Optimal. Leaf size=253 \[ \frac {b^5 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac {5 a b^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 253, 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} \[ -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {5 a b^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {b^5 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )} \]
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^{11}} \, 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^{11}} \, 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^{11}}+\frac {5 a^4 b^6}{x^8}+\frac {10 a^3 b^7}{x^5}+\frac {10 a^2 b^8}{x^2}+5 a b^9 x+b^{10} x^4\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}}{10 x^{10} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^4 \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac {5 a b^4 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {b^5 x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (7 a^5+50 a^4 b x^3+175 a^3 b^2 x^6+700 a^2 b^3 x^9-175 a b^4 x^{12}-14 b^5 x^{15}\right )}{70 x^{10} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.64, size = 59, normalized size = 0.23 \[ \frac {14 \, b^{5} x^{15} + 175 \, a b^{4} x^{12} - 700 \, a^{2} b^{3} x^{9} - 175 \, a^{3} b^{2} x^{6} - 50 \, a^{4} b x^{3} - 7 \, a^{5}}{70 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 108, normalized size = 0.43 \[ \frac {1}{5} \, b^{5} x^{5} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{2} \, a b^{4} x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) - \frac {700 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 175 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 50 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 7 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{70 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 80, normalized size = 0.32 \[ -\frac {\left (-14 b^{5} x^{15}-175 a \,b^{4} x^{12}+700 a^{2} b^{3} x^{9}+175 a^{3} b^{2} x^{6}+50 a^{4} b \,x^{3}+7 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{70 \left (b \,x^{3}+a \right )^{5} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 59, normalized size = 0.23 \[ \frac {14 \, b^{5} x^{15} + 175 \, a b^{4} x^{12} - 700 \, a^{2} b^{3} x^{9} - 175 \, a^{3} b^{2} x^{6} - 50 \, a^{4} b x^{3} - 7 \, a^{5}}{70 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}}{x^{11}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{11}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________