Optimal. Leaf size=264 \[ -\frac {8 c d \left (c d^2+a e^2\right )^3 x}{e^8}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^2}{e^9}-\frac {8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^3}{3 e^9}+\frac {c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^4}{2 e^9}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^5}{5 e^9}+\frac {2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^6}{3 e^9}-\frac {8 c^4 d (d+e x)^7}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9}+\frac {\left (c d^2+a e^2\right )^4 \log (d+e x)}{e^9} \]
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Rubi [A]
time = 0.19, antiderivative size = 264, 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^2 (d+e x)^4 \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{2 e^9}+\frac {2 c^3 (d+e x)^6 \left (a e^2+7 c d^2\right )}{3 e^9}-\frac {8 c^3 d (d+e x)^5 \left (3 a e^2+7 c d^2\right )}{5 e^9}-\frac {8 c^2 d (d+e x)^3 \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{3 e^9}+\frac {2 c (d+e x)^2 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{e^9}+\frac {\left (a e^2+c d^2\right )^4 \log (d+e x)}{e^9}-\frac {8 c d x \left (a e^2+c d^2\right )^3}{e^8}+\frac {c^4 (d+e x)^8}{8 e^9}-\frac {8 c^4 d (d+e x)^7}{7 e^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 711
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^4}{d+e x} \, dx &=\int \left (-\frac {8 c d \left (c d^2+a e^2\right )^3}{e^8}+\frac {\left (c d^2+a e^2\right )^4}{e^8 (d+e x)}+\frac {4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)}{e^8}+\frac {8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^2}{e^8}+\frac {2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^3}{e^8}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^4}{e^8}+\frac {4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^5}{e^8}-\frac {8 c^4 d (d+e x)^6}{e^8}+\frac {c^4 (d+e x)^7}{e^8}\right ) \, dx\\ &=-\frac {8 c d \left (c d^2+a e^2\right )^3 x}{e^8}+\frac {2 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^2}{e^9}-\frac {8 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^3}{3 e^9}+\frac {c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^4}{2 e^9}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^5}{5 e^9}+\frac {2 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^6}{3 e^9}-\frac {8 c^4 d (d+e x)^7}{7 e^9}+\frac {c^4 (d+e x)^8}{8 e^9}+\frac {\left (c d^2+a e^2\right )^4 \log (d+e x)}{e^9}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 227, normalized size = 0.86 \begin {gather*} \frac {c x \left (1680 a^3 e^6 (-2 d+e x)+420 a^2 c e^4 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+56 a c^2 e^2 \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+c^3 \left (-840 d^7+420 d^6 e x-280 d^5 e^2 x^2+210 d^4 e^3 x^3-168 d^3 e^4 x^4+140 d^2 e^5 x^5-120 d e^6 x^6+105 e^7 x^7\right )\right )}{840 e^8}+\frac {\left (c d^2+a e^2\right )^4 \log (d+e x)}{e^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 366, normalized size = 1.39
method | result | size |
norman | \(\frac {c^{4} x^{8}}{8 e}+\frac {c \left (4 e^{6} a^{3}+6 e^{4} d^{2} a^{2} c +4 d^{4} e^{2} c^{2} a +d^{6} c^{3}\right ) x^{2}}{2 e^{7}}+\frac {c^{2} \left (6 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{4}}{4 e^{5}}-\frac {d \,c^{4} x^{7}}{7 e^{2}}+\frac {c^{3} \left (4 e^{2} a +c \,d^{2}\right ) x^{6}}{6 e^{3}}-\frac {c d \left (4 e^{6} a^{3}+6 e^{4} d^{2} a^{2} c +4 d^{4} e^{2} c^{2} a +d^{6} c^{3}\right ) x}{e^{8}}-\frac {d \,c^{2} \left (6 a^{2} e^{4}+4 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{3}}{3 e^{6}}-\frac {d \,c^{3} \left (4 e^{2} a +c \,d^{2}\right ) x^{5}}{5 e^{4}}+\frac {\left (a^{4} e^{8}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(307\) |
