Optimal. Leaf size=94 \[ -\frac {\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}+\frac {c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2} \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ \frac {c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac {\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx &=\int \left (\frac {c \left (3 c d^2+2 a e^2\right )}{e^4}-\frac {2 c^2 d x}{e^3}+\frac {c^2 x^2}{e^2}+\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^2}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {\left (c d^2+a e^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (c d^2+a e^2\right ) \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.97 \[ \frac {3 c e x \left (2 a e^2+3 c d^2\right )-\frac {3 \left (a e^2+c d^2\right )^2}{d+e x}-12 c d \left (a e^2+c d^2\right ) \log (d+e x)-3 c^2 d e^2 x^2+c^2 e^3 x^3}{3 e^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 150, normalized size = 1.60 \[ \frac {c^{2} e^{4} x^{4} - 2 \, c^{2} d e^{3} x^{3} - 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \, {\left (c^{2} d^{4} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{6} x + d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 149, normalized size = 1.59 \[ \frac {1}{3} \, {\left (c^{2} - \frac {6 \, c^{2} d}{x e + d} + \frac {6 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-5\right )} + 4 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{2} d^{4} e^{3}}{x e + d} + \frac {2 \, a c d^{2} e^{5}}{x e + d} + \frac {a^{2} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 126, normalized size = 1.34 \[ \frac {c^{2} x^{3}}{3 e^{2}}-\frac {c^{2} d \,x^{2}}{e^{3}}-\frac {a^{2}}{\left (e x +d \right ) e}-\frac {2 a c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {4 a c d \ln \left (e x +d \right )}{e^{3}}+\frac {2 a c x}{e^{2}}-\frac {c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 c^{2} d^{2} x}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 112, normalized size = 1.19 \[ -\frac {c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{6} x + d e^{5}} + \frac {c^{2} e^{2} x^{3} - 3 \, c^{2} d e x^{2} + 3 \, {\left (3 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} x}{3 \, e^{4}} - \frac {4 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 116, normalized size = 1.23 \[ x\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,a\,c}{e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )}{e^5}+\frac {c^2\,x^3}{3\,e^2}-\frac {a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}{e\,\left (x\,e^5+d\,e^4\right )}-\frac {c^2\,d\,x^2}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 107, normalized size = 1.14 \[ - \frac {c^{2} d x^{2}}{e^{3}} + \frac {c^{2} x^{3}}{3 e^{2}} - \frac {4 c d \left (a e^{2} + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + x \left (\frac {2 a c}{e^{2}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {- a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4}}{d e^{5} + e^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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