{2}+\frac {3}{2} b^{2} c d \,x^{2}+a^{2} d^{2} x -3 a b c d x +3 b^{2} c^{2} x \right )}{b^{3}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{4}}\) | \(109\) |
risch | \(\frac {d^{3} x^{3}}{3 b}-\frac {d^{3} a \,x^{2}}{2 b^{2}}+\frac {3 d^{2} c \,x^{2}}{2 b}+\frac {d^{3} a^{2} x}{b^{3}}-\frac {3 d^{2} a c x}{b^{2}}+\frac {3 d \,c^{2} x}{b}-\frac {\ln \left (b x +a \right ) a^{3} d^{3}}{b^{4}}+\frac {3 \ln \left (b x +a \right ) a^{2} c \,d^{2}}{b^{3}}-\frac {3 \ln \left (b x +a \right ) a \,c^{2} d}{b^{2}}+\frac {\ln \left (b x +a \right ) c^{3}}{b}\) | \(133\) |
norman | \(\frac {\left (\frac {1}{2} a b \,d^{3}+\frac {3}{2} b^{2} c \,d^{2}\right ) x^{5}+\left (\frac {1}{2} a^{2} d^{3}+\frac {3}{2} a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{4}-\frac {a^{3} \left (11 a^{3} d^{3}-27 a^{2} b c \,d^{2}+54 a \,b^{2} c^{2} d \right )}{6 b^{4}}+\frac {b^{2} d^{3} x^{6}}{3}-\frac {3 a \left (a^{3} d^{3}-2 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d \right ) x^{2}}{b^{2}}-\frac {3 a^{2} \left (3 a^{3} d^{3}-7 a^{2} b c \,d^{2}+16 a \,b^{2} c^{2} d \right ) x}{2 b^{3}}}{\left (b x +a \right )^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{4}}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 114, normalized size = 1.56 \begin {gather*} \frac {2 \, b^{2} d^{3} x^{3} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.02, size = 116, normalized size = 1.59 \begin {gather*} \frac {2 \, b^{3} d^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 83, normalized size = 1.14 \begin {gather*} x^{2} \left (- \frac {a d^{3}}{2 b^{2}} + \frac {3 c d^{2}}{2 b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \frac {d^{3} x^{3}}{3 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.69, size = 115, normalized size = 1.58 \begin {gather*} \frac {2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 118, normalized size = 1.62 \begin {gather*} x\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )-x^2\,\left (\frac {a\,d^3}{2\,b^2}-\frac {3\,c\,d^2}{2\,b}\right )+\frac {d^3\,x^3}{3\,b}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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