Optimal. Leaf size=630 \[ \frac {27\ 3^{3/4} a^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{a+b x^2}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1}\right ),4 \sqrt {3}-7\right )}{8 \sqrt {2} b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {81 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{32 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {81 a^2 x}{16 b^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}} \]
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Rubi [A] time = 0.59, antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {288, 321, 238, 198, 235, 304, 219, 1879} \[ -\frac {81 a^2 x}{16 b^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{7/6} \left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )}+\frac {27\ 3^{3/4} a^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{8 \sqrt {2} b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}-\frac {81 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {\left (\frac {a}{a+b x^2}\right )^{2/3}+\sqrt [3]{\frac {a}{a+b x^2}}+1}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {a}{b x^2+a}}+\sqrt {3}+1}{-\sqrt [3]{\frac {a}{b x^2+a}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{32 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (-\sqrt [3]{\frac {a}{a+b x^2}}-\sqrt {3}+1\right )^2}}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 198
Rule 219
Rule 235
Rule 238
Rule 288
Rule 304
Rule 321
Rule 1879
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^{7/6}} \, dx &=-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {9 \int \frac {x^2}{\sqrt [6]{a+b x^2}} \, dx}{b}\\ &=-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac {(27 a) \int \frac {1}{\sqrt [6]{a+b x^2}} \, dx}{8 b^2}\\ &=-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac {\left (27 a^2\right ) \int \frac {1}{\left (a+b x^2\right )^{7/6}} \, dx}{16 b^2}\\ &=-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac {\left (27 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-b x^2}} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{16 b^2 \left (\frac {a}{a+b x^2}\right )^{2/3} \left (a+b x^2\right )^{2/3}}\\ &=-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}-\frac {\left (81 a^2 \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{32 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac {\left (81 a^2 \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{32 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}-\frac {\left (81 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} a^2 \sqrt {-\frac {b x^2}{a+b x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{\frac {a}{a+b x^2}}\right )}{16 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2}}\\ &=-\frac {81 a x}{16 b^2 \sqrt [6]{a+b x^2}}-\frac {3 x^3}{b \sqrt [6]{a+b x^2}}+\frac {27 x \left (a+b x^2\right )^{5/6}}{8 b^2}+\frac {81 a^2 \sqrt {-\frac {b x^2}{a+b x^2}} \sqrt {-1+\frac {a}{a+b x^2}}}{16 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )}-\frac {81 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^2 \sqrt {-\frac {b x^2}{a+b x^2}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{32 b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}+\frac {27\ 3^{3/4} a^2 \sqrt {-\frac {b x^2}{a+b x^2}} \left (1-\sqrt [3]{\frac {a}{a+b x^2}}\right ) \sqrt {\frac {1+\sqrt [3]{\frac {a}{a+b x^2}}+\left (\frac {a}{a+b x^2}\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}{1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}}\right )|-7+4 \sqrt {3}\right )}{8 \sqrt {2} b^3 x \left (\frac {a}{a+b x^2}\right )^{2/3} \sqrt [6]{a+b x^2} \sqrt {-\frac {1-\sqrt [3]{\frac {a}{a+b x^2}}}{\left (1-\sqrt {3}-\sqrt [3]{\frac {a}{a+b x^2}}\right )^2}} \sqrt {-1+\frac {a}{a+b x^2}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 65, normalized size = 0.10 \[ \frac {3 x \left (9 a \sqrt [6]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {3}{2};-\frac {b x^2}{a}\right )-9 a+2 b x^2\right )}{16 b^2 \sqrt [6]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {5}{6}} x^{4}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (b \,x^{2}+a \right )^{\frac {7}{6}}}\, dx \]
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 \[ \int \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{6}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4}{{\left (b\,x^2+a\right )}^{7/6}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.04, size = 27, normalized size = 0.04 \[ \frac {x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{6}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {7}{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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