Optimal. Leaf size=105 \[ -\frac {12 b^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 105, 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 = {283, 281, 335, 275, 196} \[ -\frac {12 b^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 196
Rule 275
Rule 281
Rule 283
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^4\right )^{5/4}} \, dx &=-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}}-\frac {(6 b) \int \frac {1}{x^2 \left (a+b x^4\right )^{5/4}} \, dx}{5 a}\\ &=-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}+\frac {\left (12 b^2\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{5 a^2}\\ &=-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}+\frac {\left (12 b \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{5/4} x^3} \, dx}{5 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (12 b \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{5 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (6 b \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{5 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {1}{5 a x^5 \sqrt [4]{a+b x^4}}+\frac {6 b}{5 a^2 x \sqrt [4]{a+b x^4}}-\frac {12 b^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} \sqrt [4]{a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 54, normalized size = 0.51 \[ -\frac {\sqrt [4]{\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {5}{4},\frac {5}{4};-\frac {1}{4};-\frac {b x^4}{a}\right )}{5 a x^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b^{2} x^{14} + 2 \, a b x^{10} + a^{2} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {5}{4}} x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^6\,{\left (b\,x^4+a\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 2.06, size = 44, normalized size = 0.42 \[ \frac {\Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {5}{4} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{4}} x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________