Optimal. Leaf size=213 \[ -\frac {16 \sqrt {a+b x^2} (A b-8 a C)}{35 a b^5}-\frac {x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt {a+b x^2}}-\frac {x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1804, 819, 641, 217, 206} \begin {gather*} -\frac {x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 \sqrt {a+b x^2} (A b-8 a C)}{35 a b^5}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 641
Rule 819
Rule 1804
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^6 (-7 a B+(A b-8 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^4 \left (-35 a^2 B+6 a (A b-8 a C) x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {x^2 \left (-105 a^3 B+24 a^2 (A b-8 a C) x\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^3}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {\int \frac {-105 a^4 B+48 a^3 (A b-8 a C) x}{\sqrt {a+b x^2}} \, dx}{105 a^4 b^4}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b^4}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b^4}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 165, normalized size = 0.77 \begin {gather*} \frac {384 a^4 C-3 a^3 b (16 A+7 x (5 B-64 C x))+14 a^2 b^2 x^2 (5 x (24 C x-5 B)-12 A)+14 a b^3 x^4 (x (60 C x-29 B)-15 A)+105 \sqrt {a} \sqrt {b} B \left (a+b x^2\right )^3 \sqrt {\frac {b x^2}{a}+1} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+b^4 x^6 (x (105 C x-176 B)-105 A)}{105 b^5 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.44, size = 173, normalized size = 0.81 \begin {gather*} \frac {384 a^4 C-48 a^3 A b-105 a^3 b B x+1344 a^3 b C x^2-168 a^2 A b^2 x^2-350 a^2 b^2 B x^3+1680 a^2 b^2 C x^4-210 a A b^3 x^4-406 a b^3 B x^5+840 a b^3 C x^6-105 A b^4 x^6-176 b^4 B x^7+105 b^4 C x^8}{105 b^5 \left (a+b x^2\right )^{7/2}}-\frac {B \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 522, normalized size = 2.45 \begin {gather*} \left [\frac {105 \, {\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \, {\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \, {\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \, {\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac {105 \, {\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \, {\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \, {\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \, {\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 204, normalized size = 0.96 \begin {gather*} \frac {{\left ({\left ({\left ({\left ({\left ({\left ({\left (\frac {105 \, C x}{b} - \frac {176 \, B}{b}\right )} x + \frac {105 \, {\left (8 \, C a^{4} b^{7} - A a^{3} b^{8}\right )}}{a^{3} b^{9}}\right )} x - \frac {406 \, B a}{b^{2}}\right )} x + \frac {210 \, {\left (8 \, C a^{5} b^{6} - A a^{4} b^{7}\right )}}{a^{3} b^{9}}\right )} x - \frac {350 \, B a^{2}}{b^{3}}\right )} x + \frac {168 \, {\left (8 \, C a^{6} b^{5} - A a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x - \frac {105 \, B a^{3}}{b^{4}}\right )} x + \frac {48 \, {\left (8 \, C a^{7} b^{4} - A a^{6} b^{5}\right )}}{a^{3} b^{9}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {B \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 265, normalized size = 1.24 \begin {gather*} \frac {C \,x^{8}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {B \,x^{7}}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}-\frac {A \,x^{6}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {8 C a \,x^{6}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {2 A a \,x^{4}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}-\frac {B \,x^{5}}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} b^{2}}+\frac {16 C \,a^{2} x^{4}}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}-\frac {8 A \,a^{2} x^{2}}{5 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{3}}+\frac {64 C \,a^{3} x^{2}}{5 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{4}}-\frac {B \,x^{3}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}-\frac {16 A \,a^{3}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{4}}+\frac {128 C \,a^{4}}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{5}}-\frac {B x}{\sqrt {b \,x^{2}+a}\, b^{4}}+\frac {B \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.68, size = 435, normalized size = 2.04 \begin {gather*} \frac {C x^{8}}{{\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 {8 \, C a x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \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 {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 {16 \, C a^{2} x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {2 \, A a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} + \frac {64 \, C a^{3} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} - \frac {8 \, A a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \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 {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} + \frac {128 \, C a^{4}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}} - \frac {16 \, A a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \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^7\,\left (C\,x^2+B\,x+A\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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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