Optimal. Leaf size=111 \[ -\frac {8 d x \sqrt {c+\frac {d}{x^2}} (5 b c-6 a d)}{15 c^4}+\frac {4 d x (5 b c-6 a d)}{15 c^3 \sqrt {c+\frac {d}{x^2}}}+\frac {x^3 (5 b c-6 a d)}{15 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^5}{5 c \sqrt {c+\frac {d}{x^2}}} \]
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Rubi [A] time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {453, 271, 192, 191} \[ \frac {x^3 (5 b c-6 a d)}{15 c^2 \sqrt {c+\frac {d}{x^2}}}-\frac {8 d x \sqrt {c+\frac {d}{x^2}} (5 b c-6 a d)}{15 c^4}+\frac {4 d x (5 b c-6 a d)}{15 c^3 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^5}{5 c \sqrt {c+\frac {d}{x^2}}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) x^4}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx &=\frac {a x^5}{5 c \sqrt {c+\frac {d}{x^2}}}+\frac {(5 b c-6 a d) \int \frac {x^2}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx}{5 c}\\ &=\frac {(5 b c-6 a d) x^3}{15 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^5}{5 c \sqrt {c+\frac {d}{x^2}}}-\frac {(4 d (5 b c-6 a d)) \int \frac {1}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx}{15 c^2}\\ &=\frac {4 d (5 b c-6 a d) x}{15 c^3 \sqrt {c+\frac {d}{x^2}}}+\frac {(5 b c-6 a d) x^3}{15 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^5}{5 c \sqrt {c+\frac {d}{x^2}}}-\frac {(8 d (5 b c-6 a d)) \int \frac {1}{\sqrt {c+\frac {d}{x^2}}} \, dx}{15 c^3}\\ &=\frac {4 d (5 b c-6 a d) x}{15 c^3 \sqrt {c+\frac {d}{x^2}}}-\frac {8 d (5 b c-6 a d) \sqrt {c+\frac {d}{x^2}} x}{15 c^4}+\frac {(5 b c-6 a d) x^3}{15 c^2 \sqrt {c+\frac {d}{x^2}}}+\frac {a x^5}{5 c \sqrt {c+\frac {d}{x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 80, normalized size = 0.72 \[ \frac {3 a \left (c^3 x^6-2 c^2 d x^4+8 c d^2 x^2+16 d^3\right )+5 b c \left (c^2 x^4-4 c d x^2-8 d^2\right )}{15 c^4 x \sqrt {c+\frac {d}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 95, normalized size = 0.86 \[ \frac {{\left (3 \, a c^{3} x^{7} + {\left (5 \, b c^{3} - 6 \, a c^{2} d\right )} x^{5} - 4 \, {\left (5 \, b c^{2} d - 6 \, a c d^{2}\right )} x^{3} - 8 \, {\left (5 \, b c d^{2} - 6 \, a d^{3}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, {\left (c^{5} x^{2} + c^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 91, normalized size = 0.82 \[ \frac {\left (3 a \,x^{6} c^{3}-6 a \,c^{2} d \,x^{4}+5 b \,c^{3} x^{4}+24 a c \,d^{2} x^{2}-20 b \,c^{2} d \,x^{2}+48 a \,d^{3}-40 b c \,d^{2}\right ) \left (c \,x^{2}+d \right )}{15 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} c^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 128, normalized size = 1.15 \[ \frac {1}{3} \, b {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} - 6 \, \sqrt {c + \frac {d}{x^{2}}} d x}{c^{3}} - \frac {3 \, d^{2}}{\sqrt {c + \frac {d}{x^{2}}} c^{3} x}\right )} + \frac {1}{5} \, a {\left (\frac {5 \, d^{3}}{\sqrt {c + \frac {d}{x^{2}}} c^{4} x} + \frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} x^{5} - 5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d x^{3} + 15 \, \sqrt {c + \frac {d}{x^{2}}} d^{2} x}{c^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.77, size = 79, normalized size = 0.71 \[ \frac {3\,a\,c^3\,x^6+5\,b\,c^3\,x^4-6\,a\,c^2\,d\,x^4-20\,b\,c^2\,d\,x^2+24\,a\,c\,d^2\,x^2-40\,b\,c\,d^2+48\,a\,d^3}{15\,c^4\,x\,\sqrt {c+\frac {d}{x^2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.83, size = 561, normalized size = 5.05 \[ a \left (\frac {c^{5} d^{\frac {19}{2}} x^{10} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac {5 c^{3} d^{\frac {23}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac {30 c^{2} d^{\frac {25}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac {40 c d^{\frac {27}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}} + \frac {16 d^{\frac {29}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c^{7} d^{9} x^{6} + 15 c^{6} d^{10} x^{4} + 15 c^{5} d^{11} x^{2} + 5 c^{4} d^{12}}\right ) + b \left (\frac {c^{3} d^{\frac {9}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac {3 c^{2} d^{\frac {11}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac {12 c d^{\frac {13}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}} - \frac {8 d^{\frac {15}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c^{5} d^{4} x^{4} + 6 c^{4} d^{5} x^{2} + 3 c^{3} d^{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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