Optimal. Leaf size=117 \[ -\frac {\left (c d^2+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {4 c d \left (c d^2+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {c \left (3 c d^2+a e^2\right )}{2 e^5 (d+e x)^4}+\frac {4 c^2 d}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 117, 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 {c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac {4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac {c^2}{2 e^5 (d+e x)^2}+\frac {4 c^2 d}{3 e^5 (d+e x)^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)^7} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^7}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^6}+\frac {2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^5}-\frac {4 c^2 d}{e^4 (d+e x)^4}+\frac {c^2}{e^4 (d+e x)^3}\right ) \, dx\\ &=-\frac {\left (c d^2+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {4 c d \left (c d^2+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {c \left (3 c d^2+a e^2\right )}{2 e^5 (d+e x)^4}+\frac {4 c^2 d}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 89, normalized size = 0.76 \begin {gather*} -\frac {5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 120, normalized size = 1.03
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{3 e^{2}}-\frac {c \left (e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {d c \left (e^{2} a +c \,d^{2}\right ) x}{5 e^{4}}-\frac {5 a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{6}}\) | \(102\) |
gosper | \(-\frac {15 c^{2} x^{4} e^{4}+20 c^{2} d \,e^{3} x^{3}+15 a c \,e^{4} x^{2}+15 c^{2} d^{2} e^{2} x^{2}+6 a c d \,e^{3} x +6 c^{2} d^{3} e x +5 a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{5} \left (e x +d \right )^{6}}\) | \(104\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{3 e^{2}}-\frac {\left (e^{3} a c +d^{2} e \,c^{2}\right ) x^{2}}{2 e^{4}}-\frac {d \left (e^{3} a c +d^{2} e \,c^{2}\right ) x}{5 e^{5}}-\frac {5 a^{2} e^{5}+a c \,d^{2} e^{3}+c^{2} d^{4} e}{30 e^{6}}}{\left (e x +d \right )^{6}}\) | \(109\) |
default | \(\frac {4 c^{2} d}{3 e^{5} \left (e x +d \right )^{3}}+\frac {4 c d \left (e^{2} a +c \,d^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{2 e^{5} \left (e x +d \right )^{2}}-\frac {c \left (e^{2} a +3 c \,d^{2}\right )}{2 e^{5} \left (e x +d \right )^{4}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{6 e^{5} \left (e x +d \right )^{6}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 146, normalized size = 1.25 \begin {gather*} -\frac {15 \, c^{2} x^{4} e^{4} + 20 \, c^{2} d x^{3} e^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 15 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 146, normalized size = 1.25 \begin {gather*} -\frac {6 \, c^{2} d^{3} x e + c^{2} d^{4} + 5 \, {\left (3 \, c^{2} x^{4} + 3 \, a c x^{2} + a^{2}\right )} e^{4} + 2 \, {\left (10 \, c^{2} d x^{3} + 3 \, a c d x\right )} e^{3} + {\left (15 \, c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}}{30 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.26, size = 172, normalized size = 1.47 \begin {gather*} \frac {- 5 a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 20 c^{2} d e^{3} x^{3} - 15 c^{2} e^{4} x^{4} + x^{2} \left (- 15 a c e^{4} - 15 c^{2} d^{2} e^{2}\right ) + x \left (- 6 a c d e^{3} - 6 c^{2} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.59, size = 97, normalized size = 0.83 \begin {gather*} -\frac {{\left (15 \, c^{2} x^{4} e^{4} + 20 \, c^{2} d x^{3} e^{3} + 15 \, c^{2} d^{2} x^{2} e^{2} + 6 \, c^{2} d^{3} x e + c^{2} d^{4} + 15 \, a c x^{2} e^{4} + 6 \, a c d x e^{3} + a c d^{2} e^{2} + 5 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 157, normalized size = 1.34 \begin {gather*} -\frac {\frac {5\,a^2\,e^4+a\,c\,d^2\,e^2+c^2\,d^4}{30\,e^5}+\frac {c^2\,x^4}{2\,e}+\frac {2\,c^2\,d\,x^3}{3\,e^2}+\frac {c\,x^2\,\left (c\,d^2+a\,e^2\right )}{2\,e^3}+\frac {c\,d\,x\,\left (c\,d^2+a\,e^2\right )}{5\,e^4}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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