Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}+\frac {\left (a+b x^2\right )^{11/2}}{11 b^4}-\frac {a \left (a+b x^2\right )^{9/2}}{3 b^4} \]
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Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \begin {gather*} \frac {3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac {a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {\left (a+b x^2\right )^{11/2}}{11 b^4}-\frac {a \left (a+b x^2\right )^{9/2}}{3 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^3 (a+b x)^{3/2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{3/2}}{b^3}+\frac {3 a^2 (a+b x)^{5/2}}{b^3}-\frac {3 a (a+b x)^{7/2}}{b^3}+\frac {(a+b x)^{9/2}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {3 a^2 \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac {a \left (a+b x^2\right )^{9/2}}{3 b^4}+\frac {\left (a+b x^2\right )^{11/2}}{11 b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (a+b x^2\right )^{5/2} \left (-16 a^3+40 a^2 b x^2-70 a b^2 x^4+105 b^3 x^6\right )}{1155 b^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 72, normalized size = 0.90 \begin {gather*} \frac {\sqrt {a+b x^2} \left (-16 a^5+8 a^4 b x^2-6 a^3 b^2 x^4+5 a^2 b^3 x^6+140 a b^4 x^8+105 b^5 x^{10}\right )}{1155 b^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 68, normalized size = 0.85 \begin {gather*} \frac {{\left (105 \, b^{5} x^{10} + 140 \, a b^{4} x^{8} + 5 \, a^{2} b^{3} x^{6} - 6 \, a^{3} b^{2} x^{4} + 8 \, a^{4} b x^{2} - 16 \, a^{5}\right )} \sqrt {b x^{2} + a}}{1155 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 57, normalized size = 0.71 \begin {gather*} \frac {105 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} - 231 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}}{1155 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.59 \begin {gather*} -\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (-105 b^{3} x^{6}+70 a \,b^{2} x^{4}-40 a^{2} b \,x^{2}+16 a^{3}\right )}{1155 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 73, normalized size = 0.91 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{6}}{11 \, b} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a x^{4}}{33 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} x^{2}}{231 \, b^{3}} - \frac {16 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}}{1155 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.60, size = 64, normalized size = 0.80 \begin {gather*} \sqrt {b\,x^2+a}\,\left (\frac {4\,a\,x^8}{33}+\frac {b\,x^{10}}{11}-\frac {16\,a^5}{1155\,b^4}+\frac {a^2\,x^6}{231\,b}-\frac {2\,a^3\,x^4}{385\,b^2}+\frac {8\,a^4\,x^2}{1155\,b^3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.81, size = 133, normalized size = 1.66 \begin {gather*} \begin {cases} - \frac {16 a^{5} \sqrt {a + b x^{2}}}{1155 b^{4}} + \frac {8 a^{4} x^{2} \sqrt {a + b x^{2}}}{1155 b^{3}} - \frac {2 a^{3} x^{4} \sqrt {a + b x^{2}}}{385 b^{2}} + \frac {a^{2} x^{6} \sqrt {a + b x^{2}}}{231 b} + \frac {4 a x^{8} \sqrt {a + b x^{2}}}{33} + \frac {b x^{10} \sqrt {a + b x^{2}}}{11} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{2}} x^{8}}{8} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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