Optimal. Leaf size=69 \[ -\frac {2 \sqrt {c+d x^4} F_1\left (-\frac {1}{8};1,-\frac {1}{2};\frac {7}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {e x} \sqrt {1+\frac {d x^4}{c}}} \]
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Rubi [A]
time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {477, 525, 524}
\begin {gather*} -\frac {2 \sqrt {c+d x^4} F_1\left (-\frac {1}{8};1,-\frac {1}{2};\frac {7}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {e x} \sqrt {\frac {d x^4}{c}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 477
Rule 524
Rule 525
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\sqrt {c+\frac {d x^8}{e^4}}}{x^2 \left (a+\frac {b x^8}{e^4}\right )} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {\left (2 \sqrt {c+d x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {d x^8}{c e^4}}}{x^2 \left (a+\frac {b x^8}{e^4}\right )} \, dx,x,\sqrt {e x}\right )}{e \sqrt {1+\frac {d x^4}{c}}}\\ &=-\frac {2 \sqrt {c+d x^4} F_1\left (-\frac {1}{8};1,-\frac {1}{2};\frac {7}{8};-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {e x} \sqrt {1+\frac {d x^4}{c}}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(69)=138\).
time = 10.09, size = 143, normalized size = 2.07 \begin {gather*} \frac {x \left (-70 a \left (c+d x^4\right )-10 (b c-4 a d) x^4 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {7}{8};\frac {1}{2},1;\frac {15}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+14 b d x^8 \sqrt {1+\frac {d x^4}{c}} F_1\left (\frac {15}{8};\frac {1}{2},1;\frac {23}{8};-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )}{35 a^2 (e x)^{3/2} \sqrt {c+d x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \,x^{4}+c}}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{4}+a \right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d x^{4}}}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{4}\right )}\, dx \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 {\sqrt {d\,x^4+c}}{{\left (e\,x\right )}^{3/2}\,\left (b\,x^4+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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