Optimal. Leaf size=73 \[ -\frac {a (A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {(A b-2 a B) \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, 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 {\left (a+b x^2\right )^{9/2} (A b-2 a B)}{9 b^3}-\frac {a \left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^3}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^{5/2} (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a (-A b+a B) (a+b x)^{5/2}}{b^2}+\frac {(A b-2 a B) (a+b x)^{7/2}}{b^2}+\frac {B (a+b x)^{9/2}}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {a (A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {(A b-2 a B) \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {B \left (a+b x^2\right )^{11/2}}{11 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 56, normalized size = 0.77 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-22 a A b+8 a^2 B+77 A b^2 x^2-28 a b B x^2+63 b^2 B x^4\right )}{693 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 96, normalized size = 1.32
method | result | size |
gosper | \(-\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (-63 b^{2} B \,x^{4}-77 A \,b^{2} x^{2}+28 B a b \,x^{2}+22 a b A -8 a^{2} B \right )}{693 b^{3}}\) | \(53\) |
default | \(B \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )+A \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )\) | \(96\) |
trager | \(-\frac {\left (-63 b^{5} B \,x^{10}-77 A \,b^{5} x^{8}-161 B a \,b^{4} x^{8}-209 A a \,b^{4} x^{6}-113 B \,a^{2} b^{3} x^{6}-165 A \,a^{2} b^{3} x^{4}-3 B \,a^{3} b^{2} x^{4}-11 A \,a^{3} b^{2} x^{2}+4 B \,a^{4} b \,x^{2}+22 a^{4} b A -8 a^{5} B \right ) \sqrt {b \,x^{2}+a}}{693 b^{3}}\) | \(125\) |
risch | \(-\frac {\left (-63 b^{5} B \,x^{10}-77 A \,b^{5} x^{8}-161 B a \,b^{4} x^{8}-209 A a \,b^{4} x^{6}-113 B \,a^{2} b^{3} x^{6}-165 A \,a^{2} b^{3} x^{4}-3 B \,a^{3} b^{2} x^{4}-11 A \,a^{3} b^{2} x^{2}+4 B \,a^{4} b \,x^{2}+22 a^{4} b A -8 a^{5} B \right ) \sqrt {b \,x^{2}+a}}{693 b^{3}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 90, normalized size = 1.23 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{4}}{11 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a x^{2}}{99 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A x^{2}}{9 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2}}{693 \, b^{3}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a}{63 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 122, normalized size = 1.67 \begin {gather*} \frac {{\left (63 \, B b^{5} x^{10} + 7 \, {\left (23 \, B a b^{4} + 11 \, A b^{5}\right )} x^{8} + {\left (113 \, B a^{2} b^{3} + 209 \, A a b^{4}\right )} x^{6} + 8 \, B a^{5} - 22 \, A a^{4} b + 3 \, {\left (B a^{3} b^{2} + 55 \, A a^{2} b^{3}\right )} x^{4} - {\left (4 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{693 \, b^{3}} \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. 260 vs.
\(2 (65) = 130\).
time = 0.53, size = 260, normalized size = 3.56 \begin {gather*} \begin {cases} - \frac {2 A a^{4} \sqrt {a + b x^{2}}}{63 b^{2}} + \frac {A a^{3} x^{2} \sqrt {a + b x^{2}}}{63 b} + \frac {5 A a^{2} x^{4} \sqrt {a + b x^{2}}}{21} + \frac {19 A a b x^{6} \sqrt {a + b x^{2}}}{63} + \frac {A b^{2} x^{8} \sqrt {a + b x^{2}}}{9} + \frac {8 B a^{5} \sqrt {a + b x^{2}}}{693 b^{3}} - \frac {4 B a^{4} x^{2} \sqrt {a + b x^{2}}}{693 b^{2}} + \frac {B a^{3} x^{4} \sqrt {a + b x^{2}}}{231 b} + \frac {113 B a^{2} x^{6} \sqrt {a + b x^{2}}}{693} + \frac {23 B a b x^{8} \sqrt {a + b x^{2}}}{99} + \frac {B b^{2} x^{10} \sqrt {a + b x^{2}}}{11} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{4}}{4} + \frac {B x^{6}}{6}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 73, normalized size = 1.00 \begin {gather*} \frac {63 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} B - 154 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B a + 99 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a^{2} + 77 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} A b - 99 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A a b}{693 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 115, normalized size = 1.58 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {8\,B\,a^5-22\,A\,a^4\,b}{693\,b^3}+\frac {B\,b^2\,x^{10}}{11}+\frac {x^8\,\left (77\,A\,b^5+161\,B\,a\,b^4\right )}{693\,b^3}+\frac {a\,x^6\,\left (209\,A\,b+113\,B\,a\right )}{693}+\frac {a^3\,x^2\,\left (11\,A\,b-4\,B\,a\right )}{693\,b^2}+\frac {a^2\,x^4\,\left (55\,A\,b+B\,a\right )}{231\,b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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