Optimal. Leaf size=131 \[ -\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}-\frac {7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac {7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac {7 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{5/2} \sqrt [4]{a-b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {331, 235, 234}
\begin {gather*} -\frac {7 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{5/2} \sqrt [4]{a-b x^2}}-\frac {7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac {7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 234
Rule 235
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt [4]{a-b x^2}} \, dx &=-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}+\frac {(7 b) \int \frac {1}{x^4 \sqrt [4]{a-b x^2}} \, dx}{10 a}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}-\frac {7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}+\frac {\left (7 b^2\right ) \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx}{20 a^2}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}-\frac {7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac {7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac {\left (7 b^3\right ) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{40 a^3}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}-\frac {7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac {7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac {\left (7 b^3 \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{40 a^3 \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{5 a x^5}-\frac {7 b \left (a-b x^2\right )^{3/4}}{30 a^2 x^3}-\frac {7 b^2 \left (a-b x^2\right )^{3/4}}{20 a^3 x}-\frac {7 b^{5/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{20 a^{5/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 52, normalized size = 0.40 \begin {gather*} -\frac {\sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {5}{2},\frac {1}{4};-\frac {3}{2};\frac {b x^2}{a}\right )}{5 x^5 \sqrt [4]{a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{6} \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 34, normalized size = 0.26 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 \sqrt [4]{a} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^6\,{\left (a-b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________