Optimal. Leaf size=210 \[ \frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {(4 b C-15 a D) x}{3 b^5 \sqrt {a+b x^2}}+\frac {D x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b C-9 a D) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {1818, 1599,
1277, 1598, 466, 1171, 396, 223, 212} \begin {gather*} \frac {x^5 \left (a \left (19 a^2 D-12 a b C+5 b^2 B\right )+2 A b^3\right )}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {x^5 \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 {(2 b C-9 a D) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}}-\frac {x (4 b C-15 a D)}{3 b^5 \sqrt {a+b x^2}}+\frac {a x (b C-3 a D)}{3 b^5 \left (a+b x^2\right )^{3/2}}+\frac {D x \sqrt {a+b x^2}}{2 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 396
Rule 466
Rule 1171
Rule 1277
Rule 1598
Rule 1599
Rule 1818
Rubi steps
\begin {align*} \int \frac {x^4 \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^5}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^3 \left (-\left (2 A b+\frac {5 a \left (b^2 B-a b C+a^2 D\right )}{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^5}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^4 \left (-2 A b-\frac {5 a \left (b^2 B-a b C+a^2 D\right )}{b^2}-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^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^3 \left (\frac {35 a^2 (b C-2 a D) x}{b^2}+\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^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {x^4 \left (\frac {35 a^2 (b C-2 a D)}{b^2}+\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^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {\frac {35 a^3 (b C-3 a D)}{b}-105 a^2 (b C-3 a D) x^2-105 a^2 b D x^4}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^2 b^4}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {(4 b C-15 a D) x}{3 b^5 \sqrt {a+b x^2}}+\frac {\int \frac {\frac {105 a^3 (b C-4 a D)}{b}+105 a^3 D x^2}{\sqrt {a+b x^2}} \, dx}{105 a^3 b^4}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {(4 b C-15 a D) x}{3 b^5 \sqrt {a+b x^2}}+\frac {D x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b C-9 a D) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 b^5}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {(4 b C-15 a D) x}{3 b^5 \sqrt {a+b x^2}}+\frac {D x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b C-9 a D) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 b^5}\\ &=\frac {\left (A-\frac {a \left (b^2 B-a b C+a^2 D\right )}{b^3}\right ) x^5}{7 a \left (a+b x^2\right )^{7/2}}+\frac {\left (2 A b^3+a \left (5 b^2 B-12 a b C+19 a^2 D\right )\right ) x^5}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac {a (b C-3 a D) x}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {(4 b C-15 a D) x}{3 b^5 \sqrt {a+b x^2}}+\frac {D x \sqrt {a+b x^2}}{2 b^5}+\frac {(2 b C-9 a D) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.66, size = 173, normalized size = 0.82 \begin {gather*} \frac {x \left (945 a^6 D+12 A b^6 x^6+6 a b^5 x^4 \left (7 A+5 B x^2\right )-210 a^5 b \left (C-15 D x^2\right )+a^2 b^4 x^6 \left (-352 C+105 D x^2\right )+14 a^4 b^2 x^2 \left (-50 C+261 D x^2\right )+4 a^3 b^3 x^4 \left (-203 C+396 D x^2\right )\right )}{210 a^2 b^5 \left (a+b x^2\right )^{7/2}}+\frac {(-2 b C+9 a D) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 b^{11/2}} \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. \(499\) vs.
\(2(182)=364\).
time = 0.11, size = 500, normalized size = 2.38
method | result | size |
default | \(D \left (\frac {x^{9}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {9 a \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )}{2 b}\right )+C \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+B \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+A \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )\) | \(500\) |
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. 753 vs.
