Optimal. Leaf size=257 \[ -\frac {2 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {4 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {4 a^2 \sqrt {a+\frac {b}{x^4}}}{15 \sqrt {b} x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )}-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {335, 279, 305, 220, 1196} \[ -\frac {2 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {4 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {4 a^2 \sqrt {a+\frac {b}{x^4}}}{15 \sqrt {b} x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )}-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 279
Rule 305
Rule 335
Rule 1196
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 \left (a+b x^4\right )^{3/2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {1}{3} (2 a) \operatorname {Subst}\left (\int x^2 \sqrt {a+b x^4} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {1}{15} \left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {\left (4 a^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{15 \sqrt {b}}+\frac {\left (4 a^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{15 \sqrt {b}}\\ &=-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {4 a^2 \sqrt {a+\frac {b}{x^4}}}{15 \sqrt {b} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}+\frac {4 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {2 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 52, normalized size = 0.20 \[ -\frac {b \sqrt {a+\frac {b}{x^4}} \, _2F_1\left (-\frac {9}{4},-\frac {3}{2};-\frac {5}{4};-\frac {a x^4}{b}\right )}{9 x^7 \sqrt {\frac {a x^4}{b}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x^{4} + b\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{x^{8}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 257, normalized size = 1.00 \[ -\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (12 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{3} \sqrt {b}\, x^{12}+12 i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a^{\frac {5}{2}} b \,x^{9} \EllipticE \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )-12 i \sqrt {-\frac {i \sqrt {a}\, x^{2}-\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, a^{\frac {5}{2}} b \,x^{9} \EllipticF \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x , i\right )+23 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} b^{\frac {3}{2}} x^{8}+16 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,b^{\frac {5}{2}} x^{4}+5 \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{\frac {7}{2}}\right )}{45 \left (a \,x^{4}+b \right )^{2} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b^{\frac {3}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{x^4}\right )}^{3/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.51, size = 41, normalized size = 0.16 \[ - \frac {a^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________