Optimal. Leaf size=133 \[ \frac {40 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}-\frac {20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b} \]
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Rubi [A] time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {275, 321, 233, 232} \[ -\frac {20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}+\frac {40 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}-\frac {10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b} \]
Antiderivative was successfully verified.
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Rule 232
Rule 233
Rule 275
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{13}}{\left (a-b x^4\right )^{3/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^6}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {x^4}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{11 b}\\ &=-\frac {10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac {\left (30 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^2}\\ &=-\frac {20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac {\left (20 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right )}{77 b^3}\\ &=-\frac {20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac {\left (20 a^3 \left (1-\frac {b x^4}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{77 b^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {20 a^2 x^2 \sqrt [4]{a-b x^4}}{77 b^3}-\frac {10 a x^6 \sqrt [4]{a-b x^4}}{77 b^2}-\frac {x^{10} \sqrt [4]{a-b x^4}}{11 b}+\frac {40 a^{7/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a-b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 92, normalized size = 0.69 \[ \frac {x^2 \left (20 a^3 \left (1-\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {b x^4}{a}\right )-20 a^3+10 a^2 b x^4+3 a b^2 x^8+7 b^3 x^{12}\right )}{77 b^3 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{13}}{b x^{4} - a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{13}}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{13}}{\left (-b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{13}}{{\left (-b x^{4} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{13}}{{\left (a-b\,x^4\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.76, size = 29, normalized size = 0.22 \[ \frac {x^{14} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{14 a^{\frac {3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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