Optimal. Leaf size=46 \[ \frac {\left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^2}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^2}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^{5/2} (A+B x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(A b-a B) (a+b x)^{5/2}}{b}+\frac {B (a+b x)^{7/2}}{b}\right ) \, dx,x,x^2\right )\\ &=\frac {(A b-a B) \left (a+b x^2\right )^{7/2}}{7 b^2}+\frac {B \left (a+b x^2\right )^{9/2}}{9 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.74 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-2 a B+9 A b+7 b B x^2\right )}{63 b^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 34, normalized size = 0.74 \begin {gather*} \frac {\left (a+b x^2\right )^{7/2} \left (-2 a B+9 A b+7 b B x^2\right )}{63 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 97, normalized size = 2.11 \begin {gather*} \frac {{\left (7 \, B b^{4} x^{8} + {\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{6} - 2 \, B a^{4} + 9 \, A a^{3} b + 3 \, {\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{4} + {\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{63 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 44, normalized size = 0.96 \begin {gather*} \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} B - 9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a + 9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} A b}{63 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.67 \begin {gather*} \frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (7 B b \,x^{2}+9 A b -2 B a \right )}{63 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 50, normalized size = 1.09 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} B x^{2}}{9 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} B a}{63 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} A}{7 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 44, normalized size = 0.96 \begin {gather*} \frac {7\,B\,{\left (b\,x^2+a\right )}^{9/2}+9\,A\,b\,{\left (b\,x^2+a\right )}^{7/2}-9\,B\,a\,{\left (b\,x^2+a\right )}^{7/2}}{63\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.66, size = 209, normalized size = 4.54 \begin {gather*} \begin {cases} \frac {A a^{3} \sqrt {a + b x^{2}}}{7 b} + \frac {3 A a^{2} x^{2} \sqrt {a + b x^{2}}}{7} + \frac {3 A a b x^{4} \sqrt {a + b x^{2}}}{7} + \frac {A b^{2} x^{6} \sqrt {a + b x^{2}}}{7} - \frac {2 B a^{4} \sqrt {a + b x^{2}}}{63 b^{2}} + \frac {B a^{3} x^{2} \sqrt {a + b x^{2}}}{63 b} + \frac {5 B a^{2} x^{4} \sqrt {a + b x^{2}}}{21} + \frac {19 B a b x^{6} \sqrt {a + b x^{2}}}{63} + \frac {B b^{2} x^{8} \sqrt {a + b x^{2}}}{9} & \text {for}\: b \neq 0 \\a^{\frac {5}{2}} \left (\frac {A x^{2}}{2} + \frac {B x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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