Optimal. Leaf size=68 \[ -\frac {32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac {\sqrt [4]{a+b x^4}}{9 a x^9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac {32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac {\sqrt [4]{a+b x^4}}{9 a x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}-\frac {(8 b) \int \frac {1}{x^6 \left (a+b x^4\right )^{3/4}} \, dx}{9 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}+\frac {\left (32 b^2\right ) \int \frac {1}{x^2 \left (a+b x^4\right )^{3/4}} \, dx}{45 a^2}\\ &=-\frac {\sqrt [4]{a+b x^4}}{9 a x^9}+\frac {8 b \sqrt [4]{a+b x^4}}{45 a^2 x^5}-\frac {32 b^2 \sqrt [4]{a+b x^4}}{45 a^3 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 42, normalized size = 0.62 \[ -\frac {\sqrt [4]{a+b x^4} \left (5 a^2-8 a b x^4+32 b^2 x^8\right )}{45 a^3 x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 38, normalized size = 0.56 \[ -\frac {{\left (32 \, b^{2} x^{8} - 8 \, a b x^{4} + 5 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{45 \, a^{3} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 39, normalized size = 0.57 \[ -\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{8}-8 a b \,x^{4}+5 a^{2}\right )}{45 a^{3} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 52, normalized size = 0.76 \[ -\frac {\frac {45 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}}{x} - \frac {18 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} b}{x^{5}} + \frac {5 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}}}{x^{9}}}{45 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.27, size = 38, normalized size = 0.56 \[ -\frac {{\left (b\,x^4+a\right )}^{1/4}\,\left (5\,a^2-8\,a\,b\,x^4+32\,b^2\,x^8\right )}{45\,a^3\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.63, size = 406, normalized size = 5.97 \[ \frac {5 a^{4} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {2 a^{3} b^{\frac {21}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {21 a^{2} b^{\frac {25}{4}} x^{8} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {56 a b^{\frac {29}{4}} x^{12} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} + \frac {32 b^{\frac {33}{4}} x^{16} \sqrt [4]{\frac {a}{b x^{4}} + 1} \Gamma \left (- \frac {9}{4}\right )}{64 a^{5} b^{4} x^{8} \Gamma \left (\frac {3}{4}\right ) + 128 a^{4} b^{5} x^{12} \Gamma \left (\frac {3}{4}\right ) + 64 a^{3} b^{6} x^{16} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________