Optimal. Leaf size=111 \[ \frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}-\frac{14 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4} \]
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Rubi [A] time = 0.0291233, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {195, 229, 227, 196} \[ \frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}-\frac{14 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4} \]
Antiderivative was successfully verified.
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Rule 195
Rule 229
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^{7/4} \, dx &=\frac{2}{9} x \left (a+b x^2\right )^{7/4}+\frac{1}{9} (7 a) \int \left (a+b x^2\right )^{3/4} \, dx\\ &=\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4}+\frac{1}{15} \left (7 a^2\right ) \int \frac{1}{\sqrt [4]{a+b x^2}} \, dx\\ &=\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4}+\frac{\left (7 a^2 \sqrt [4]{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{15 \sqrt [4]{a+b x^2}}\\ &=\frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4}-\frac{\left (7 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}{15 \sqrt [4]{a+b x^2}}\\ &=\frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4}-\frac{14 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 )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0071599, size = 47, normalized size = 0.42 \[ \frac{a x \left (a+b x^2\right )^{3/4} \, _2F_1\left (-\frac{7}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\left (\frac{b x^2}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{{\frac{7}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{7}{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{7}{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.20992, size = 26, normalized size = 0.23 \begin{align*} a^{\frac{7}{4}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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