Optimal. Leaf size=153 \[ \frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}-\frac{112 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{9/2} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0808795, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {283, 281, 335, 275, 196} \[ \frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}-\frac{112 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{9/2} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 283
Rule 281
Rule 335
Rule 275
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{x^{14} \left (a+b x^4\right )^{5/4}} \, dx &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}-\frac{(14 b) \int \frac{1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx}{13 a}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}+\frac{\left (140 b^2\right ) \int \frac{1}{x^6 \left (a+b x^4\right )^{5/4}} \, dx}{117 a^2}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}-\frac{\left (56 b^3\right ) \int \frac{1}{x^2 \left (a+b x^4\right )^{5/4}} \, dx}{39 a^3}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}+\frac{\left (112 b^4\right ) \int \frac{x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{39 a^4}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}+\frac{\left (112 b^3 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \int \frac{1}{\left (1+\frac{a}{b x^4}\right )^{5/4} x^3} \, dx}{39 a^4 \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{\left (112 b^3 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a x^4}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{39 a^4 \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{\left (56 b^3 \sqrt [4]{1+\frac{a}{b x^4}} x\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x^2}\right )}{39 a^4 \sqrt [4]{a+b x^4}}\\ &=-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{112 b^{7/2} \sqrt [4]{1+\frac{a}{b x^4}} x E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{9/2} \sqrt [4]{a+b x^4}}\\ \end{align*}
Mathematica [C] time = 0.0093738, size = 54, normalized size = 0.35 \[ -\frac{\sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{13}{4},\frac{5}{4};-\frac{9}{4};-\frac{b x^4}{a}\right )}{13 a x^{13} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{14}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{b^{2} x^{22} + 2 \, a b x^{18} + a^{2} x^{14}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 7.65682, size = 44, normalized size = 0.29 \begin{align*} \frac{\Gamma \left (- \frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{13}{4}, \frac{5}{4} \\ - \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x^{13} \Gamma \left (- \frac{9}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]