Optimal. Leaf size=113 \[ -\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}-\frac {d \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 113, 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 = {462, 335, 338,
304, 211, 214} \begin {gather*} -\frac {d \text {ArcTan}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 338
Rule 462
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {d \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{e^3}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {(2 d) \text {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 )}{e^3}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}+\frac {d \text {Subst}\left (\int \frac {1}{e-\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt {b} e}-\frac {d \text {Subst}\left (\int \frac {1}{e+\sqrt {b} x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt {b} e}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{a e \sqrt {e x}}-\frac {d \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 100, normalized size = 0.88 \begin {gather*} \frac {x \left (-2 b^{3/4} c \sqrt [4]{a+b x^2}-a d \sqrt {x} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )+a d \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a+b x^2}}\right )\right )}{a b^{3/4} (e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 93, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, {\left (d {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {3}{4}}} - \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {3}{4}}}\right )} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} c}{a \sqrt {x}}\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.89, size = 85, normalized size = 0.75 \begin {gather*} \frac {\sqrt [4]{b} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a e^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {d\,x^2+c}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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