Optimal. Leaf size=152 \[ \frac {7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}-\frac {7 b^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{5/2} \sqrt [4]{a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 331, 235,
233, 202} \begin {gather*} -\frac {7 b^{5/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{5/2} \sqrt [4]{a+b x^4}}+\frac {7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 202
Rule 233
Rule 235
Rule 281
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^{11} \sqrt [4]{a+b x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^6 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}-\frac {(7 b) \text {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{20 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}+\frac {\left (7 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{40 a^2}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}+\frac {\left (7 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )}{80 a^3}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}+\frac {\left (7 b^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{80 a^3 \sqrt [4]{a+b x^4}}\\ &=\frac {7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}-\frac {\left (7 b^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{80 a^3 \sqrt [4]{a+b x^4}}\\ &=\frac {7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{10 a x^{10}}+\frac {7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac {7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}-\frac {7 b^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{5/2} \sqrt [4]{a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 51, normalized size = 0.34 \begin {gather*} -\frac {\sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {5}{2},\frac {1}{4};-\frac {3}{2};-\frac {b x^4}{a}\right )}{10 x^{10} \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{11} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
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 = 0.07, size = 25, normalized size = 0.16 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{15} + a x^{11}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.77, size = 32, normalized size = 0.21 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 \sqrt [4]{a} x^{10}} \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^{11}\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________