Optimal. Leaf size=77 \[ \frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 x}{35 a^4 \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {198, 197}
\begin {gather*} \frac {16 x}{35 a^4 \sqrt {a+c x^2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {x}{7 a \left (a+c x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rubi steps
\begin {align*} \int \frac {1}{\left (a+c x^2\right )^{9/2}} \, dx &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 \int \frac {1}{\left (a+c x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (a+c x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac {x}{7 a \left (a+c x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+c x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+c x^2\right )^{3/2}}+\frac {16 x}{35 a^4 \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 51, normalized size = 0.66 \begin {gather*} \frac {35 a^3 x+70 a^2 c x^3+56 a c^2 x^5+16 c^3 x^7}{35 a^4 \left (a+c x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 74, normalized size = 0.96
method | result | size |
gosper | \(\frac {x \left (16 c^{3} x^{6}+56 a \,c^{2} x^{4}+70 a^{2} c \,x^{2}+35 a^{3}\right )}{35 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(48\) |
trager | \(\frac {x \left (16 c^{3} x^{6}+56 a \,c^{2} x^{4}+70 a^{2} c \,x^{2}+35 a^{3}\right )}{35 \left (c \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(48\) |
default | \(\frac {x}{7 a \left (c \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {c \,x^{2}+a}}\right )}{7 a}}{a}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 61, normalized size = 0.79 \begin {gather*} \frac {16 \, x}{35 \, \sqrt {c x^{2} + a} a^{4}} + \frac {8 \, x}{35 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, x}{35 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {x}{7 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.56, size = 91, normalized size = 1.18 \begin {gather*} \frac {{\left (16 \, c^{3} x^{7} + 56 \, a c^{2} x^{5} + 70 \, a^{2} c x^{3} + 35 \, a^{3} x\right )} \sqrt {c x^{2} + a}}{35 \, {\left (a^{4} c^{4} x^{8} + 4 \, a^{5} c^{3} x^{6} + 6 \, a^{6} c^{2} x^{4} + 4 \, a^{7} c x^{2} + a^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1265 vs.
\(2 (70) = 140\).
time = 1.04, size = 1265, normalized size = 16.43 \begin {gather*} \frac {35 a^{14} x}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {175 a^{13} c x^{3}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {371 a^{12} c^{2} x^{5}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {429 a^{11} c^{3} x^{7}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {286 a^{10} c^{4} x^{9}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {104 a^{9} c^{5} x^{11}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {16 a^{8} c^{6} x^{13}}{35 a^{\frac {37}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {35}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {33}{2}} c^{2} x^{4} \sqrt {1 + \frac {c x^{2}}{a}} + 700 a^{\frac {31}{2}} c^{3} x^{6} \sqrt {1 + \frac {c x^{2}}{a}} + 525 a^{\frac {29}{2}} c^{4} x^{8} \sqrt {1 + \frac {c x^{2}}{a}} + 210 a^{\frac {27}{2}} c^{5} x^{10} \sqrt {1 + \frac {c x^{2}}{a}} + 35 a^{\frac {25}{2}} c^{6} x^{12} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 55, normalized size = 0.71 \begin {gather*} \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, c^{3} x^{2}}{a^{4}} + \frac {7 \, c^{2}}{a^{3}}\right )} + \frac {35 \, c}{a^{2}}\right )} x^{2} + \frac {35}{a}\right )} x}{35 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 61, normalized size = 0.79 \begin {gather*} \frac {16\,x}{35\,a^4\,\sqrt {c\,x^2+a}}+\frac {8\,x}{35\,a^3\,{\left (c\,x^2+a\right )}^{3/2}}+\frac {6\,x}{35\,a^2\,{\left (c\,x^2+a\right )}^{5/2}}+\frac {x}{7\,a\,{\left (c\,x^2+a\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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