Optimal. Leaf size=165 \[ -\frac {4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}+\frac {c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac {3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac {\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac {2 c^3 d x^2}{e^5}+\frac {c^3 x^3}{3 e^4} \]
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Rubi [A] time = 0.16, antiderivative size = 165, 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 = {697} \begin {gather*} \frac {c^2 x \left (3 a e^2+10 c d^2\right )}{e^6}-\frac {4 c^2 d \left (3 a e^2+5 c d^2\right ) \log (d+e x)}{e^7}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7 (d+e x)}+\frac {3 c d \left (a e^2+c d^2\right )^2}{e^7 (d+e x)^2}-\frac {\left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^3}-\frac {2 c^3 d x^2}{e^5}+\frac {c^3 x^3}{3 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac {c^2 \left (10 c d^2+3 a e^2\right )}{e^6}-\frac {4 c^3 d x}{e^5}+\frac {c^3 x^2}{e^4}+\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^4}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^3}+\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)^2}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)}\right ) \, dx\\ &=\frac {c^2 \left (10 c d^2+3 a e^2\right ) x}{e^6}-\frac {2 c^3 d x^2}{e^5}+\frac {c^3 x^3}{3 e^4}-\frac {\left (c d^2+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 c d \left (c d^2+a e^2\right )^2}{e^7 (d+e x)^2}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^7 (d+e x)}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) \log (d+e x)}{e^7}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 197, normalized size = 1.19 \begin {gather*} \frac {-a^3 e^6-3 a^2 c e^4 \left (d^2+3 d e x+3 e^2 x^2\right )-12 c^2 d (d+e x)^3 \left (3 a e^2+5 c d^2\right ) \log (d+e x)+3 a c^2 e^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 )+c^3 \left (-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^2\right )^3}{(d+e x)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 334, normalized size = 2.02 \begin {gather*} \frac {c^{3} e^{6} x^{6} - 3 \, c^{3} d e^{5} x^{5} - 37 \, c^{3} d^{6} - 39 \, a c^{2} d^{4} e^{2} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 3 \, {\left (5 \, c^{3} d^{2} e^{4} + 3 \, a c^{2} e^{6}\right )} x^{4} + {\left (73 \, c^{3} d^{3} e^{3} + 27 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (13 \, c^{3} d^{4} e^{2} - 9 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} x^{2} - 3 \, {\left (17 \, c^{3} d^{5} e + 27 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x - 12 \, {\left (5 \, c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + {\left (5 \, c^{3} d^{3} e^{3} + 3 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (5 \, c^{3} d^{4} e^{2} + 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \, {\left (5 \, c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 192, normalized size = 1.16 \begin {gather*} -4 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{3} \, {\left (c^{3} x^{3} e^{8} - 6 \, c^{3} d x^{2} e^{7} + 30 \, c^{3} d^{2} x e^{6} + 9 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac {{\left (37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \, {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} e^{\left (-7\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 258, normalized size = 1.56 \begin {gather*} -\frac {a^{3}}{3 \left (e x +d \right )^{3} e}-\frac {a^{2} c \,d^{2}}{\left (e x +d \right )^{3} e^{3}}-\frac {a \,c^{2} d^{4}}{\left (e x +d \right )^{3} e^{5}}-\frac {c^{3} d^{6}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {c^{3} x^{3}}{3 e^{4}}+\frac {3 a^{2} c d}{\left (e x +d \right )^{2} e^{3}}+\frac {6 a \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{5}}+\frac {3 c^{3} d^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {2 c^{3} d \,x^{2}}{e^{5}}-\frac {3 a^{2} c}{\left (e x +d \right ) e^{3}}-\frac {18 a \,c^{2} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {12 a \,c^{2} d \ln \left (e x +d \right )}{e^{5}}+\frac {3 a \,c^{2} x}{e^{4}}-\frac {15 c^{3} d^{4}}{\left (e x +d \right ) e^{7}}-\frac {20 c^{3} d^{3} \ln \left (e x +d \right )}{e^{7}}+\frac {10 c^{3} d^{2} x}{e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 226, normalized size = 1.37 \begin {gather*} -\frac {37 \, c^{3} d^{6} + 39 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 9 \, {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (9 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{3 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {c^{3} e^{2} x^{3} - 6 \, c^{3} d e x^{2} + 3 \, {\left (10 \, c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x}{3 \, e^{6}} - \frac {4 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 228, normalized size = 1.38 \begin {gather*} x\,\left (\frac {3\,a\,c^2}{e^4}+\frac {10\,c^3\,d^2}{e^6}\right )-\frac {x^2\,\left (3\,a^2\,c\,e^5+18\,a\,c^2\,d^2\,e^3+15\,c^3\,d^4\,e\right )+\frac {a^3\,e^6+3\,a^2\,c\,d^2\,e^4+39\,a\,c^2\,d^4\,e^2+37\,c^3\,d^6}{3\,e}+x\,\left (3\,a^2\,c\,d\,e^4+30\,a\,c^2\,d^3\,e^2+27\,c^3\,d^5\right )}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}-\frac {\ln \left (d+e\,x\right )\,\left (20\,c^3\,d^3+12\,a\,c^2\,d\,e^2\right )}{e^7}+\frac {c^3\,x^3}{3\,e^4}-\frac {2\,c^3\,d\,x^2}{e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 238, normalized size = 1.44 \begin {gather*} - \frac {2 c^{3} d x^{2}}{e^{5}} + \frac {c^{3} x^{3}}{3 e^{4}} - \frac {4 c^{2} d \left (3 a e^{2} + 5 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x \left (\frac {3 a c^{2}}{e^{4}} + \frac {10 c^{3} d^{2}}{e^{6}}\right ) + \frac {- a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 39 a c^{2} d^{4} e^{2} - 37 c^{3} d^{6} + x^{2} \left (- 9 a^{2} c e^{6} - 54 a c^{2} d^{2} e^{4} - 45 c^{3} d^{4} e^{2}\right ) + x \left (- 9 a^{2} c d e^{5} - 90 a c^{2} d^{3} e^{3} - 81 c^{3} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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