Optimal. Leaf size=251 \[ \frac {b^5 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {5 a b^4 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
[Out]
________________________________________________________________________________________
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} \[ \frac {b^5 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}+\frac {5 a b^4 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {5 a^2 b^3 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 x^5 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
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^6} \, 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^6} \, 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 (10 a^3 b^7+\frac {a^5 b^5}{x^6}+\frac {5 a^4 b^6}{x^3}+10 a^2 b^8 x^3+5 a b^9 x^6+b^{10} x^9\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}}{5 x^5 \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac {10 a^3 b^2 x \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^2 b^3 x^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {5 a b^4 x^7 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac {b^5 x^{10} \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 \left (a+b x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ \frac {\sqrt {\left (a+b x^3\right )^2} \left (-14 a^5-175 a^4 b x^3+700 a^3 b^2 x^6+175 a^2 b^3 x^9+50 a b^4 x^{12}+7 b^5 x^{15}\right )}{70 x^5 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 59, normalized size = 0.24 \[ \frac {7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 106, normalized size = 0.42 \[ \frac {1}{10} \, b^{5} x^{10} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{7} \, a b^{4} x^{7} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{2} \, a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + 10 \, a^{3} b^{2} x \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {25 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 2 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{10 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 80, normalized size = 0.32 \[ -\frac {\left (-7 b^{5} x^{15}-50 a \,b^{4} x^{12}-175 a^{2} b^{3} x^{9}-700 a^{3} b^{2} x^{6}+175 a^{4} b \,x^{3}+14 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{70 \left (b \,x^{3}+a \right )^{5} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.67, size = 59, normalized size = 0.24 \[ \frac {7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________