Optimal. Leaf size=195 \[ \frac {2 a b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac {2 a^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1112, 270} \begin {gather*} \frac {2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {2 a^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1112
Rubi steps
\begin {align*} \int (d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^{5/2} \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 (d x)^{5/2}+\frac {3 a^2 b^4 (d x)^{9/2}}{d^2}+\frac {3 a b^5 (d x)^{13/2}}{d^4}+\frac {b^6 (d x)^{17/2}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {2 a^3 (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac {6 a^2 b (d x)^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac {2 a b^2 (d x)^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d^5 \left (a+b x^2\right )}+\frac {2 b^3 (d x)^{19/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{19 d^7 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.34 \begin {gather*} \frac {2 x (d x)^{5/2} \sqrt {\left (a+b x^2\right )^2} \left (1045 a^3+1995 a^2 b x^2+1463 a b^2 x^4+385 b^3 x^6\right )}{7315 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 117.05, size = 105, normalized size = 0.54 \begin {gather*} \frac {2 \left (a d^2+b d^2 x^2\right ) \left (1045 a^3 d^6 (d x)^{7/2}+1995 a^2 b d^4 (d x)^{11/2}+1463 a b^2 d^2 (d x)^{15/2}+385 b^3 (d x)^{19/2}\right )}{7315 d^9 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.65, size = 54, normalized size = 0.28 \begin {gather*} \frac {2}{7315} \, {\left (385 \, b^{3} d^{2} x^{9} + 1463 \, a b^{2} d^{2} x^{7} + 1995 \, a^{2} b d^{2} x^{5} + 1045 \, a^{3} d^{2} x^{3}\right )} \sqrt {d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 99, normalized size = 0.51 \begin {gather*} \frac {2}{19} \, \sqrt {d x} b^{3} d^{2} x^{9} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{5} \, \sqrt {d x} a b^{2} d^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {6}{11} \, \sqrt {d x} a^{2} b d^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{7} \, \sqrt {d x} a^{3} d^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 61, normalized size = 0.31 \begin {gather*} \frac {2 \left (385 b^{3} x^{6}+1463 a \,b^{2} x^{4}+1995 a^{2} b \,x^{2}+1045 a^{3}\right ) \left (d x \right )^{\frac {5}{2}} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x}{7315 \left (b \,x^{2}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 83, normalized size = 0.43 \begin {gather*} \frac {2}{285} \, {\left (15 \, b^{3} d^{\frac {5}{2}} x^{3} + 19 \, a b^{2} d^{\frac {5}{2}} x\right )} x^{\frac {13}{2}} + \frac {4}{165} \, {\left (11 \, a b^{2} d^{\frac {5}{2}} x^{3} + 15 \, a^{2} b d^{\frac {5}{2}} x\right )} x^{\frac {9}{2}} + \frac {2}{77} \, {\left (7 \, a^{2} b d^{\frac {5}{2}} x^{3} + 11 \, a^{3} d^{\frac {5}{2}} x\right )} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{\frac {5}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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