Optimal. Leaf size=253 \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0580731, antiderivative size = 253, normalized size of antiderivative = 1., 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} \[ -\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 x^{17} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1355
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{x^{18}} \, 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^{18}} \, 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^{18}}+\frac{5 a^4 b^6}{x^{15}}+\frac{10 a^3 b^7}{x^{12}}+\frac{10 a^2 b^8}{x^9}+\frac{5 a b^9}{x^6}+\frac{b^{10}}{x^3}\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}}{17 x^{17} \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{14 x^{14} \left (a+b x^3\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 x^{11} \left (a+b x^3\right )}-\frac{5 a^2 b^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^8 \left (a+b x^3\right )}-\frac{a b^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5 \left (a+b x^3\right )}-\frac{b^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0193082, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (6545 a^2 b^3 x^9+4760 a^3 b^2 x^6+1870 a^4 b x^3+308 a^5+5236 a b^4 x^{12}+2618 b^5 x^{15}\right )}{5236 x^{17} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 80, normalized size = 0.3 \begin{align*} -{\frac{2618\,{b}^{5}{x}^{15}+5236\,a{b}^{4}{x}^{12}+6545\,{a}^{2}{b}^{3}{x}^{9}+4760\,{a}^{3}{b}^{2}{x}^{6}+1870\,{a}^{4}b{x}^{3}+308\,{a}^{5}}{5236\,{x}^{17} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.993811, size = 80, normalized size = 0.32 \begin{align*} -\frac{2618 \, b^{5} x^{15} + 5236 \, a b^{4} x^{12} + 6545 \, a^{2} b^{3} x^{9} + 4760 \, a^{3} b^{2} x^{6} + 1870 \, a^{4} b x^{3} + 308 \, a^{5}}{5236 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.68313, size = 153, normalized size = 0.6 \begin{align*} -\frac{2618 \, b^{5} x^{15} + 5236 \, a b^{4} x^{12} + 6545 \, a^{2} b^{3} x^{9} + 4760 \, a^{3} b^{2} x^{6} + 1870 \, a^{4} b x^{3} + 308 \, a^{5}}{5236 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{18}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12291, size = 144, normalized size = 0.57 \begin{align*} -\frac{2618 \, b^{5} x^{15} \mathrm{sgn}\left (b x^{3} + a\right ) + 5236 \, a b^{4} x^{12} \mathrm{sgn}\left (b x^{3} + a\right ) + 6545 \, a^{2} b^{3} x^{9} \mathrm{sgn}\left (b x^{3} + a\right ) + 4760 \, a^{3} b^{2} x^{6} \mathrm{sgn}\left (b x^{3} + a\right ) + 1870 \, a^{4} b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + 308 \, a^{5} \mathrm{sgn}\left (b x^{3} + a\right )}{5236 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]