Optimal. Leaf size=90 \[ \frac {2 \sqrt {b} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a+b x^2}}-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {316, 284, 335, 196} \[ \frac {2 \sqrt {b} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a+b x^2}}-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 284
Rule 316
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{3/2} \sqrt [4]{a+b x^2}} \, dx &=-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {b \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{c^2}\\ &=-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {\left (\sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{c^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {\left (\sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{c^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2}{c \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {2 \sqrt {b} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.60 \[ -\frac {2 x \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {3}{4};-\frac {b x^2}{a}\right )}{(c x)^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{b c^{2} x^{4} + a c^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.04, size = 31, normalized size = 0.34 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{\sqrt [4]{b} c^{\frac {3}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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