Optimal. Leaf size=73 \[ \frac {\left (a+b x^2\right )^{7/2} (A b-2 a B)}{7 b^3}-\frac {a \left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^3}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^3} \]
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Rubi [A] time = 0.06, 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 = {446, 77} \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} (A b-2 a B)}{7 b^3}-\frac {a \left (a+b x^2\right )^{5/2} (A b-a B)}{5 b^3}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int x^3 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^{3/2} (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a (-A b+a B) (a+b x)^{3/2}}{b^2}+\frac {(A b-2 a B) (a+b x)^{5/2}}{b^2}+\frac {B (a+b x)^{7/2}}{b^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {a (A b-a B) \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {(A b-2 a B) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.78 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2} \left (8 a^2 B-2 a b \left (9 A+10 B x^2\right )+5 b^2 x^2 \left (9 A+7 B x^2\right )\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 56, normalized size = 0.77 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2} \left (8 a^2 B-18 a A b-20 a b B x^2+45 A b^2 x^2+35 b^2 B x^4\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 99, normalized size = 1.36 \begin {gather*} \frac {{\left (35 \, B b^{4} x^{8} + 5 \, {\left (10 \, B a b^{3} + 9 \, A b^{4}\right )} x^{6} + 8 \, B a^{4} - 18 \, A a^{3} b + 3 \, {\left (B a^{2} b^{2} + 24 \, A a b^{3}\right )} x^{4} - {\left (4 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 73, normalized size = 1.00 \begin {gather*} \frac {35 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B - 90 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a + 63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2} + 45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b - 63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a b}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.73 \begin {gather*} -\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (-35 B \,b^{2} x^{4}-45 A \,b^{2} x^{2}+20 B a b \,x^{2}+18 a b A -8 a^{2} B \right )}{315 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 90, normalized size = 1.23 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B x^{4}}{9 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a x^{2}}{63 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A x^{2}}{7 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} B a^{2}}{315 \, b^{3}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a}{35 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 96, normalized size = 1.32 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {8\,B\,a^4-18\,A\,a^3\,b}{315\,b^3}+\frac {x^6\,\left (45\,A\,b^4+50\,B\,a\,b^3\right )}{315\,b^3}+\frac {B\,b\,x^8}{9}+\frac {a^2\,x^2\,\left (9\,A\,b-4\,B\,a\right )}{315\,b^2}+\frac {a\,x^4\,\left (24\,A\,b+B\,a\right )}{105\,b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.72, size = 209, normalized size = 2.86 \begin {gather*} \begin {cases} - \frac {2 A a^{3} \sqrt {a + b x^{2}}}{35 b^{2}} + \frac {A a^{2} x^{2} \sqrt {a + b x^{2}}}{35 b} + \frac {8 A a x^{4} \sqrt {a + b x^{2}}}{35} + \frac {A b x^{6} \sqrt {a + b x^{2}}}{7} + \frac {8 B a^{4} \sqrt {a + b x^{2}}}{315 b^{3}} - \frac {4 B a^{3} x^{2} \sqrt {a + b x^{2}}}{315 b^{2}} + \frac {B a^{2} x^{4} \sqrt {a + b x^{2}}}{105 b} + \frac {10 B a x^{6} \sqrt {a + b x^{2}}}{63} + \frac {B b x^{8} \sqrt {a + b x^{2}}}{9} & \text {for}\: b \neq 0 \\a^{\frac {3}{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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