Optimal. Leaf size=100 \[ -\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac {2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5} \]
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Rubi [A]
time = 0.06, 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 = {711}
\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 711
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 = 0.03, size = 111, normalized size = 1.11 \begin {gather*} \frac {-a^2 e^4+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 )+4 c \left (3 c d^2+a e^2\right ) (d+e x)^2 \log (d+e x)}{2 e^5 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 107, normalized size = 1.07
method | result | size |
default | \(-\frac {c^{2} \left (-\frac {1}{2} e \,x^{2}+3 d x \right )}{e^{4}}+\frac {4 c d \left (e^{2} a +c \,d^{2}\right )}{e^{5} \left (e x +d \right )}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}\) | \(107\) |
norman | \(\frac {-\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}-18 c^{2} d^{4}}{2 e^{5}}+\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{e^{2}}+\frac {2 d \left (2 a c \,e^{2}+6 d^{2} c^{2}\right ) x}{e^{4}}}{\left (e x +d \right )^{2}}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(112\) |
risch | \(\frac {c^{2} x^{2}}{2 e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\left (4 d \,e^{2} a c +4 c^{2} d^{3}\right ) x -\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}-7 c^{2} d^{4}}{2 e}}{e^{4} \left (e x +d \right )^{2}}+\frac {2 c \ln \left (e x +d \right ) a}{e^{3}}+\frac {6 c^{2} \ln \left (e x +d \right ) d^{2}}{e^{5}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 114, normalized size = 1.14 \begin {gather*} 2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (c^{2} x^{2} e - 6 \, c^{2} d x\right )} e^{\left (-4\right )} + \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 (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.96, size = 167, normalized size = 1.67 \begin {gather*} \frac {2 \, c^{2} d^{3} x e + 7 \, c^{2} d^{4} + {\left (c^{2} x^{4} - a^{2}\right )} e^{4} - 4 \, {\left (c^{2} d x^{3} - 2 \, a c d x\right )} e^{3} - {\left (11 \, c^{2} d^{2} x^{2} - 6 \, a c d^{2}\right )} e^{2} + 4 \, {\left (6 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + a c x^{2} e^{4} + 2 \, a c d x e^{3} + {\left (3 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (x^{2} e^{7} + 2 \, d x e^{6} + d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.38, 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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Giac [A]
time = 1.27, 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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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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