Optimal. Leaf size=210 \[ \frac {16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {a x}{d \left (d+e x^2\right )^{11/2}} \]
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Rubi [A] time = 0.22, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1155, 1803, 12, 271, 264} \[ \frac {16 e^3 x^{11} \left (8 e (10 a e+b d)+3 c d^2\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 x^9 \left (8 e (10 a e+b d)+3 c d^2\right )}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {2 e x^7 \left (8 e (10 a e+b d)+3 c d^2\right )}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {x^5 \left (8 e (10 a e+b d)+3 c d^2\right )}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {x^3 (10 a e+b d)}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {a x}{d \left (d+e x^2\right )^{11/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 1155
Rule 1803
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {\int \frac {x^2 \left (10 a e+d \left (b+c x^2\right )\right )}{\left (d+e x^2\right )^{13/2}} \, dx}{d}\\ &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\int \frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^4}{\left (d+e x^2\right )^{13/2}} \, dx}{3 d^2}\\ &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {1}{3} \left (3 c+\frac {8 e (b d+10 a e)}{d^2}\right ) \int \frac {x^4}{\left (d+e x^2\right )^{13/2}} \, dx\\ &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {\left (2 e \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac {x^6}{\left (d+e x^2\right )^{13/2}} \, dx}{5 d^3}\\ &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {\left (8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac {x^8}{\left (d+e x^2\right )^{13/2}} \, dx}{35 d^4}\\ &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {\left (16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right )\right ) \int \frac {x^{10}}{\left (d+e x^2\right )^{13/2}} \, dx}{315 d^5}\\ &=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right ) x^{11}}{3465 d^6 \left (d+e x^2\right )^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 167, normalized size = 0.80 \[ \frac {5 a \left (693 d^5 x+2310 d^4 e x^3+3696 d^3 e^2 x^5+3168 d^2 e^3 x^7+1408 d e^4 x^9+256 e^5 x^{11}\right )+d x^3 \left (b \left (1155 d^4+1848 d^3 e x^2+1584 d^2 e^2 x^4+704 d e^3 x^6+128 e^4 x^8\right )+3 c d x^2 \left (231 d^3+198 d^2 e x^2+88 d e^2 x^4+16 e^3 x^6\right )\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 224, normalized size = 1.07 \[ \frac {{\left (16 \, {\left (3 \, c d^{2} e^{3} + 8 \, b d e^{4} + 80 \, a e^{5}\right )} x^{11} + 88 \, {\left (3 \, c d^{3} e^{2} + 8 \, b d^{2} e^{3} + 80 \, a d e^{4}\right )} x^{9} + 198 \, {\left (3 \, c d^{4} e + 8 \, b d^{3} e^{2} + 80 \, a d^{2} e^{3}\right )} x^{7} + 3465 \, a d^{5} x + 231 \, {\left (3 \, c d^{5} + 8 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} x^{5} + 1155 \, {\left (b d^{5} + 10 \, a d^{4} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{3465 \, {\left (d^{6} e^{6} x^{12} + 6 \, d^{7} e^{5} x^{10} + 15 \, d^{8} e^{4} x^{8} + 20 \, d^{9} e^{3} x^{6} + 15 \, d^{10} e^{2} x^{4} + 6 \, d^{11} e x^{2} + d^{12}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 189, normalized size = 0.90 \[ \frac {{\left ({\left ({\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, {\left (3 \, c d^{2} e^{8} + 8 \, b d e^{9} + 80 \, a e^{10}\right )} x^{2} e^{\left (-5\right )}}{d^{6}} + \frac {11 \, {\left (3 \, c d^{3} e^{7} + 8 \, b d^{2} e^{8} + 80 \, a d e^{9}\right )} e^{\left (-5\right )}}{d^{6}}\right )} + \frac {99 \, {\left (3 \, c d^{4} e^{6} + 8 \, b d^{3} e^{7} + 80 \, a d^{2} e^{8}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac {231 \, {\left (3 \, c d^{5} e^{5} + 8 \, b d^{4} e^{6} + 80 \, a d^{3} e^{7}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac {1155 \, {\left (b d^{5} e^{5} + 10 \, a d^{4} e^{6}\right )} e^{\left (-5\right )}}{d^{6}}\right )} x^{2} + \frac {3465 \, a}{d}\right )} x}{3465 \, {\left (x^{2} e + d\right )}^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 172, normalized size = 0.82 \[ \frac {\left (1280 a \,e^{5} x^{10}+128 b d \,e^{4} x^{10}+48 c \,d^{2} e^{3} x^{10}+7040 a d \,e^{4} x^{8}+704 b \,d^{2} e^{3} x^{8}+264 c \,d^{3} e^{2} x^{8}+15840 a \,d^{2} e^{3} x^{6}+1584 b \,d^{3} e^{2} x^{6}+594 c \,d^{4} e \,x^{6}+18480 a \,d^{3} e^{2} x^{4}+1848 b \,d^{4} e \,x^{4}+693 c \,d^{5} x^{4}+11550 a \,d^{4} e \,x^{2}+1155 b \,d^{5} x^{2}+3465 a \,d^{5}\right ) x}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 335, normalized size = 1.60 \[ -\frac {c x^{3}}{8 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {256 \, a x}{693 \, \sqrt {e x^{2} + d} d^{6}} + \frac {128 \, a x}{693 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{5}} + \frac {32 \, a x}{231 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{4}} + \frac {80 \, a x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{3}} + \frac {10 \, a x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d^{2}} + \frac {a x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} d} + \frac {c x}{264 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} + \frac {16 \, c x}{1155 \, \sqrt {e x^{2} + d} d^{4} e^{2}} + \frac {8 \, c x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e^{2}} + \frac {2 \, c x}{385 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e^{2}} + \frac {c x}{231 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e^{2}} - \frac {3 \, c d x}{88 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{2}} - \frac {b x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {128 \, b x}{3465 \, \sqrt {e x^{2} + d} d^{5} e} + \frac {64 \, b x}{3465 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4} e} + \frac {16 \, b x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3} e} + \frac {8 \, b x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2} e} + \frac {b x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.76, size = 226, normalized size = 1.08 \[ \frac {x\,\left (\frac {a}{11\,d}-\frac {d\,\left (\frac {b}{11\,d}-\frac {c}{11\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{11/2}}-\frac {x\,\left (\frac {c}{9\,e^2}-\frac {-c\,d^2+b\,d\,e+10\,a\,e^2}{99\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{9/2}}+\frac {x\,\left (3\,c\,d^2+8\,b\,d\,e+80\,a\,e^2\right )}{693\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (6\,c\,d^2+16\,b\,d\,e+160\,a\,e^2\right )}{1155\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (24\,c\,d^2+64\,b\,d\,e+640\,a\,e^2\right )}{3465\,d^5\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (48\,c\,d^2+128\,b\,d\,e+1280\,a\,e^2\right )}{3465\,d^6\,e^2\,\sqrt {e\,x^2+d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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