Optimal. Leaf size=83 \[ \frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\sqrt {a} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a+b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 83, 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 = {314, 284, 335, 196} \[ \frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\sqrt {a} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 196
Rule 284
Rule 314
Rule 335
Rubi steps
\begin {align*} \int \frac {\sqrt {c x}}{\sqrt [4]{a+b x^2}} \, dx &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}-\frac {1}{2} a \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx\\ &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}-\frac {\left (a \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}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\left (a \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 )}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac {x \sqrt {c x}}{\sqrt [4]{a+b x^2}}+\frac {\sqrt {a} \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 {b} \sqrt [4]{a+b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 56, normalized size = 0.67 \[ \frac {2 x \sqrt {c x} \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};-\frac {b x^2}{a}\right )}{3 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,x}}{{\left (b\,x^2+a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.01, size = 44, normalized size = 0.53 \[ \frac {\sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________