Optimal. Leaf size=251 \[ \frac {b^5 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {5 a b^4 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {5 a^4 b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {2 a^3 b^2 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1355, 270} \[ \frac {b^5 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {5 a b^4 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {2 a^3 b^2 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^4 b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1355
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^2} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^5}{x^2} \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^5 b^5}{x^2}+5 a^4 b^6 x+10 a^3 b^7 x^4+10 a^2 b^8 x^7+5 a b^9 x^{10}+b^{10} x^{13}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )}\\ &=-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {5 a^4 b x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {2 a^3 b^2 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^2 b^3 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {5 a b^4 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {b^5 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ \frac {\sqrt {\left (a+b x^3\right )^2} \left (-308 a^5+770 a^4 b x^3+616 a^3 b^2 x^6+385 a^2 b^3 x^9+140 a b^4 x^{12}+22 b^5 x^{15}\right )}{308 x \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 59, normalized size = 0.24 \[ \frac {22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 105, normalized size = 0.42 \[ \frac {1}{14} \, b^{5} x^{14} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{11} \, a b^{4} x^{11} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{4} \, a^{2} b^{3} x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a^{3} b^{2} x^{5} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{2} \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.32 \[ -\frac {\left (-22 b^{5} x^{15}-140 a \,b^{4} x^{12}-385 a^{2} b^{3} x^{9}-616 a^{3} b^{2} x^{6}-770 a^{4} b \,x^{3}+308 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{308 \left (b \,x^{3}+a \right )^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 59, normalized size = 0.24 \[ \frac {22 \, b^{5} x^{15} + 140 \, a b^{4} x^{12} + 385 \, a^{2} b^{3} x^{9} + 616 \, a^{3} b^{2} x^{6} + 770 \, a^{4} b x^{3} - 308 \, a^{5}}{308 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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