Optimal. Leaf size=128 \[ \frac {4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac {2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}}-\frac {8 a^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {281, 291, 203,
202} \begin {gather*} -\frac {8 a^{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 )}{3 b^{7/2} \sqrt [4]{a+b x^4}}+\frac {4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac {2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 202
Rule 203
Rule 281
Rule 291
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^6}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )\\ &=\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}}-\frac {(5 a) \text {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{9 b}\\ &=-\frac {2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{3 b^2}\\ &=\frac {4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac {2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}}-\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{3 b^3}\\ &=\frac {4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac {2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}}-\frac {\left (4 a^2 \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 )}{3 b^3 \sqrt [4]{a+b x^4}}\\ &=\frac {4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac {2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac {x^{10}}{9 b \sqrt [4]{a+b x^4}}-\frac {8 a^{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 )}{3 b^{7/2} \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 = 8.77, size = 79, normalized size = 0.62 \begin {gather*} \frac {x^2 \left (-12 a^2-2 a b x^4+b^2 x^8+12 a^2 \sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^4}{a}\right )\right )}{9 b^3 \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 \[\int \frac {x^{13}}{\left (b \,x^{4}+a \right )^{\frac {5}{4}}}\, dx\]
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 = 0.08, size = 35, normalized size = 0.27 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{13}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, 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 = 0.74, size = 27, normalized size = 0.21 \begin {gather*} \frac {x^{14} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{14 a^{\frac {5}{4}}} \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 {x^{13}}{{\left (b\,x^4+a\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________