Optimal. Leaf size=122 \[ \frac {d \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}+\frac {2 \sqrt {e x} (b c-a d)}{a b e \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 122, 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, 240, 212, 208, 205} \[ \frac {d \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}+\frac {2 \sqrt {e x} (b c-a d)}{a b e \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 240
Rule 329
Rule 452
Rubi steps
\begin {align*} \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx &=\frac {2 (b c-a d) \sqrt {e x}}{a b e \sqrt [4]{a+b x^2}}+\frac {d \int \frac {1}{\sqrt {e x} \sqrt [4]{a+b x^2}} \, dx}{b}\\ &=\frac {2 (b c-a d) \sqrt {e x}}{a b e \sqrt [4]{a+b x^2}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b e}\\ &=\frac {2 (b c-a d) \sqrt {e x}}{a b e \sqrt [4]{a+b x^2}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{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) \sqrt {e x}}{a b e \sqrt [4]{a+b x^2}}+\frac {d \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}+\frac {d \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}\\ &=\frac {2 (b c-a d) \sqrt {e x}}{a b e \sqrt [4]{a+b x^2}}+\frac {d \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{5/4} \sqrt {e}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 68, normalized size = 0.56 \[ \frac {2 \left (d x^3 \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {5}{4},\frac {5}{4};\frac {9}{4};-\frac {b x^2}{a}\right )+5 c x\right )}{5 a \sqrt {e x} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 384, normalized size = 3.15 \[ \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (b c - a d\right )} \sqrt {e x} - 4 \, {\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} b^{4} d e \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {3}{4}} - {\left (b^{5} e x^{2} + a b^{4} e\right )} \sqrt {\frac {\sqrt {b x^{2} + a} d^{2} e x + {\left (b^{3} e^{2} x^{2} + a b^{2} e^{2}\right )} \sqrt {\frac {d^{4}}{b^{5} e^{2}}}}{b x^{2} + a}} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {3}{4}}}{b d^{4} x^{2} + a d^{4}}\right ) + {\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d + {\left (b^{2} e x^{2} + a b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right ) - {\left (a b^{2} e x^{2} + a^{2} b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x} d - {\left (b^{2} e x^{2} + a b e\right )} \left (\frac {d^{4}}{b^{5} e^{2}}\right )^{\frac {1}{4}}}{b x^{2} + a}\right )}{2 \, {\left (a b^{2} e x^{2} + a^{2} b e\right )}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {d \,x^{2}+c}{\sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \sqrt {e x}}\,{d x} \]
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 \[ \int \frac {d\,x^2+c}{\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.06, size = 83, normalized size = 0.68 \[ \frac {c \Gamma \left (\frac {1}{4}\right )}{2 a \sqrt [4]{b} \sqrt {e} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {5}{4}\right )} + \frac {d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} \sqrt {e} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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