Optimal. Leaf size=92 \[ \frac{128 b^3 \sqrt [4]{a+b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}} \]
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Rubi [A] time = 0.0286821, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{128 b^3 \sqrt [4]{a+b x^4}}{195 a^4 x}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^{14} \left (a+b x^4\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}-\frac{(12 b) \int \frac{1}{x^{10} \left (a+b x^4\right )^{3/4}} \, dx}{13 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}+\frac{\left (32 b^2\right ) \int \frac{1}{x^6 \left (a+b x^4\right )^{3/4}} \, dx}{39 a^2}\\ &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}-\frac{\left (128 b^3\right ) \int \frac{1}{x^2 \left (a+b x^4\right )^{3/4}} \, dx}{195 a^3}\\ &=-\frac{\sqrt [4]{a+b x^4}}{13 a x^{13}}+\frac{4 b \sqrt [4]{a+b x^4}}{39 a^2 x^9}-\frac{32 b^2 \sqrt [4]{a+b x^4}}{195 a^3 x^5}+\frac{128 b^3 \sqrt [4]{a+b x^4}}{195 a^4 x}\\ \end{align*}
Mathematica [A] time = 0.0189832, size = 53, normalized size = 0.58 \[ \frac{\sqrt [4]{a+b x^4} \left (20 a^2 b x^4-15 a^3-32 a b^2 x^8+128 b^3 x^{12}\right )}{195 a^4 x^{13}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 50, normalized size = 0.5 \begin{align*} -{\frac{-128\,{b}^{3}{x}^{12}+32\,a{b}^{2}{x}^{8}-20\,{a}^{2}b{x}^{4}+15\,{a}^{3}}{195\,{x}^{13}{a}^{4}}\sqrt [4]{b{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992936, size = 93, normalized size = 1.01 \begin{align*} \frac{\frac{195 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}}{x} - \frac{117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} b^{2}}{x^{5}} + \frac{65 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} b}{x^{9}} - \frac{15 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{x^{13}}}{195 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4831, size = 122, normalized size = 1.33 \begin{align*} \frac{{\left (128 \, b^{3} x^{12} - 32 \, a b^{2} x^{8} + 20 \, a^{2} b x^{4} - 15 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{195 \, a^{4} x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.76619, size = 692, normalized size = 7.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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