Optimal. Leaf size=128 \[ \frac {a^3 (A b-a B)}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a+b x^2}}-\frac {3 a (A b-2 a B) \sqrt {a+b x^2}}{b^5}+\frac {(A b-4 a B) \left (a+b x^2\right )^{3/2}}{3 b^5}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^5} \]
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Rubi [A]
time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 78}
\begin {gather*} \frac {a^3 (A b-a B)}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a+b x^2}}-\frac {3 a \sqrt {a+b x^2} (A b-2 a B)}{b^5}+\frac {\left (a+b x^2\right )^{3/2} (A b-4 a B)}{3 b^5}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^3 (-A b+a B)}{b^4 (a+b x)^{5/2}}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^{3/2}}+\frac {3 a (-A b+2 a B)}{b^4 \sqrt {a+b x}}+\frac {(A b-4 a B) \sqrt {a+b x}}{b^4}+\frac {B (a+b x)^{3/2}}{b^4}\right ) \, dx,x,x^2\right )\\ &=\frac {a^3 (A b-a B)}{3 b^5 \left (a+b x^2\right )^{3/2}}-\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a+b x^2}}-\frac {3 a (A b-2 a B) \sqrt {a+b x^2}}{b^5}+\frac {(A b-4 a B) \left (a+b x^2\right )^{3/2}}{3 b^5}+\frac {B \left (a+b x^2\right )^{5/2}}{5 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 98, normalized size = 0.77 \begin {gather*} \frac {128 a^4 B+24 a^2 b^2 x^2 \left (-5 A+2 B x^2\right )+b^4 x^6 \left (5 A+3 B x^2\right )-2 a b^3 x^4 \left (15 A+4 B x^2\right )+a^3 \left (-80 A b+192 b B x^2\right )}{15 b^5 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 190, normalized size = 1.48
method | result | size |
gosper | \(-\frac {-3 B \,b^{4} x^{8}-5 A \,b^{4} x^{6}+8 B a \,b^{3} x^{6}+30 A a \,b^{3} x^{4}-48 B \,a^{2} b^{2} x^{4}+120 a^{2} A \,b^{2} x^{2}-192 B \,a^{3} b \,x^{2}+80 A \,a^{3} b -128 B \,a^{4}}{15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}\) | \(101\) |
trager | \(-\frac {-3 B \,b^{4} x^{8}-5 A \,b^{4} x^{6}+8 B a \,b^{3} x^{6}+30 A a \,b^{3} x^{4}-48 B \,a^{2} b^{2} x^{4}+120 a^{2} A \,b^{2} x^{2}-192 B \,a^{3} b \,x^{2}+80 A \,a^{3} b -128 B \,a^{4}}{15 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{5}}\) | \(101\) |
risch | \(-\frac {\left (-3 b^{2} B \,x^{4}-5 A \,b^{2} x^{2}+14 B a b \,x^{2}+40 a b A -73 a^{2} B \right ) \sqrt {b \,x^{2}+a}}{15 b^{5}}-\frac {\sqrt {b \,x^{2}+a}\, \left (9 A \,b^{2} x^{2}-12 B a b \,x^{2}+8 a b A -11 a^{2} B \right ) a^{2}}{3 b^{5} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(120\) |
default | \(B \left (\frac {x^{8}}{5 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {8 a \left (\frac {x^{6}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a \left (\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{b}\right )}{b}\right )}{5 b}\right )+A \left (\frac {x^{6}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a \left (\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{b}\right )}{b}\right )\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 174, normalized size = 1.36 \begin {gather*} \frac {B x^{8}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} - \frac {8 \, B a x^{6}}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{6}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {16 \, B a^{2} x^{4}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} - \frac {2 \, A a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {64 \, B a^{3} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {8 \, A a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {128 \, B a^{4}}{15 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} - \frac {16 \, A a^{3}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.77, size = 123, normalized size = 0.96 \begin {gather*} \frac {{\left (3 \, B b^{4} x^{8} - {\left (8 \, B a b^{3} - 5 \, A b^{4}\right )} x^{6} + 128 \, B a^{4} - 80 \, A a^{3} b + 6 \, {\left (8 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{4} + 24 \, {\left (8 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\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. 437 vs.
\(2 (119) = 238\).
time = 0.58, size = 437, normalized size = 3.41 \begin {gather*} \begin {cases} - \frac {80 A a^{3} b}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} - \frac {120 A a^{2} b^{2} x^{2}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} - \frac {30 A a b^{3} x^{4}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} + \frac {5 A b^{4} x^{6}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} + \frac {128 B a^{4}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} + \frac {192 B a^{3} b x^{2}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} + \frac {48 B a^{2} b^{2} x^{4}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} - \frac {8 B a b^{3} x^{6}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} + \frac {3 B b^{4} x^{8}}{15 a b^{5} \sqrt {a + b x^{2}} + 15 b^{6} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{8}}{8} + \frac {B x^{10}}{10}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 141, normalized size = 1.10 \begin {gather*} \frac {12 \, {\left (b x^{2} + a\right )} B a^{3} - B a^{4} - 9 \, {\left (b x^{2} + a\right )} A a^{2} b + A a^{3} b}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{5}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B b^{20} - 20 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B a b^{20} + 90 \, \sqrt {b x^{2} + a} B a^{2} b^{20} + 5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A b^{21} - 45 \, \sqrt {b x^{2} + a} A a b^{21}}{15 \, b^{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 122, normalized size = 0.95 \begin {gather*} \frac {3\,B\,{\left (b\,x^2+a\right )}^4-5\,B\,a^4+90\,B\,a^2\,{\left (b\,x^2+a\right )}^2+5\,A\,b\,{\left (b\,x^2+a\right )}^3-20\,B\,a\,{\left (b\,x^2+a\right )}^3+60\,B\,a^3\,\left (b\,x^2+a\right )+5\,A\,a^3\,b-45\,A\,a\,b\,{\left (b\,x^2+a\right )}^2-45\,A\,a^2\,b\,\left (b\,x^2+a\right )}{15\,b^5\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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