Optimal. Leaf size=67 \[ -\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (4 b c-5 a d) \sqrt [4]{a+b x^2}}{5 a^2 e^3 \sqrt {e x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {464, 270}
\begin {gather*} \frac {2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 464
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}-\frac {(4 b c-5 a d) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx}{5 a e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (4 b c-5 a d) \sqrt [4]{a+b x^2}}{5 a^2 e^3 \sqrt {e x}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 44, normalized size = 0.66 \begin {gather*} -\frac {2 x \sqrt [4]{a+b x^2} \left (a c-4 b c x^2+5 a d x^2\right )}{5 a^2 (e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 39, normalized size = 0.58
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (5 a d \,x^{2}-4 c \,x^{2} b +a c \right )}{5 a^{2} \left (e x \right )^{\frac {7}{2}}}\) | \(39\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (5 a d \,x^{2}-4 c \,x^{2} b +a c \right )}{5 e^{3} \sqrt {e x}\, a^{2} x^{2}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.87 \begin {gather*} \frac {2}{5} \, {\left (\frac {c {\left (\frac {5 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} b}{\sqrt {x}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{x^{\frac {5}{2}}}\right )}}{a^{2}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} d}{a \sqrt {x}}\right )} e^{\left (-\frac {7}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.98, size = 37, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left ({\left (4 \, b c - 5 \, a d\right )} x^{2} - a c\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} e^{\left (-\frac {7}{2}\right )}}{5 \, a^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (60) = 120\).
time = 38.40, size = 121, normalized size = 1.81 \begin {gather*} - \frac {\sqrt [4]{b} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{8 a e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{b} d \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a e^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {b^{\frac {5}{4}} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} e^{\frac {7}{2}} \Gamma \left (\frac {3}{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 [B]
time = 0.64, size = 49, normalized size = 0.73 \begin {gather*} -\frac {\left (\frac {2\,c}{5\,a\,e^3}+\frac {x^2\,\left (10\,a\,d-8\,b\,c\right )}{5\,a^2\,e^3}\right )\,{\left (b\,x^2+a\right )}^{1/4}}{x^2\,\sqrt {e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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