Optimal. Leaf size=124 \[ -\frac {80 a^{7/2} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{77 b^{7/2} \left (a+b x^2\right )^{3/4}}+\frac {40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b} \]
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Rubi [A] time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 233, 231} \[ \frac {40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac {80 a^{7/2} \left (\frac {b x^2}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a+b x^2\right )^{3/4}}-\frac {20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b} \]
Antiderivative was successfully verified.
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Rule 231
Rule 233
Rule 321
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b}-\frac {(10 a) \int \frac {x^4}{\left (a+b x^2\right )^{3/4}} \, dx}{11 b}\\ &=-\frac {20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b}+\frac {\left (60 a^2\right ) \int \frac {x^2}{\left (a+b x^2\right )^{3/4}} \, dx}{77 b^2}\\ &=\frac {40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b}-\frac {\left (40 a^3\right ) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{77 b^3}\\ &=\frac {40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b}-\frac {\left (40 a^3 \left (1+\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{77 b^3 \left (a+b x^2\right )^{3/4}}\\ &=\frac {40 a^2 x \sqrt [4]{a+b x^2}}{77 b^3}-\frac {20 a x^3 \sqrt [4]{a+b x^2}}{77 b^2}+\frac {2 x^5 \sqrt [4]{a+b x^2}}{11 b}-\frac {80 a^{7/2} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{77 b^{7/2} \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 90, normalized size = 0.73 \[ \frac {2 \left (-20 a^3 x \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {b x^2}{a}\right )+20 a^3 x+10 a^2 b x^3-3 a b^2 x^5+7 b^3 x^7\right )}{77 b^3 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (b\,x^2+a\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.96, size = 27, normalized size = 0.22 \[ \frac {x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac {3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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