Optimal. Leaf size=101 \[ \frac {c^2 x}{e^4}-\frac {\left (c d^2+a e^2\right )^2}{3 e^5 (d+e x)^3}+\frac {2 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)^2}-\frac {2 c \left (3 c d^2+a e^2\right )}{e^5 (d+e x)}-\frac {4 c^2 d \log (d+e x)}{e^5} \]
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Rubi [A]
time = 0.05, antiderivative size = 101, 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 = {711}
\begin {gather*} -\frac {2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac {2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac {\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac {4 c^2 d \log (d+e x)}{e^5}+\frac {c^2 x}{e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^2}{(d+e x)^4} \, dx &=\int \left (\frac {c^2}{e^4}+\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^4}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^3}+\frac {2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^2}-\frac {4 c^2 d}{e^4 (d+e x)}\right ) \, dx\\ &=\frac {c^2 x}{e^4}-\frac {\left (c d^2+a e^2\right )^2}{3 e^5 (d+e x)^3}+\frac {2 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)^2}-\frac {2 c \left (3 c d^2+a e^2\right )}{e^5 (d+e x)}-\frac {4 c^2 d \log (d+e x)}{e^5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 110, normalized size = 1.09 \begin {gather*} -\frac {a^2 e^4+2 a c e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+c^2 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )+12 c^2 d (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 112, normalized size = 1.11
method | result | size |
risch | \(\frac {c^{2} x}{e^{4}}+\frac {\left (-2 e^{3} a c -6 d^{2} e \,c^{2}\right ) x^{2}-2 c d \left (e^{2} a +5 c \,d^{2}\right ) x -\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+13 c^{2} d^{4}}{3 e}}{e^{4} \left (e x +d \right )^{3}}-\frac {4 c^{2} d \ln \left (e x +d \right )}{e^{5}}\) | \(106\) |
default | \(\frac {c^{2} x}{e^{4}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 c \left (e^{2} a +3 c \,d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {4 c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {2 c d \left (e^{2} a +c \,d^{2}\right )}{e^{5} \left (e x +d \right )^{2}}\) | \(112\) |
norman | \(\frac {\frac {c^{2} x^{4}}{e}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+22 c^{2} d^{4}}{3 e^{5}}-\frac {\left (2 a c \,e^{2}+12 d^{2} c^{2}\right ) x^{2}}{e^{3}}-\frac {d \left (2 a c \,e^{2}+18 d^{2} c^{2}\right ) x}{e^{4}}}{\left (e x +d \right )^{3}}-\frac {4 c^{2} d \ln \left (e x +d \right )}{e^{5}}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 121, normalized size = 1.20 \begin {gather*} -4 \, c^{2} d e^{\left (-5\right )} \log \left (x e + d\right ) + c^{2} x e^{\left (-4\right )} - \frac {13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 6 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + a^{2} e^{4} + 6 \, {\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.79, size = 171, normalized size = 1.69 \begin {gather*} -\frac {27 \, c^{2} d^{3} x e + 13 \, c^{2} d^{4} - {\left (3 \, c^{2} x^{4} - 6 \, a c x^{2} - a^{2}\right )} e^{4} - 3 \, {\left (3 \, c^{2} d x^{3} - 2 \, a c d x\right )} e^{3} + {\left (9 \, c^{2} d^{2} x^{2} + 2 \, a c d^{2}\right )} e^{2} + 12 \, {\left (c^{2} d x^{3} e^{3} + 3 \, c^{2} d^{2} x^{2} e^{2} + 3 \, c^{2} d^{3} x e + c^{2} d^{4}\right )} \log \left (x e + d\right )}{3 \, {\left (x^{3} e^{8} + 3 \, d x^{2} e^{7} + 3 \, d^{2} x e^{6} + d^{3} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 138, normalized size = 1.37 \begin {gather*} - \frac {4 c^{2} d \log {\left (d + e x \right )}}{e^{5}} + \frac {c^{2} x}{e^{4}} + \frac {- a^{2} e^{4} - 2 a c d^{2} e^{2} - 13 c^{2} d^{4} + x^{2} \left (- 6 a c e^{4} - 18 c^{2} d^{2} e^{2}\right ) + x \left (- 6 a c d e^{3} - 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 101, normalized size = 1.00 \begin {gather*} -4 \, c^{2} d e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + c^{2} x e^{\left (-4\right )} - \frac {{\left (13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 6 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + a^{2} e^{4} + 6 \, {\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 133, normalized size = 1.32 \begin {gather*} \frac {c^2\,x}{e^4}-\frac {x\,\left (10\,c^2\,d^3+2\,a\,c\,d\,e^2\right )+x^2\,\left (6\,c^2\,d^2\,e+2\,a\,c\,e^3\right )+\frac {a^2\,e^4+2\,a\,c\,d^2\,e^2+13\,c^2\,d^4}{3\,e}}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3}-\frac {4\,c^2\,d\,\ln \left (d+e\,x\right )}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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