Optimal. Leaf size=124 \[ -\frac {16 a^{5/2} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}+\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}} \]
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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 = {285, 197, 196} \[ \frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {16 a^{5/2} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 197
Rule 285
Rubi steps
\begin {align*} \int \frac {x^6}{\left (a+b x^2\right )^{5/4}} \, dx &=\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {(10 a) \int \frac {x^4}{\left (a+b x^2\right )^{5/4}} \, dx}{9 b}\\ &=-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}+\frac {\left (4 a^2\right ) \int \frac {x^2}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b^2}\\ &=\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {\left (8 a^3\right ) \int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx}{3 b^3}\\ &=\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {\left (8 a^2 \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{3 b^3 \sqrt [4]{a+b x^2}}\\ &=\frac {8 a^2 x}{3 b^3 \sqrt [4]{a+b x^2}}-\frac {4 a x^3}{9 b^2 \sqrt [4]{a+b x^2}}+\frac {2 x^5}{9 b \sqrt [4]{a+b x^2}}-\frac {16 a^{5/2} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 78, normalized size = 0.63 \[ \frac {2 \left (12 a^2 x \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )-12 a^2 x-2 a b x^3+b^2 x^5\right )}{9 b^3 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} x^{6}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (b\,x^2+a\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.97, size = 27, normalized size = 0.22 \[ \frac {x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7 a^{\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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