Optimal. Leaf size=255 \[ -\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 255, 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} \[ -\frac {a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{22 x^{22} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )} \]
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^{23}} \, 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^{23}} \, 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^{23}}+\frac {5 a^4 b^6}{x^{20}}+\frac {10 a^3 b^7}{x^{17}}+\frac {10 a^2 b^8}{x^{14}}+\frac {5 a b^9}{x^{11}}+\frac {b^{10}}{x^8}\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}}{22 x^{22} \left (a+b x^3\right )}-\frac {5 a^4 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{19 x^{19} \left (a+b x^3\right )}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{8 x^{16} \left (a+b x^3\right )}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{13 x^{13} \left (a+b x^3\right )}-\frac {a b^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 x^{10} \left (a+b x^3\right )}-\frac {b^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 83, normalized size = 0.33 \[ -\frac {\sqrt {\left (a+b x^3\right )^2} \left (6916 a^5+40040 a^4 b x^3+95095 a^3 b^2 x^6+117040 a^2 b^3 x^9+76076 a b^4 x^{12}+21736 b^5 x^{15}\right )}{152152 x^{22} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 59, normalized size = 0.23 \[ -\frac {21736 \, b^{5} x^{15} + 76076 \, a b^{4} x^{12} + 117040 \, a^{2} b^{3} x^{9} + 95095 \, a^{3} b^{2} x^{6} + 40040 \, a^{4} b x^{3} + 6916 \, a^{5}}{152152 \, x^{22}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 107, normalized size = 0.42 \[ -\frac {21736 \, b^{5} x^{15} \mathrm {sgn}\left (b x^{3} + a\right ) + 76076 \, a b^{4} x^{12} \mathrm {sgn}\left (b x^{3} + a\right ) + 117040 \, a^{2} b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 95095 \, a^{3} b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 40040 \, a^{4} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 6916 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )}{152152 \, x^{22}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 80, normalized size = 0.31 \[ -\frac {\left (21736 b^{5} x^{15}+76076 a \,b^{4} x^{12}+117040 a^{2} b^{3} x^{9}+95095 a^{3} b^{2} x^{6}+40040 a^{4} b \,x^{3}+6916 a^{5}\right ) \left (\left (b \,x^{3}+a \right )^{2}\right )^{\frac {5}{2}}}{152152 \left (b \,x^{3}+a \right )^{5} x^{22}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 59, normalized size = 0.23 \[ -\frac {21736 \, b^{5} x^{15} + 76076 \, a b^{4} x^{12} + 117040 \, a^{2} b^{3} x^{9} + 95095 \, a^{3} b^{2} x^{6} + 40040 \, a^{4} b x^{3} + 6916 \, a^{5}}{152152 \, x^{22}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.22, size = 231, normalized size = 0.91 \[ -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{22\,x^{22}\,\left (b\,x^3+a\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^7\,\left (b\,x^3+a\right )}-\frac {a\,b^4\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{2\,x^{10}\,\left (b\,x^3+a\right )}-\frac {5\,a^4\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{19\,x^{19}\,\left (b\,x^3+a\right )}-\frac {10\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{13\,x^{13}\,\left (b\,x^3+a\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{8\,x^{16}\,\left (b\,x^3+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}{x^{23}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________