Optimal. Leaf size=125 \[ -\frac {d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac {d \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac {2 (e x)^{3/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {452, 329, 331, 298, 205, 208} \[ -\frac {d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac {d \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac {2 (e x)^{3/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 298
Rule 329
Rule 331
Rule 452
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx &=\frac {2 (b c-a d) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {d \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{b}\\ &=\frac {2 (b c-a d) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{b e}\\ &=\frac {2 (b c-a d) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {b x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{b e}\\ &=\frac {2 (b c-a d) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac {(d e) \operatorname {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{b^{3/2}}-\frac {(d e) \operatorname {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{b^{3/2}}\\ &=\frac {2 (b c-a d) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac {d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac {d \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 69, normalized size = 0.55 \[ \frac {2 \sqrt {e x} \left (3 d x^3 \left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac {7}{4},\frac {7}{4};\frac {11}{4};-\frac {b x^2}{a}\right )+7 c x\right )}{21 a \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x}\, \left (d \,x^{2}+c \right )}{\left (b \,x^{2}+a \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {{\left (d x^{2} + c\right )} \sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {7}{4}}}\,{d x} \]
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 \[ \int \frac {\sqrt {e\,x}\,\left (d\,x^2+c\right )}{{\left (b\,x^2+a\right )}^{7/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 21.77, size = 87, normalized size = 0.70 \[ \frac {c \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )}{2 a^{\frac {7}{4}} \left (1 + \frac {b x^{2}}{a}\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {d \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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