Optimal. Leaf size=149 \[ -\frac {x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {8 a B+3 (2 A b+5 a C) x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 A b+5 a C) x}{35 a^2 b^3 \sqrt {a+b x^2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1818, 833, 792,
197} \begin {gather*} \frac {x (5 a C+2 A b)}{35 a^2 b^3 \sqrt {a+b x^2}}-\frac {3 x (5 a C+2 A b)+8 a B}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x^2 (x (5 a C+2 A b)+4 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^4 (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 197
Rule 792
Rule 833
Rule 1818
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^3 (-4 a B-(2 A b+5 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x \left (-8 a^2 B-3 a (2 A b+5 a C) x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac {x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {8 a B+3 (2 A b+5 a C) x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 A b+5 a C) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a b^3}\\ &=-\frac {x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {8 a B+3 (2 A b+5 a C) x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 A b+5 a C) x}{35 a^2 b^3 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 79, normalized size = 0.53 \begin {gather*} \frac {-8 a^4 B-28 a^3 b B x^2-35 a^2 b^2 B x^4+21 a A b^3 x^5+6 A b^4 x^7+15 a b^3 C x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \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. \(326\) vs.
\(2(133)=266\).
time = 0.12, size = 327, normalized size = 2.19
method | result | size |
gosper | \(\frac {6 A \,b^{4} x^{7}+15 C a \,x^{7} b^{3}+21 A a \,b^{3} x^{5}-35 B \,x^{4} a^{2} b^{2}-28 B \,a^{3} b \,x^{2}-8 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2} b^{3}}\) | \(76\) |
trager | \(\frac {6 A \,b^{4} x^{7}+15 C a \,x^{7} b^{3}+21 A a \,b^{3} x^{5}-35 B \,x^{4} a^{2} b^{2}-28 B \,a^{3} b \,x^{2}-8 B \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2} b^{3}}\) | \(76\) |
default | \(C \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )+A \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )\) | \(327\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 253, normalized size = 1.70 \begin {gather*} -\frac {C x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, C 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 {4 \, B a x^{2}}{15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {C x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {C x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, C a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, C 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 {8 \, B a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.45, size = 122, normalized size = 0.82 \begin {gather*} \frac {{\left (21 \, A a b^{3} x^{5} - 35 \, B a^{2} b^{2} x^{4} + 3 \, {\left (5 \, C a b^{3} + 2 \, A b^{4}\right )} x^{7} - 28 \, B a^{3} b x^{2} - 8 \, B a^{4}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{2} b^{7} x^{8} + 4 \, a^{3} b^{6} x^{6} + 6 \, a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 61.13, size = 575, normalized size = 3.86 \begin {gather*} A \left (\frac {7 a x^{5}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{7}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} - \frac {8 a^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {28 a b x^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {C x^{7}}{7 a^{\frac {9}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {7}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 21 a^{\frac {5}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 7 a^{\frac {3}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.24, size = 81, normalized size = 0.54 \begin {gather*} \frac {{\left ({\left (3 \, x {\left (\frac {7 \, A}{a} + \frac {{\left (5 \, C a^{2} b^{3} + 2 \, A a b^{4}\right )} x^{2}}{a^{3} b^{3}}\right )} - \frac {35 \, B}{b}\right )} x^{2} - \frac {28 \, B a}{b^{2}}\right )} x^{2} - \frac {8 \, B a^{2}}{b^{3}}}{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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Mupad [B]
time = 1.19, size = 186, normalized size = 1.25 \begin {gather*} \frac {x\,\left (\frac {C\,a^2-A\,a\,b}{35\,a\,b^3}+\frac {a\,\left (\frac {C}{5\,b^2}-\frac {7\,A\,b^2-7\,C\,a\,b}{35\,a\,b^3}\right )}{b}\right )+\frac {2\,B\,a}{5\,b^3}}{{\left (b\,x^2+a\right )}^{5/2}}-\frac {\frac {B}{3\,b^3}+x\,\left (\frac {C}{3\,b^3}-\frac {3\,A\,b-10\,C\,a}{105\,a\,b^3}\right )}{{\left (b\,x^2+a\right )}^{3/2}}-\frac {\frac {B\,a^2}{7\,b^3}-\frac {a\,x\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )}{b}}{{\left (b\,x^2+a\right )}^{7/2}}+\frac {x\,\left (2\,A\,b+5\,C\,a\right )}{35\,a^2\,b^3\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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