risch | \(\frac {c^{3} x^{4} a \,d^{2}}{e^{3}}-\frac {2 c^{2} x^{3} a^{2} d}{e^{2}}-\frac {4 c^{3} x^{3} a \,d^{3}}{3 e^{4}}+\frac {3 c^{2} a^{2} d^{2} x^{2}}{e^{3}}+\frac {2 c^{3} a \,d^{4} x^{2}}{e^{5}}+\frac {\ln \left (e x +d \right ) a^{4}}{e}+\frac {c^{4} x^{8}}{8 e}+\frac {2 c^{3} x^{6} a}{3 e}+\frac {c^{4} x^{6} d^{2}}{6 e^{3}}-\frac {c^{4} x^{5} d^{3}}{5 e^{4}}+\frac {3 c^{2} x^{4} a^{2}}{2 e}+\frac {c^{4} x^{4} d^{4}}{4 e^{5}}-\frac {c^{4} x^{3} d^{5}}{3 e^{6}}+\frac {2 c \,a^{3} x^{2}}{e}+\frac {c^{4} d^{6} x^{2}}{2 e^{7}}-\frac {c^{4} d^{7} x}{e^{8}}+\frac {\ln \left (e x +d \right ) c^{4} d^{8}}{e^{9}}-\frac {d \,c^{4} x^{7}}{7 e^{2}}-\frac {4 c \,a^{3} d x}{e^{2}}-\frac {6 c^{2} a^{2} d^{3} x}{e^{4}}-\frac {4 c^{3} a \,d^{5} x}{e^{6}}-\frac {4 c^{3} x^{5} d a}{5 e^{2}}+\frac {4 \ln \left (e x +d \right ) a^{3} c \,d^{2}}{e^{3}}+\frac {6 \ln \left (e x +d \right ) a^{2} c^{2} d^{4}}{e^{5}}+\frac {4 \ln \left (e x +d \right ) a \,c^{3} d^{6}}{e^{7}}\) | \(358\) |
default | \(-\frac {c \left (-\frac {c^{3} x^{8} e^{7}}{8}+\frac {d \,c^{3} x^{7} e^{6}}{7}+\frac {\left (-e^{5} \left (2 e^{2} a +c \,d^{2}\right ) c^{2}-2 e^{7} c^{2} a \right ) x^{6}}{6}+\frac {\left (d \left (2 e^{2} a +c \,d^{2}\right ) e^{4} c^{2}+2 d \,e^{6} c^{2} a \right ) x^{5}}{5}+\frac {\left (-2 e^{5} \left (2 e^{2} a +c \,d^{2}\right ) a c -e^{3} c \left (2 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )\right ) x^{4}}{4}+\frac {\left (2 d \left (2 e^{2} a +c \,d^{2}\right ) a c \,e^{4}+d \,e^{2} c \left (2 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )\right ) x^{3}}{3}-\frac {e \left (2 e^{2} a +c \,d^{2}\right ) \left (2 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{2}}{2}+d \left (2 e^{2} a +c \,d^{2}\right ) \left (2 a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x \right )}{e^{8}}+\frac {\left (a^{4} e^{8}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \ln \left (e x +d \right )}{e^{9}}\) | \(366\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 297, normalized size = 1.12 \begin {gather*} {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} e^{\left (-9\right )} \log \left (x e + d\right ) + \frac {1}{840} \, {\left (105 \, c^{4} x^{8} e^{7} - 120 \, c^{4} d x^{7} e^{6} + 140 \, {\left (c^{4} d^{2} e^{5} + 4 \, a c^{3} e^{7}\right )} x^{6} - 168 \, {\left (c^{4} d^{3} e^{4} + 4 \, a c^{3} d e^{6}\right )} x^{5} + 210 \, {\left (c^{4} d^{4} e^{3} + 4 \, a c^{3} d^{2} e^{5} + 6 \, a^{2} c^{2} e^{7}\right )} x^{4} - 280 \, {\left (c^{4} d^{5} e^{2} + 4 \, a c^{3} d^{3} e^{4} + 6 \, a^{2} c^{2} d e^{6}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e + 4 \, a c^{3} d^{4} e^{3} + 6 \, a^{2} c^{2} d^{2} e^{5} + 4 \, a^{3} c e^{7}\right )} x^{2} - 840 \, {\left (c^{4} d^{7} + 4 \, a c^{3} d^{5} e^{2} + 6 \, a^{2} c^{2} d^{3} e^{4} + 4 \, a^{3} c d e^{6}\right )} x\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.42, size = 306, normalized size = 1.16 \begin {gather*} \frac {1}{840} \, {\left (420 \, c^{4} d^{6} x^{2} e^{2} - 840 \, c^{4} d^{7} x e + 35 \, {\left (3 \, c^{4} x^{8} + 16 \, a c^{3} x^{6} + 36 \, a^{2} c^{2} x^{4} + 48 \, a^{3} c