Optimal. Leaf size=188 \[ \frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1819, 821, 272,
65, 214} \begin {gather*} -\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{35 a^4 \sqrt {a+b x^2}}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 1819
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{x^2 \left (a+b x^2\right )^{9/2}} \, dx &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {-7 A-7 B x+6 \left (\frac {A b}{a}-C\right ) x^2}{x^2 \left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {35 A+35 B x-4 \left (\frac {13 A b}{a}-6 C\right ) x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-105 A-105 B x+6 \left (\frac {29 A b}{a}-8 C\right ) x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{105 a^3}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}+\frac {\int \frac {105 A+105 B x}{x^2 \sqrt {a+b x^2}} \, dx}{105 a^4}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {B \int \frac {1}{x \sqrt {a+b x^2}} \, dx}{a^4}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {B \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^4}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {B \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a^4 b}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 168, normalized size = 0.89 \begin {gather*} \frac {-384 A b^4 x^8+14 a^2 b^2 x^4 (-120 A+x (25 B+12 C x))+14 a^3 b x^2 (-60 A+x (29 B+15 C x))+3 a b^3 x^6 (-448 A+x (35 B+16 C x))+a^4 (-105 A+x (176 B+105 C x))+210 \sqrt {a} B x \left (a+b x^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{105 a^5 x \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 277, normalized size = 1.47
method | result | size |
default | \(C \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )+B \left (\frac {1}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {1}{5 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\frac {1}{3 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\frac {1}{a \sqrt {b \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}}{a}}{a}}{a}\right )+A \left (-\frac {1}{a x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 b \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{a}\right )\) | \(277\) |
risch | \(\text {Expression too large to display}\) | \(2044\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 228, normalized size = 1.21 \begin {gather*} \frac {16 \, C x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {128 \, A b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, A b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a^{4}} + \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {B}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 525, normalized size = 2.79 \begin {gather*} \left [\frac {105 \, {\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \, {\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \, {\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \, {\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}, \frac {105 \, {\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \, {\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \, {\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \, {\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}\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. 6922 vs.
\(2 (155) = 310\).
time = 47.11, size = 6922, normalized size = 36.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.47, size = 239, normalized size = 1.27 \begin {gather*} \frac {{\left ({\left ({\left ({\left (3 \, {\left (x {\left (\frac {35 \, B b^{3}}{a^{4}} + \frac {{\left (16 \, C a^{20} b^{6} - 93 \, A a^{19} b^{7}\right )} x}{a^{24} b^{3}}\right )} + \frac {28 \, {\left (2 \, C a^{21} b^{5} - 11 \, A a^{20} b^{6}\right )}}{a^{24} b^{3}}\right )} x + \frac {350 \, B b^{2}}{a^{3}}\right )} x + \frac {210 \, {\left (C a^{22} b^{4} - 5 \, A a^{21} b^{5}\right )}}{a^{24} b^{3}}\right )} x + \frac {406 \, B b}{a^{2}}\right )} x + \frac {105 \, {\left (C a^{23} b^{3} - 4 \, A a^{22} b^{4}\right )}}{a^{24} b^{3}}\right )} x + \frac {176 \, B}{a}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.10, size = 225, normalized size = 1.20 \begin {gather*} \frac {\frac {B}{7\,a}+\frac {B\,{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {B\,{\left (b\,x^2+a\right )}^3}{a^4}+\frac {B\,\left (b\,x^2+a\right )}{5\,a^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {A}{a^4}+\frac {128\,A\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {16\,C\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,C\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,C\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {C\,x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {29\,A\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,A\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {A\,b\,x}{7\,a^2\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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