Optimal. Leaf size=279 \[ \frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt {a+b x^2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt {a+b x^2}}{8 a b^6}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{13/2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {1818, 1599,
1277, 1598, 470, 294, 327, 223, 212} \begin {gather*} \frac {x^7 \left (A-\frac {a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 b^{13/2}}-\frac {x \sqrt {a+b x^2} \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 a b^6}+\frac {x^3 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{12 a b^5 \sqrt {a+b x^2}}+\frac {x^5 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 294
Rule 327
Rule 470
Rule 1277
Rule 1598
Rule 1599
Rule 1818
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^5 \left (-7 a \left (B-\frac {a (b C-a D)}{b^2}\right ) x-7 a \left (C-\frac {a D}{b}\right ) x^3-7 a D x^5\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^6 \left (-7 a \left (B-\frac {a (b C-a D)}{b^2}\right )-7 a \left (C-\frac {a D}{b}\right ) x^2-7 a D x^4\right )}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^5 \left (-7 a \left (2 B-\frac {a (9 b C-16 a D)}{b^2}\right ) x+\frac {35 a^2 D x^3}{b}\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^6 \left (-7 a \left (2 B-\frac {a (9 b C-16 a D)}{b^2}\right )+\frac {35 a^2 D x^2}{b}\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac {x^6}{\left (a+b x^2\right )^{5/2}} \, dx}{20 a b^3}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac {x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{12 a b^4}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt {a+b x^2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 a b^5}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt {a+b x^2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt {a+b x^2}}{8 a b^6}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^6}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt {a+b x^2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt {a+b x^2}}{8 a b^6}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^6}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^7}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) x^7}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^5}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac {D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x^3}{12 a b^5 \sqrt {a+b x^2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) x \sqrt {a+b x^2}}{8 a b^6}+\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{13/2}}\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 207, normalized size = 0.74 \begin {gather*} \frac {x \left (-10395 a^6 D+120 A b^6 x^6+630 a^5 b \left (6 C-55 D x^2\right )+a^2 b^4 x^4 \left (-3248 B+6336 C x^2-1155 D x^4\right )-42 a^4 b^2 \left (20 B-300 C x^2+957 D x^4\right )-8 a^3 b^3 x^2 \left (350 B-1827 C x^2+2178 D x^4\right )+2 a b^5 x^6 \left (-704 B+105 \left (2 C x^2+D x^4\right )\right )\right )}{840 a b^6 \left (a+b x^2\right )^{7/2}}-\frac {\left (8 b^2 B-36 a b C+99 a^2 D\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{8 b^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(530\) vs.
\(2(247)=494\).
time = 0.12, size = 531, normalized size = 1.90
method | result | size |
default | \(D \left (\frac {x^{11}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {11 a \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )}{4 b}\right )+C \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )+B \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+A \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )\) | \(531\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 986 vs.
\(2 (247) = 494\).
