Optimal. Leaf size=135 \[ -\frac {3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1341, 646, 43} \[ -\frac {3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 646
Rule 1341
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{5/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {\left (3 b^5 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^5} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac {\left (3 b^5 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname {Subst}\left (\int \left (\frac {a^2}{b^7 (a+b x)^5}-\frac {2 a}{b^7 (a+b x)^4}+\frac {1}{b^7 (a+b x)^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac {3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 58, normalized size = 0.43 \[ \frac {-a^2-4 a b \sqrt [3]{x}-6 b^2 x^{2/3}}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.10, size = 136, normalized size = 1.01 \[ \frac {20 \, a b^{9} x^{3} - 60 \, a^{4} b^{6} x^{2} - a^{10} - 9 \, {\left (5 \, a^{2} b^{8} x^{2} - 4 \, a^{5} b^{5} x\right )} x^{\frac {2}{3}} - 3 \, {\left (2 \, b^{10} x^{3} - 20 \, a^{3} b^{7} x^{2} + 5 \, a^{6} b^{4} x\right )} x^{\frac {1}{3}}}{4 \, {\left (b^{15} x^{4} + 4 \, a^{3} b^{12} x^{3} + 6 \, a^{6} b^{9} x^{2} + 4 \, a^{9} b^{6} x + a^{12} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.55, size = 43, normalized size = 0.32 \[ -\frac {6 \, b^{2} x^{\frac {2}{3}} + 4 \, a b x^{\frac {1}{3}} + a^{2}}{4 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 54, normalized size = 0.40 \[ -\frac {\sqrt {b^{2} x^{\frac {2}{3}}+2 a b \,x^{\frac {1}{3}}+a^{2}}\, \left (6 b^{2} x^{\frac {2}{3}}+4 a b \,x^{\frac {1}{3}}+a^{2}\right )}{4 \left (b \,x^{\frac {1}{3}}+a \right )^{5} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.72, size = 53, normalized size = 0.39 \[ -\frac {3}{2 \, b^{5} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{2}} + \frac {2 \, a}{b^{6} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{2}}{4 \, b^{7} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.80, size = 53, normalized size = 0.39 \[ -\frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^2+6\,b^2\,x^{2/3}+4\,a\,b\,x^{1/3}\right )}{4\,b^3\,{\left (a+b\,x^{1/3}\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________