Optimal. Leaf size=184 \[ -\frac {\left (c d^2+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c d \left (c d^2+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Rubi [A]
time = 0.09, antiderivative size = 184, 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 {3 c^2 \left (a e^2+5 c d^2\right )}{2 e^7 (d+e x)^2}+\frac {4 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{4 e^7 (d+e x)^4}+\frac {6 c d \left (a e^2+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^7} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^7}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^5}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^4}+\frac {3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^3}-\frac {6 c^3 d}{e^6 (d+e x)^2}+\frac {c^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {\left (c d^2+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {6 c d \left (c d^2+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{4 e^7 (d+e x)^4}+\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{2 e^7 (d+e x)^2}+\frac {6 c^3 d}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 172, normalized size = 0.93 \begin {gather*} \frac {-10 a^3 e^6-3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )-6 a c^2 e^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 )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 216, normalized size = 1.17
method | result | size |
risch | \(\frac {\frac {6 c^{3} d \,x^{5}}{e^{2}}-\frac {3 c^{2} \left (e^{2} a -15 c \,d^{2}\right ) x^{4}}{2 e^{3}}-\frac {2 c^{2} d \left (3 e^{2} a -55 c \,d^{2}\right ) x^{3}}{3 e^{4}}-\frac {c \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-125 c^{2} d^{4}\right ) x^{2}}{4 e^{5}}-\frac {c d \left (3 a^{2} e^{4}+6 a c \,d^{2} e^{2}-137 c^{2} d^{4}\right ) x}{10 e^{6}}-\frac {10 e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a -147 d^{6} c^{3}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(201\) |
norman | \(\frac {-\frac {10 e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +6 d^{4} e^{2} c^{2} a -147 d^{6} c^{3}}{60 e^{7}}-\frac {3 \left (e^{2} c^{2} a -15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (3 e^{4} a^{2} c +6 d^{2} e^{2} c^{2} a -125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}+\frac {6 c^{3} d \,x^{5}}{e^{2}}-\frac {2 d \left (3 e^{2} c^{2} a -55 c^{3} d^{2}\right ) x^{3}}{3 e^{4}}-\frac {d \left (3 e^{4} a^{2} c +6 d^{2} e^{2} c^{2} a -137 d^{4} c^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(209\) |
default | \(\frac {4 c^{2} d \left (3 e^{2} a +5 c \,d^{2}\right )}{3 e^{7} \left (e x +d \right )^{3}}+\frac {6 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{5 e^{7} \left (e x +d \right )^{5}}+\frac {6 c^{3} d}{e^{7} \left (e x +d \right )}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {3 c^{2} \left (e^{2} a +5 c \,d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {3 c \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {e^{6} a^{3}+3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}\) | \(216\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 244, normalized size = 1.33 \begin {gather*} c^{3} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {360 \, c^{3} d x^{5} e^{5} + 147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - a c^{2} e^{6}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{3} - 3 \, a c^{2} d e^{5}\right )} x^{3} - 10 \, a^{3} e^{6} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} - 3 \, a^{2} c d e^{5}\right )} x}{60 \, {\left (x^{6} e^{13} + 6 \, d x^{5} e^{12} + 15 \, d^{2} x^{4} e^{11} + 20 \, d^{3} x^{3} e^{10} + 15 \, d^{4} x^{2} e^{9} + 6 \, d^{5} x e^{8} + d^{6} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.52, size = 318, normalized size = 1.73 \begin {gather*} \frac {822 \, c^{3} d^{5} x e + 147 \, c^{3} d^{6} - 5 \, {\left (18 \, a c^{2} x^{4} + 9 \, a^{2} c x^{2} + 2 \, a^{3}\right )} e^{6} + 6 \, {\left (60 \, c^{3} d x^{5} - 20 \, a c^{2} d x^{3} - 3 \, a^{2} c d x\right )} e^{5} + 3 \, {\left (450 \, c^{3} d^{2} x^{4} - 30 \, a c^{2} d^{2} x^{2} - a^{2} c d^{2}\right )} e^{4} + 4 \, {\left (550 \, c^{3} d^{3} x^{3} - 9 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (625 \, c^{3} d^{4} x^{2} - 2 \, a c^{2} d^{4}\right )} e^{2} + 60 \, {\left (c^{3} x^{6} e^{6} + 6 \, c^{3} d x^{5} e^{5} + 15 \, c^{3} d^{2} x^{4} e^{4} + 20 \, c^{3} d^{3} x^{3} e^{3} + 15 \, c^{3} d^{4} x^{2} e^{2} + 6 \, c^{3} d^{5} x e + c^{3} d^{6}\right )} \log \left (x e + d\right )}{60 \, {\left (x^{6} e^{13} + 6 \, d x^{5} e^{12} + 15 \, d^{2} x^{4} e^{11} + 20 \, d^{3} x^{3} e^{10} + 15 \, d^{4} x^{2} e^{9} + 6 \, d^{5} x e^{8} + d^{6} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 24.23, size = 272, normalized size = 1.48 \begin {gather*} \frac {c^{3} \log {\left (d + e x \right )}}{e^{7}} + \frac {- 10 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} + 147 c^{3} d^{6} + 360 c^{3} d e^{5} x^{5} + x^{4} \left (- 90 a c^{2} e^{6} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 120 a c^{2} d e^{5} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 45 a^{2} c e^{6} - 90 a c^{2} d^{2} e^{4} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 18 a^{2} c d e^{5} - 36 a c^{2} d^{3} e^{3} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 196, normalized size = 1.07 \begin {gather*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (360 \, c^{3} d x^{5} e^{4} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - a c^{2} e^{5}\right )} x^{4} + 40 \, {\left (55 \, c^{3} d^{3} e^{2} - 3 \, a c^{2} d e^{4}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 6 \, a c^{2} d^{2} e^{3} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}\right )} x + {\left (147 \, c^{3} d^{6} - 6 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\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 = 0.37, size = 262, normalized size = 1.42 \begin {gather*} \frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {\frac {10\,a^3\,e^6+3\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2-147\,c^3\,d^6}{60\,e^7}+\frac {x^2\,\left (3\,a^2\,c\,e^4+6\,a\,c^2\,d^2\,e^2-125\,c^3\,d^4\right )}{4\,e^5}+\frac {x\,\left (3\,a^2\,c\,d\,e^4+6\,a\,c^2\,d^3\,e^2-137\,c^3\,d^5\right )}{10\,e^6}-\frac {2\,x^3\,\left (55\,c^3\,d^3-3\,a\,c^2\,d\,e^2\right )}{3\,e^4}-\frac {6\,c^3\,d\,x^5}{e^2}+\frac {3\,c^2\,x^4\,\left (a\,e^2-15\,c\,d^2\right )}{2\,e^3}}{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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