Optimal. Leaf size=117 \[ \frac {a^2 (b c-a d)^2 x}{b^5}-\frac {a (b c-a d)^2 x^2}{2 b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^4}{4 b^2}+\frac {d^2 x^5}{5 b}-\frac {a^3 (b c-a d)^2 \log (a+b x)}{b^6} \]
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Rubi [A]
time = 0.08, antiderivative size = 117, 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 {a^3 (b c-a d)^2 \log (a+b x)}{b^6}+\frac {a^2 x (b c-a d)^2}{b^5}-\frac {a x^2 (b c-a d)^2}{2 b^4}+\frac {x^3 (b c-a d)^2}{3 b^3}+\frac {d x^4 (2 b c-a d)}{4 b^2}+\frac {d^2 x^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {x^3 (c+d x)^2}{a+b x} \, dx &=\int \left (\frac {a^2 (-b c+a d)^2}{b^5}-\frac {a (-b c+a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^2}{b^3}+\frac {d (2 b c-a d) x^3}{b^2}+\frac {d^2 x^4}{b}-\frac {a^3 (-b c+a d)^2}{b^5 (a+b x)}\right ) \, dx\\ &=\frac {a^2 (b c-a d)^2 x}{b^5}-\frac {a (b c-a d)^2 x^2}{2 b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^4}{4 b^2}+\frac {d^2 x^5}{5 b}-\frac {a^3 (b c-a d)^2 \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 112, normalized size = 0.96 \begin {gather*} \frac {60 a^2 b (b c-a d)^2 x-30 a b^2 (b c-a d)^2 x^2+20 b^3 (b c-a d)^2 x^3+15 b^4 d (2 b c-a d) x^4+12 b^5 d^2 x^5-60 a^3 (b c-a d)^2 \log (a+b x)}{60 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 175, normalized size = 1.50
method | result | size |
norman | \(\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{b^{5}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{3}}{3 b^{3}}+\frac {d^{2} x^{5}}{5 b}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 b^{4}}-\frac {d \left (a d -2 b c \right ) x^{4}}{4 b^{2}}-\frac {a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(153\) |
default | \(\frac {\frac {1}{5} d^{2} x^{5} b^{4}-\frac {1}{4} a \,b^{3} d^{2} x^{4}+\frac {1}{2} b^{4} c d \,x^{4}+\frac {1}{3} a^{2} b^{2} d^{2} x^{3}-\frac {2}{3} a \,b^{3} c d \,x^{3}+\frac {1}{3} b^{4} c^{2} x^{3}-\frac {1}{2} a^{3} b \,d^{2} x^{2}+a^{2} b^{2} c d \,x^{2}-\frac {1}{2} a \,b^{3} c^{2} x^{2}+a^{4} d^{2} x -2 a^{3} b c d x +a^{2} b^{2} c^{2} x}{b^{5}}-\frac {a^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(175\) |
risch | \(\frac {d^{2} x^{5}}{5 b}-\frac {a \,d^{2} x^{4}}{4 b^{2}}+\frac {c d \,x^{4}}{2 b}+\frac {a^{2} d^{2} x^{3}}{3 b^{3}}-\frac {2 a c d \,x^{3}}{3 b^{2}}+\frac {c^{2} x^{3}}{3 b}-\frac {a^{3} d^{2} x^{2}}{2 b^{4}}+\frac {a^{2} c d \,x^{2}}{b^{3}}-\frac {a \,c^{2} x^{2}}{2 b^{2}}+\frac {a^{4} d^{2} x}{b^{5}}-\frac {2 a^{3} c d x}{b^{4}}+\frac {a^{2} c^{2} x}{b^{3}}-\frac {a^{5} \ln \left (b x +a \right ) d^{2}}{b^{6}}+\frac {2 a^{4} \ln \left (b x +a \right ) c d}{b^{5}}-\frac {a^{3} \ln \left (b x +a \right ) c^{2}}{b^{4}}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 169, normalized size = 1.44 \begin {gather*} \frac {12 \, b^{4} d^{2} x^{5} + 15 \, {\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{4} + 20 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{3} - 30 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 60 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} x}{60 \, b^{5}} - \frac {{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.51, size = 170, normalized size = 1.45 \begin {gather*} \frac {12 \, b^{5} d^{2} x^{5} + 15 \, {\left (2 \, b^{5} c d - a b^{4} d^{2}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} - 30 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{2} + 60 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x - 60 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 155, normalized size = 1.32 \begin {gather*} - \frac {a^{3} \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{6}} + x^{4} \left (- \frac {a d^{2}}{4 b^{2}} + \frac {c d}{2 b}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3 b^{3}} - \frac {2 a c d}{3 b^{2}} + \frac {c^{2}}{3 b}\right ) + x^{2} \left (- \frac {a^{3} d^{2}}{2 b^{4}} + \frac {a^{2} c d}{b^{3}} - \frac {a c^{2}}{2 b^{2}}\right ) + x \left (\frac {a^{4} d^{2}}{b^{5}} - \frac {2 a^{3} c d}{b^{4}} + \frac {a^{2} c^{2}}{b^{3}}\right ) + \frac {d^{2} x^{5}}{5 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.66, size = 181, normalized size = 1.55 \begin {gather*} \frac {12 \, b^{4} d^{2} x^{5} + 30 \, b^{4} c d x^{4} - 15 \, a b^{3} d^{2} x^{4} + 20 \, b^{4} c^{2} x^{3} - 40 \, a b^{3} c d x^{3} + 20 \, a^{2} b^{2} d^{2} x^{3} - 30 \, a b^{3} c^{2} x^{2} + 60 \, a^{2} b^{2} c d x^{2} - 30 \, a^{3} b d^{2} x^{2} + 60 \, a^{2} b^{2} c^{2} x - 120 \, a^{3} b c d x + 60 \, a^{4} d^{2} x}{60 \, b^{5}} - \frac {{\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 181, normalized size = 1.55 \begin {gather*} x^3\,\left (\frac {c^2}{3\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{3\,b}\right )-x^4\,\left (\frac {a\,d^2}{4\,b^2}-\frac {c\,d}{2\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^5\,d^2-2\,a^4\,b\,c\,d+a^3\,b^2\,c^2\right )}{b^6}+\frac {d^2\,x^5}{5\,b}-\frac {a\,x^2\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{2\,b}+\frac {a^2\,x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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