Optimal. Leaf size=90 \[ -\frac {b \log (x) (b c-a d)^2}{a^4}+\frac {b (b c-a d)^2 \log (a+b x)}{a^4}-\frac {(b c-a d)^2}{a^3 x}+\frac {c (b c-2 a d)}{2 a^2 x^2}-\frac {c^2}{3 a x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 90, 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 = {88} \[ \frac {c (b c-2 a d)}{2 a^2 x^2}-\frac {(b c-a d)^2}{a^3 x}-\frac {b \log (x) (b c-a d)^2}{a^4}+\frac {b (b c-a d)^2 \log (a+b x)}{a^4}-\frac {c^2}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{x^4 (a+b x)} \, dx &=\int \left (\frac {c^2}{a x^4}+\frac {c (-b c+2 a d)}{a^2 x^3}+\frac {(-b c+a d)^2}{a^3 x^2}-\frac {b (-b c+a d)^2}{a^4 x}+\frac {b^2 (-b c+a d)^2}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac {c^2}{3 a x^3}+\frac {c (b c-2 a d)}{2 a^2 x^2}-\frac {(b c-a d)^2}{a^3 x}-\frac {b (b c-a d)^2 \log (x)}{a^4}+\frac {b (b c-a d)^2 \log (a+b x)}{a^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 99, normalized size = 1.10 \[ \frac {-2 a^3 \left (c^2+3 c d x+3 d^2 x^2\right )+3 a^2 b c x (c+4 d x)-6 a b^2 c^2 x^2-6 b x^3 \log (x) (b c-a d)^2+6 b x^3 (b c-a d)^2 \log (a+b x)}{6 a^4 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 131, normalized size = 1.46 \[ -\frac {2 \, a^{3} c^{2} - 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} \log \left (b x + a\right ) + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} \log \relax (x) + 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 138, normalized size = 1.53 \[ -\frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, a^{3} c^{2} + 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 3 \, {\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} x}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 153, normalized size = 1.70 \[ -\frac {b \,d^{2} \ln \relax (x )}{a^{2}}+\frac {b \,d^{2} \ln \left (b x +a \right )}{a^{2}}+\frac {2 b^{2} c d \ln \relax (x )}{a^{3}}-\frac {2 b^{2} c d \ln \left (b x +a \right )}{a^{3}}-\frac {b^{3} c^{2} \ln \relax (x )}{a^{4}}+\frac {b^{3} c^{2} \ln \left (b x +a \right )}{a^{4}}-\frac {d^{2}}{a x}+\frac {2 b c d}{a^{2} x}-\frac {b^{2} c^{2}}{a^{3} x}-\frac {c d}{a \,x^{2}}+\frac {b \,c^{2}}{2 a^{2} x^{2}}-\frac {c^{2}}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 126, normalized size = 1.40 \[ \frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left (b x + a\right )}{a^{4}} - \frac {{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \relax (x)}{a^{4}} - \frac {2 \, a^{2} c^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} - 3 \, {\left (a b c^{2} - 2 \, a^{2} c d\right )} x}{6 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 123, normalized size = 1.37 \[ \frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a\,d-b\,c\right )}^2\,\left (a+2\,b\,x\right )}{a\,\left (a^2\,b\,d^2-2\,a\,b^2\,c\,d+b^3\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^4}-\frac {\frac {c^2}{3\,a}+\frac {x^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{a^3}+\frac {c\,x\,\left (2\,a\,d-b\,c\right )}{2\,a^2}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.94, size = 240, normalized size = 2.67 \[ \frac {- 2 a^{2} c^{2} + x^{2} \left (- 6 a^{2} d^{2} + 12 a b c d - 6 b^{2} c^{2}\right ) + x \left (- 6 a^{2} c d + 3 a b c^{2}\right )}{6 a^{3} x^{3}} - \frac {b \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{3} b d^{2} - 2 a^{2} b^{2} c d + a b^{3} c^{2} - a b \left (a d - b c\right )^{2}}{2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}} \right )}}{a^{4}} + \frac {b \left (a d - b c\right )^{2} \log {\left (x + \frac {a^{3} b d^{2} - 2 a^{2} b^{2} c d + a b^{3} c^{2} + a b \left (a d - b c\right )^{2}}{2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}} \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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