Integrand size = 26, antiderivative size = 289 \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=-\frac {4 b^5 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}}}{\left (a+\frac {b}{\sqrt [4]{x}}\right ) \sqrt [4]{x}}+\frac {40 a^2 b^3 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} \sqrt [4]{x}}{a+\frac {b}{\sqrt [4]{x}}}+\frac {20 a^3 b^2 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} \sqrt {x}}{a+\frac {b}{\sqrt [4]{x}}}+\frac {20 a^4 b \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} x^{3/4}}{3 \left (a+\frac {b}{\sqrt [4]{x}}\right )}+\frac {a^5 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} x}{a+\frac {b}{\sqrt [4]{x}}}+\frac {20 a b^4 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} \log \left (\sqrt [4]{x}\right )}{a+\frac {b}{\sqrt [4]{x}}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 1369, 269, 45} \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=-\frac {4 b^5 \sqrt {a^2+\frac {2 a b}{\sqrt [4]{x}}+\frac {b^2}{\sqrt {x}}}}{\sqrt [4]{x} \left (a+\frac {b}{\sqrt [4]{x}}\right )}+\frac {20 a b^4 \log \left (\sqrt [4]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [4]{x}}+\frac {b^2}{\sqrt {x}}}}{a+\frac {b}{\sqrt [4]{x}}}+\frac {40 a^2 b^3 \sqrt [4]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [4]{x}}+\frac {b^2}{\sqrt {x}}}}{a+\frac {b}{\sqrt [4]{x}}}+\frac {a^5 x \sqrt {a^2+\frac {2 a b}{\sqrt [4]{x}}+\frac {b^2}{\sqrt {x}}}}{a+\frac {b}{\sqrt [4]{x}}}+\frac {20 a^4 b x^{3/4} \sqrt {a^2+\frac {2 a b}{\sqrt [4]{x}}+\frac {b^2}{\sqrt {x}}}}{3 \left (a+\frac {b}{\sqrt [4]{x}}\right )}+\frac {20 a^3 b^2 \sqrt {x} \sqrt {a^2+\frac {2 a b}{\sqrt [4]{x}}+\frac {b^2}{\sqrt {x}}}}{a+\frac {b}{\sqrt [4]{x}}} \]
[In]
[Out]
Rule 45
Rule 269
Rule 1355
Rule 1369
Rubi steps \begin{align*} \text {integral}& = 4 \text {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{5/2} x^3 \, dx,x,\sqrt [4]{x}\right ) \\ & = \frac {\left (4 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}}\right ) \text {Subst}\left (\int \left (a b+\frac {b^2}{x}\right )^5 x^3 \, dx,x,\sqrt [4]{x}\right )}{b^4 \left (a b+\frac {b^2}{\sqrt [4]{x}}\right )} \\ & = \frac {\left (4 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}}\right ) \text {Subst}\left (\int \frac {\left (b^2+a b x\right )^5}{x^2} \, dx,x,\sqrt [4]{x}\right )}{b^4 \left (a b+\frac {b^2}{\sqrt [4]{x}}\right )} \\ & = \frac {\left (4 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}}\right ) \text {Subst}\left (\int \left (10 a^2 b^8+\frac {b^{10}}{x^2}+\frac {5 a b^9}{x}+10 a^3 b^7 x+5 a^4 b^6 x^2+a^5 b^5 x^3\right ) \, dx,x,\sqrt [4]{x}\right )}{b^4 \left (a b+\frac {b^2}{\sqrt [4]{x}}\right )} \\ & = -\frac {4 b^6 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}}}{\left (a b+\frac {b^2}{\sqrt [4]{x}}\right ) \sqrt [4]{x}}+\frac {40 a^2 b^4 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} \sqrt [4]{x}}{a b+\frac {b^2}{\sqrt [4]{x}}}+\frac {20 a^3 b^3 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} \sqrt {x}}{a b+\frac {b^2}{\sqrt [4]{x}}}+\frac {20 a^4 b^2 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} x^{3/4}}{3 \left (a b+\frac {b^2}{\sqrt [4]{x}}\right )}+\frac {a^5 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} x}{a+\frac {b}{\sqrt [4]{x}}}+\frac {5 a b^5 \sqrt {a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}} \log (x)}{a b+\frac {b^2}{\sqrt [4]{x}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.34 \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=\frac {\sqrt {\frac {\left (b+a \sqrt [4]{x}\right )^2}{\sqrt {x}}} \left (-12 b^5+120 a^2 b^3 \sqrt {x}+60 a^3 b^2 x^{3/4}+20 a^4 b x+3 a^5 x^{5/4}+15 a b^4 \sqrt [4]{x} \log (x)\right )}{3 \left (b+a \sqrt [4]{x}\right )} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.31
method | result | size |
derivativedivides | \(\frac {\left (\frac {a^{2} \sqrt {x}+b^{2}+2 a b \,x^{\frac {1}{4}}}{\sqrt {x}}\right )^{\frac {5}{2}} x \left (3 a^{5} x^{\frac {5}{4}}+20 x \,a^{4} b +60 a^{3} b^{2} x^{\frac {3}{4}}+15 b^{4} a \ln \left (x \right ) x^{\frac {1}{4}}+120 a^{2} b^{3} \sqrt {x}-12 b^{5}\right )}{3 \left (a \,x^{\frac {1}{4}}+b \right )^{5}}\) | \(91\) |
default | \(\frac {\left (\frac {a^{2} x^{\frac {3}{4}}+b^{2} x^{\frac {1}{4}}+2 a b \sqrt {x}}{x^{\frac {3}{4}}}\right )^{\frac {5}{2}} x \left (3 a^{5} x^{\frac {5}{4}}+20 x \,a^{4} b +60 a^{3} b^{2} x^{\frac {3}{4}}+15 b^{4} a \ln \left (x \right ) x^{\frac {1}{4}}+120 a^{2} b^{3} \sqrt {x}-12 b^{5}\right )}{3 \left (a \,x^{\frac {1}{4}}+b \right )^{5}}\) | \(95\) |
[In]
[Out]
Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.20 \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=5 \, a b^{4} \log \left (x\right ) + \frac {3 \, a^{5} x^{\frac {5}{4}} + 20 \, a^{4} b x + 60 \, a^{3} b^{2} x^{\frac {3}{4}} + 120 \, a^{2} b^{3} \sqrt {x} - 12 \, b^{5}}{3 \, x^{\frac {1}{4}}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.44 \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=a^{5} x \mathrm {sgn}\left (a x + b x^{\frac {3}{4}}\right ) \mathrm {sgn}\left (x\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {3}{4}}\right ) \mathrm {sgn}\left (x\right ) + \frac {20}{3} \, a^{4} b x^{\frac {3}{4}} \mathrm {sgn}\left (a x + b x^{\frac {3}{4}}\right ) \mathrm {sgn}\left (x\right ) + 20 \, a^{3} b^{2} \sqrt {x} \mathrm {sgn}\left (a x + b x^{\frac {3}{4}}\right ) \mathrm {sgn}\left (x\right ) + 40 \, a^{2} b^{3} x^{\frac {1}{4}} \mathrm {sgn}\left (a x + b x^{\frac {3}{4}}\right ) \mathrm {sgn}\left (x\right ) - \frac {4 \, b^{5} \mathrm {sgn}\left (a x + b x^{\frac {3}{4}}\right ) \mathrm {sgn}\left (x\right )}{x^{\frac {1}{4}}} \]
[In]
[Out]
Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2 a b}{\sqrt [4]{x}}\right )^{5/2} \, dx=\int {\left (a^2+\frac {b^2}{\sqrt {x}}+\frac {2\,a\,b}{x^{1/4}}\right )}^{5/2} \,d x \]
[In]
[Out]