Optimal. Leaf size=100 \[ \frac {4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3} \]
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Rubi [A] time = 0.08, antiderivative size = 100, 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} \begin {gather*} \frac {4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3} \end {gather*}
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)^3} \, dx &=\int \left (-\frac {3 c^2 d}{e^4}+\frac {c^2 x}{e^3}+\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^3}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^2}+\frac {2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx\\ &=-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac {2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 111, normalized size = 1.11 \begin {gather*} \frac {-a^2 e^4+4 c (d+e x)^2 \left (a e^2+3 c d^2\right ) \log (d+e x)+2 a c d e^2 (3 d+4 e x)+c^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )}{2 e^5 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 177, normalized size = 1.77 \begin {gather*} \frac {c^{2} e^{4} x^{4} - 4 \, c^{2} d e^{3} x^{3} - 11 \, c^{2} d^{2} e^{2} x^{2} + 7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x + 4 \, {\left (3 \, c^{2} d^{4} + a c d^{2} e^{2} + {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 106, normalized size = 1.06 \begin {gather*} 2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c^{2} x^{2} e^{3} - 6 \, c^{2} d x e^{2}\right )} e^{\left (-6\right )} + \frac {{\left (7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 136, normalized size = 1.36 \begin {gather*} -\frac {a^{2}}{2 \left (e x +d \right )^{2} e}-\frac {a c \,d^{2}}{\left (e x +d \right )^{2} e^{3}}-\frac {c^{2} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {c^{2} x^{2}}{2 e^{3}}+\frac {4 a c d}{\left (e x +d \right ) e^{3}}+\frac {2 a c \ln \left (e x +d \right )}{e^{3}}+\frac {4 c^{2} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 c^{2} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 c^{2} d x}{e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 120, normalized size = 1.20 \begin {gather*} \frac {7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac {c^{2} e x^{2} - 6 \, c^{2} d x}{2 \, e^{4}} + \frac {2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 125, normalized size = 1.25 \begin {gather*} \frac {x\,\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )+\frac {-a^2\,e^4+6\,a\,c\,d^2\,e^2+7\,c^2\,d^4}{2\,e}}{d^2\,e^4+2\,d\,e^5\,x+e^6\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (6\,c^2\,d^2+2\,a\,c\,e^2\right )}{e^5}+\frac {c^2\,x^2}{2\,e^3}-\frac {3\,c^2\,d\,x}{e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 122, normalized size = 1.22 \begin {gather*} - \frac {3 c^{2} d x}{e^{4}} + \frac {c^{2} x^{2}}{2 e^{3}} + \frac {2 c \left (a e^{2} + 3 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4} + x \left (8 a c d e^{3} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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