Optimal. Leaf size=190 \[ -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac {2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac {\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac {c^3}{3 e^7 (d+e x)^3}+\frac {3 c^3 d}{2 e^7 (d+e x)^4} \]
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Rubi [A] time = 0.12, antiderivative size = 190, 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} \[ -\frac {3 c^2 \left (a e^2+5 c d^2\right )}{5 e^7 (d+e x)^5}+\frac {2 c^2 d \left (3 a e^2+5 c d^2\right )}{3 e^7 (d+e x)^6}-\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{7 e^7 (d+e x)^7}+\frac {3 c d \left (a e^2+c d^2\right )^2}{4 e^7 (d+e x)^8}-\frac {\left (a e^2+c d^2\right )^3}{9 e^7 (d+e x)^9}-\frac {c^3}{3 e^7 (d+e x)^3}+\frac {3 c^3 d}{2 e^7 (d+e x)^4} \]
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)^{10}} \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{10}}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^9}+\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)^8}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right )}{e^6 (d+e x)^7}+\frac {3 c^2 \left (5 c d^2+a e^2\right )}{e^6 (d+e x)^6}-\frac {6 c^3 d}{e^6 (d+e x)^5}+\frac {c^3}{e^6 (d+e x)^4}\right ) \, dx\\ &=-\frac {\left (c d^2+a e^2\right )^3}{9 e^7 (d+e x)^9}+\frac {3 c d \left (c d^2+a e^2\right )^2}{4 e^7 (d+e x)^8}-\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{7 e^7 (d+e x)^7}+\frac {2 c^2 d \left (5 c d^2+3 a e^2\right )}{3 e^7 (d+e x)^6}-\frac {3 c^2 \left (5 c d^2+a e^2\right )}{5 e^7 (d+e x)^5}+\frac {3 c^3 d}{2 e^7 (d+e x)^4}-\frac {c^3}{3 e^7 (d+e x)^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 163, normalized size = 0.86 \[ -\frac {140 a^3 e^6+15 a^2 c e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{1260 e^7 (d+e x)^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 293, normalized size = 1.54 \[ -\frac {420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \, {\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \, {\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \, {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 191, normalized size = 1.01 \[ -\frac {{\left (420 \, c^{3} x^{6} e^{6} + 630 \, c^{3} d x^{5} e^{5} + 630 \, c^{3} d^{2} x^{4} e^{4} + 420 \, c^{3} d^{3} x^{3} e^{3} + 180 \, c^{3} d^{4} x^{2} e^{2} + 45 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 756 \, a c^{2} x^{4} e^{6} + 504 \, a c^{2} d x^{3} e^{5} + 216 \, a c^{2} d^{2} x^{2} e^{4} + 54 \, a c^{2} d^{3} x e^{3} + 6 \, a c^{2} d^{4} e^{2} + 540 \, a^{2} c x^{2} e^{6} + 135 \, a^{2} c d x e^{5} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{1260 \, {\left (x e + d\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 218, normalized size = 1.15 \[ \frac {3 c^{3} d}{2 \left (e x +d \right )^{4} e^{7}}-\frac {c^{3}}{3 \left (e x +d \right )^{3} e^{7}}+\frac {2 \left (3 a \,e^{2}+5 c \,d^{2}\right ) c^{2} d}{3 \left (e x +d \right )^{6} e^{7}}-\frac {3 \left (a \,e^{2}+5 c \,d^{2}\right ) c^{2}}{5 \left (e x +d \right )^{5} e^{7}}+\frac {3 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{4 \left (e x +d \right )^{8} e^{7}}-\frac {3 \left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+5 c^{2} d^{4}\right ) c}{7 \left (e x +d \right )^{7} e^{7}}-\frac {e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 a \,c^{2} d^{4} e^{2}+c^{3} d^{6}}{9 \left (e x +d \right )^{9} e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.59, size = 293, normalized size = 1.54 \[ -\frac {420 \, c^{3} e^{6} x^{6} + 630 \, c^{3} d e^{5} x^{5} + 5 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 140 \, a^{3} e^{6} + 126 \, {\left (5 \, c^{3} d^{2} e^{4} + 6 \, a c^{2} e^{6}\right )} x^{4} + 84 \, {\left (5 \, c^{3} d^{3} e^{3} + 6 \, a c^{2} d e^{5}\right )} x^{3} + 36 \, {\left (5 \, c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} + 15 \, a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (5 \, c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x}{1260 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 287, normalized size = 1.51 \[ -\frac {\frac {140\,a^3\,e^6+15\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2+5\,c^3\,d^6}{1260\,e^7}+\frac {c^3\,x^6}{3\,e}+\frac {c^3\,d\,x^5}{2\,e^2}+\frac {c^2\,x^4\,\left (5\,c\,d^2+6\,a\,e^2\right )}{10\,e^3}+\frac {c\,x^2\,\left (15\,a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{35\,e^5}+\frac {c\,d\,x\,\left (15\,a^2\,e^4+6\,a\,c\,d^2\,e^2+5\,c^2\,d^4\right )}{140\,e^6}+\frac {c^2\,d\,x^3\,\left (5\,c\,d^2+6\,a\,e^2\right )}{15\,e^4}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.18, size = 313, normalized size = 1.65 \[ \frac {- 140 a^{3} e^{6} - 15 a^{2} c d^{2} e^{4} - 6 a c^{2} d^{4} e^{2} - 5 c^{3} d^{6} - 630 c^{3} d e^{5} x^{5} - 420 c^{3} e^{6} x^{6} + x^{4} \left (- 756 a c^{2} e^{6} - 630 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 504 a c^{2} d e^{5} - 420 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 540 a^{2} c e^{6} - 216 a c^{2} d^{2} e^{4} - 180 c^{3} d^{4} e^{2}\right ) + x \left (- 135 a^{2} c d e^{5} - 54 a c^{2} d^{3} e^{3} - 45 c^{3} d^{5} e\right )}{1260 d^{9} e^{7} + 11340 d^{8} e^{8} x + 45360 d^{7} e^{9} x^{2} + 105840 d^{6} e^{10} x^{3} + 158760 d^{5} e^{11} x^{4} + 158760 d^{4} e^{12} x^{5} + 105840 d^{3} e^{13} x^{6} + 45360 d^{2} e^{14} x^{7} + 11340 d e^{15} x^{8} + 1260 e^{16} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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