Optimal. Leaf size=127 \[ \frac {8 x (a C+6 A b)}{105 a^4 b \sqrt {a+b x^2}}+\frac {4 x (a C+6 A b)}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {x (a C+6 A b)}{35 a^2 b \left (a+b x^2\right )^{5/2}}-\frac {a B-x (A b-a C)}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1814, 12, 192, 191} \begin {gather*} \frac {8 x (a C+6 A b)}{105 a^4 b \sqrt {a+b x^2}}+\frac {4 x (a C+6 A b)}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {x (a C+6 A b)}{35 a^2 b \left (a+b x^2\right )^{5/2}}-\frac {a B-x (A b-a C)}{7 a b \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 192
Rule 1814
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {-6 A-\frac {a C}{b}}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) x}{35 a^2 b \left (a+b x^2\right )^{5/2}}+\frac {(4 (6 A b+a C)) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) x}{35 a^2 b \left (a+b x^2\right )^{5/2}}+\frac {4 (6 A b+a C) x}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {(8 (6 A b+a C)) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b}\\ &=-\frac {a B-(A b-a C) x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a C) x}{35 a^2 b \left (a+b x^2\right )^{5/2}}+\frac {4 (6 A b+a C) x}{105 a^3 b \left (a+b x^2\right )^{3/2}}+\frac {8 (6 A b+a C) x}{105 a^4 b \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 92, normalized size = 0.72 \begin {gather*} \frac {-15 a^4 B+35 a^3 b x \left (3 A+C x^2\right )+14 a^2 b^2 x^3 \left (15 A+2 C x^2\right )+8 a b^3 x^5 \left (21 A+C x^2\right )+48 A b^4 x^7}{105 a^4 b \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.76, size = 99, normalized size = 0.78 \begin {gather*} \frac {-15 a^4 B+105 a^3 A b x+35 a^3 b C x^3+210 a^2 A b^2 x^3+28 a^2 b^2 C x^5+168 a A b^3 x^5+8 a b^3 C x^7+48 A b^4 x^7}{105 a^4 b \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 137, normalized size = 1.08 \begin {gather*} \frac {{\left (8 \, {\left (C a b^{3} + 6 \, A b^{4}\right )} x^{7} + 105 \, A a^{3} b x + 28 \, {\left (C a^{2} b^{2} + 6 \, A a b^{3}\right )} x^{5} - 15 \, B a^{4} + 35 \, {\left (C a^{3} b + 6 \, A a^{2} b^{2}\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{4} b^{5} x^{8} + 4 \, a^{5} b^{4} x^{6} + 6 \, a^{6} b^{3} x^{4} + 4 \, a^{7} b^{2} x^{2} + a^{8} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 112, normalized size = 0.88 \begin {gather*} \frac {{\left ({\left (4 \, x^{2} {\left (\frac {2 \, {\left (C a b^{5} + 6 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac {7 \, {\left (C a^{2} b^{4} + 6 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac {35 \, {\left (C a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac {105 \, A}{a}\right )} x - \frac {15 \, B}{b}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 0.76 \begin {gather*} \frac {48 A \,b^{4} x^{7}+8 C a \,b^{3} x^{7}+168 A \,x^{5} a \,b^{3}+28 C \,a^{2} b^{2} x^{5}+210 A \,x^{3} a^{2} b^{2}+35 C \,a^{3} b \,x^{3}+105 A x \,a^{3} b -15 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 153, normalized size = 1.20 \begin {gather*} \frac {16 \, A x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, C x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, C x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 115, normalized size = 0.91 \begin {gather*} \frac {x\,\left (6\,A\,b+C\,a\right )}{35\,a^2\,b\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {B}{7\,b}-x\,\left (\frac {A}{7\,a}-\frac {C}{7\,b}\right )}{{\left (b\,x^2+a\right )}^{7/2}}+\frac {x\,\left (24\,A\,b+4\,C\,a\right )}{105\,a^3\,b\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {x\,\left (48\,A\,b+8\,C\,a\right )}{105\,a^4\,b\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 94.22, size = 1880, normalized size = 14.80
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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