time = 0.31, size = 986, normalized size = 3.53 \begin {gather*} \frac {D x^{11}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {11 \, D a x^{9}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {C x^{9}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {1}{35} \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} B x - \frac {99 \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} D a^{2} x}{280 \, b^{2}} + \frac {9 \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} C a x}{70 \, b} - \frac {33 \, D a^{2} x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{40 \, b^{3}} + \frac {3 \, C a x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{10 \, b^{2}} - \frac {B x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} - \frac {A x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {33 \, D a^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{8 \, b^{4}} + \frac {3 \, C a x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{2 \, b^{3}} - \frac {B x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b^{2}} - \frac {99 \, D a^{3} x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}} + \frac {9 \, C a^{2} x^{3}}{2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} - \frac {B a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {5 \, A a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {4587 \, D a^{2} x}{280 \, \sqrt {b x^{2} + a} b^{6}} + \frac {561 \, D a^{3} x}{280 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{6}} - \frac {2871 \, D a^{4} x}{280 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{6}} - \frac {417 \, C a x}{70 \, \sqrt {b x^{2} + a} b^{5}} - \frac {51 \, C a^{2} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} + \frac {261 \, C a^{3} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}} + \frac {139 \, B x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, B a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, B a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {A x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {A x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, A a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, A a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {99 \, D a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {13}{2}}} - \frac {9 \, C a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {11}{2}}} + \frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.32, size = 816, normalized size = 2.92 \begin {gather*} \left [\frac {105 \, {\left ({\left (99 \, D a^{3} b^{4} - 36 \, C a^{2} b^{5} + 8 \, B a b^{6}\right )} x^{8} + 99 \, D a^{7} - 36 \, C a^{6} b + 8 \, B a^{5} b^{2} + 4 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{6} + 6 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{4} + 4 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x^{2}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (210 \, D a b^{6} x^{11} - 105 \, {\left (11 \, D a^{2} b^{5} - 4 \, C a b^{6}\right )} x^{9} - 8 \, {\left (2178 \, D a^{3} b^{4} - 792 \, C a^{2} b^{5} + 176 \, B a b^{6} - 15 \, A b^{7}\right )} x^{7} - 406 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{1680 \, {\left (a b^{11} x^{8} + 4 \, a^{2} b^{10} x^{6} + 6 \, a^{3} b^{9} x^{4} + 4 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, -\frac {105 \, {\left ({\left (99 \, D a^{3} b^{4} - 36 \, C a^{2} b^{5} + 8 \, B a b^{6}\right )} x^{8} + 99 \, D a^{7} - 36 \, C a^{6} b + 8 \, B a^{5} b^{2} + 4 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{6} + 6 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{4} + 4 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (210 \, D a b^{6} x^{11} - 105 \, {\left (11 \, D a^{2} b^{5} - 4 \, C a b^{6}\right )} x^{9} - 8 \, {\left (2178 \, D a^{3} b^{4} - 792 \, C a^{2} b^{5} + 176 \, B a b^{6} - 15 \, A b^{7}\right )} x^{7} - 406 \, {\left (99 \, D a^{4} b^{3} - 36 \, C a^{3} b^{4} + 8 \, B a^{2} b^{5}\right )} x^{5} - 350 \, {\left (99 \, D a^{5} b^{2} - 36 \, C a^{4} b^{3} + 8 \, B a^{3} b^{4}\right )} x^{3} - 105 \, {\left (99 \, D a^{6} b - 36 \, C a^{5} b^{2} + 8 \, B a^{4} b^{3}\right )} x\right )} \sqrt {b x^{2} + a}}{840 \, {\left (a b^{11} x^{8} + 4 \, a^{2} b^{10} x^{6} + 6 \, a^{3} b^{9} x^{4} + 4 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\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. 9649 vs.
\(2 (274) = 548\).
time = 211.75, size = 9649, normalized size = 34.58 \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 = 1.20, size = 265, normalized size = 0.95 \begin {gather*} \frac {{\left ({\left ({\left ({\left (105 \, {\left (\frac {2 \, D x^{2}}{b} - \frac {11 \, D a^{4} b^{9} - 4 \, C a^{3} b^{10}}{a^{3} b^{11}}\right )} x^{2} - \frac {8 \, {\left (2178 \, D a^{5} b^{8} - 792 \, C a^{4} b^{9} + 176 \, B a^{3} b^{10} - 15 \, A a^{2} b^{11}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac {406 \, {\left (99 \, D a^{6} b^{7} - 36 \, C a^{5} b^{8} + 8 \, B a^{4} b^{9}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac {350 \, {\left (99 \, D a^{7} b^{6} - 36 \, C a^{6} b^{7} + 8 \, B a^{5} b^{8}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac {105 \, {\left (99 \, D a^{8} b^{5} - 36 \, C a^{7} b^{6} + 8 \, B a^{6} b^{7}\right )}}{a^{3} b^{11}}\right )} x}{840 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {{\left (99 \, D a^{2} - 36 \, C a b + 8 \, B b^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {13}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6\,\left (A+B\,x^2+C\,x^4+x^6\,D\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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