Optimal. Leaf size=97 \[ \frac {5}{12} a x \sqrt [4]{a+b x^4}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}-\frac {5 a^{3/2} \sqrt {b} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {201, 243, 342,
281, 237} \begin {gather*} -\frac {5 a^{3/2} \sqrt {b} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}+\frac {5}{12} a x \sqrt [4]{a+b x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 237
Rule 243
Rule 281
Rule 342
Rubi steps
\begin {align*} \int \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{6} x \left (a+b x^4\right )^{5/4}+\frac {1}{6} (5 a) \int \sqrt [4]{a+b x^4} \, dx\\ &=\frac {5}{12} a x \sqrt [4]{a+b x^4}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}+\frac {1}{12} \left (5 a^2\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=\frac {5}{12} a x \sqrt [4]{a+b x^4}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}+\frac {\left (5 a^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{12 \left (a+b x^4\right )^{3/4}}\\ &=\frac {5}{12} a x \sqrt [4]{a+b x^4}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}-\frac {\left (5 a^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{12 \left (a+b x^4\right )^{3/4}}\\ &=\frac {5}{12} a x \sqrt [4]{a+b x^4}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}-\frac {\left (5 a^2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{24 \left (a+b x^4\right )^{3/4}}\\ &=\frac {5}{12} a x \sqrt [4]{a+b x^4}+\frac {1}{6} x \left (a+b x^4\right )^{5/4}-\frac {5 a^{3/2} \sqrt {b} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 8.38, size = 47, normalized size = 0.48 \begin {gather*} \frac {a x \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {5}{4},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )}{\sqrt [4]{1+\frac {b x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b \,x^{4}+a \right )^{\frac {5}{4}}\, dx\]
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 = 11, normalized size = 0.11 \begin {gather*} {\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac {5}{4}}, 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 = 0.54, size = 37, normalized size = 0.38 \begin {gather*} \frac {a^{\frac {5}{4}} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \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 [B]
time = 1.07, size = 37, normalized size = 0.38 \begin {gather*} \frac {x\,{\left (b\,x^4+a\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (\frac {b\,x^4}{a}+1\right )}^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________