x^{2}\right )} e^{8} - 24 \, {\left (5 \, c^{4} d x^{7} + 28 \, a c^{3} d x^{5} + 70 \, a^{2} c^{2} d x^{3} + 140 \, a^{3} c d x\right )} e^{7} + 140 \, {\left (c^{4} d^{2} x^{6} + 6 \, a c^{3} d^{2} x^{4} + 18 \, a^{2} c^{2} d^{2} x^{2}\right )} e^{6} - 56 \, {\left (3 \, c^{4} d^{3} x^{5} + 20 \, a c^{3} d^{3} x^{3} + 90 \, a^{2} c^{2} d^{3} x\right )} e^{5} + 210 \, {\left (c^{4} d^{4} x^{4} + 8 \, a c^{3} d^{4} x^{2}\right )} e^{4} - 280 \, {\left (c^{4} d^{5} x^{3} + 12 \, a c^{3} d^{5} x\right )} e^{3} + 840 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (x e + d\right )\right )} e^{\left (-9\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 292, normalized size = 1.11 \begin {gather*} - \frac {c^{4} d x^{7}}{7 e^{2}} + \frac {c^{4} x^{8}}{8 e} + x^{6} \cdot \left (\frac {2 a c^{3}}{3 e} + \frac {c^{4} d^{2}}{6 e^{3}}\right ) + x^{5} \left (- \frac {4 a c^{3} d}{5 e^{2}} - \frac {c^{4} d^{3}}{5 e^{4}}\right ) + x^{4} \cdot \left (\frac {3 a^{2} c^{2}}{2 e} + \frac {a c^{3} d^{2}}{e^{3}} + \frac {c^{4} d^{4}}{4 e^{5}}\right ) + x^{3} \left (- \frac {2 a^{2} c^{2} d}{e^{2}} - \frac {4 a c^{3} d^{3}}{3 e^{4}} - \frac {c^{4} d^{5}}{3 e^{6}}\right ) + x^{2} \cdot \left (\frac {2 a^{3} c}{e} + \frac {3 a^{2} c^{2} d^{2}}{e^{3}} + \frac {2 a c^{3} d^{4}}{e^{5}} + \frac {c^{4} d^{6}}{2 e^{7}}\right ) + x \left (- \frac {4 a^{3} c d}{e^{2}} - \frac {6 a^{2} c^{2} d^{3}}{e^{4}} - \frac {4 a c^{3} d^{5}}{e^{6}} - \frac {c^{4} d^{7}}{e^{8}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{4} \log {\left (d + e x \right )}}{e^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.81, size = 316, normalized size = 1.20 \begin {gather*} {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{840} \, {\left (105 \, c^{4} x^{8} e^{7} - 120 \, c^{4} d x^{7} e^{6} + 140 \, c^{4} d^{2} x^{6} e^{5} - 168 \, c^{4} d^{3} x^{5} e^{4} + 210 \, c^{4} d^{4} x^{4} e^{3} - 280 \, c^{4} d^{5} x^{3} e^{2} + 420 \, c^{4} d^{6} x^{2} e - 840 \, c^{4} d^{7} x + 560 \, a c^{3} x^{6} e^{7} - 672 \, a c^{3} d x^{5} e^{6} + 840 \, a c^{3} d^{2} x^{4} e^{5} - 1120 \, a c^{3} d^{3} x^{3} e^{4} + 1680 \, a c^{3} d^{4} x^{2} e^{3} - 3360 \, a c^{3} d^{5} x e^{2} + 1260 \, a^{2} c^{2} x^{4} e^{7} - 1680 \, a^{2} c^{2} d x^{3} e^{6} + 2520 \, a^{2} c^{2} d^{2} x^{2} e^{5} - 5040 \, a^{2} c^{2} d^{3} x e^{4} + 1680 \, a^{3} c x^{2} e^{7} - 3360 \, a^{3} c d x e^{6}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 357, normalized size = 1.35 \begin {gather*} x^2\,\left (\frac {d^2\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{e^2}+\frac {6\,a^2\,c^2}{e}\right )}{2\,e^2}+\frac {2\,a^3\,c}{e}\right )+x^4\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{4\,e^2}+\frac {3\,a^2\,c^2}{2\,e}\right )+x^6\,\left (\frac {2\,a\,c^3}{3\,e}+\frac {c^4\,d^2}{6\,e^3}\right )+\frac {\ln \left (d+e\,x\right )\,\left (a^4\,e^8+4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{e^9}+\frac {c^4\,x^8}{8\,e}-\frac {c^4\,d\,x^7}{7\,e^2}-\frac {d\,x^3\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{e^2}+\frac {6\,a^2\,c^2}{e}\right )}{3\,e}-\frac {d\,x^5\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{5\,e}-\frac {d\,x\,\left (\frac {d^2\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e}+\frac {c^4\,d^2}{e^3}\right )}{e^2}+\frac {6\,a^2\,c^2}{e}\right )}{e^2}+\frac {4\,a^3\,c}{e}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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