\(2 (183) = 366\).
time = 0.32, size = 753, normalized size = 3.59 \begin {gather*} \frac {D 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 )} C x + \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 )} D a x}{70 \, b} + \frac {3 \, D 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 {C 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 {B x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {3 \, D 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 {C 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 {9 \, D a^{2} x^{3}}{2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} - \frac {C a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {5 \, B a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {417 \, D a x}{70 \, \sqrt {b x^{2} + a} b^{5}} - \frac {51 \, D a^{2} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} + \frac {261 \, D a^{3} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{5}} + \frac {139 \, C x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, C a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, C a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {B x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {B x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, B a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, B a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {3 \, A x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, A x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, A a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {9 \, D a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {11}{2}}} + \frac {C \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 = 5.55, size = 653, normalized size = 3.11 \begin {gather*} \left [\frac {105 \, {\left ({\left (9 \, D a^{3} b^{4} - 2 \, C a^{2} b^{5}\right )} x^{8} + 9 \, D a^{7} - 2 \, C a^{6} b + 4 \, {\left (9 \, D a^{4} b^{3} - 2 \, C a^{3} b^{4}\right )} x^{6} + 6 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{4} + 4 \, {\left (9 \, D a^{6} b - 2 \, C a^{5} b^{2}\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 (105 \, D a^{2} b^{5} x^{9} + 2 \, {\left (792 \, D a^{3} b^{4} - 176 \, C a^{2} b^{5} + 15 \, B a b^{6} + 6 \, A b^{7}\right )} x^{7} + 14 \, {\left (261 \, D a^{4} b^{3} - 58 \, C a^{3} b^{4} + 3 \, A a b^{6}\right )} x^{5} + 350 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{3} + 105 \, {\left (9 \, D a^{6} b - 2 \, C a^{5} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{420 \, {\left (a^{2} b^{10} x^{8} + 4 \, a^{3} b^{9} x^{6} + 6 \, a^{4} b^{8} x^{4} + 4 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}, \frac {105 \, {\left ({\left (9 \, D a^{3} b^{4} - 2 \, C a^{2} b^{5}\right )} x^{8} + 9 \, D a^{7} - 2 \, C a^{6} b + 4 \, {\left (9 \, D a^{4} b^{3} - 2 \, C a^{3} b^{4}\right )} x^{6} + 6 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{4} + 4 \, {\left (9 \, D a^{6} b - 2 \, C a^{5} b^{2}\right )} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, D a^{2} b^{5} x^{9} + 2 \, {\left (792 \, D a^{3} b^{4} - 176 \, C a^{2} b^{5} + 15 \, B a b^{6} + 6 \, A b^{7}\right )} x^{7} + 14 \, {\left (261 \, D a^{4} b^{3} - 58 \, C a^{3} b^{4} + 3 \, A a b^{6}\right )} x^{5} + 350 \, {\left (9 \, D a^{5} b^{2} - 2 \, C a^{4} b^{3}\right )} x^{3} + 105 \, {\left (9 \, D a^{6} b - 2 \, C a^{5} b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{2} b^{10} x^{8} + 4 \, a^{3} b^{9} x^{6} + 6 \, a^{4} b^{8} x^{4} + 4 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\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. 6467 vs.
\(2 (199) = 398\).
time = 98.69, size = 6467, normalized size = 30.80 \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 = 2.45, size = 203, normalized size = 0.97 \begin {gather*} \frac {{\left ({\left ({\left ({\left (\frac {105 \, D x^{2}}{b} + \frac {2 \, {\left (792 \, D a^{4} b^{7} - 176 \, C a^{3} b^{8} + 15 \, B a^{2} b^{9} + 6 \, A a b^{10}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {14 \, {\left (261 \, D a^{5} b^{6} - 58 \, C a^{4} b^{7} + 3 \, A a^{2} b^{9}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {350 \, {\left (9 \, D a^{6} b^{5} - 2 \, C a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x^{2} + \frac {105 \, {\left (9 \, D a^{7} b^{4} - 2 \, C a^{6} b^{5}\right )}}{a^{3} b^{9}}\right )} x}{210 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {{\left (9 \, D a - 2 \, C b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{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^4\,\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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