Optimal. Leaf size=101 \[ -\frac {24 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a-b x^2}}+\frac {12 x \left (a-b x^2\right )^{3/4}}{5 b^2}+\frac {2 x^3}{b \sqrt [4]{a-b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {288, 321, 229, 228} \[ -\frac {24 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a-b x^2}}+\frac {12 x \left (a-b x^2\right )^{3/4}}{5 b^2}+\frac {2 x^3}{b \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 228
Rule 229
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a-b x^2\right )^{5/4}} \, dx &=\frac {2 x^3}{b \sqrt [4]{a-b x^2}}-\frac {6 \int \frac {x^2}{\sqrt [4]{a-b x^2}} \, dx}{b}\\ &=\frac {2 x^3}{b \sqrt [4]{a-b x^2}}+\frac {12 x \left (a-b x^2\right )^{3/4}}{5 b^2}-\frac {(12 a) \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{5 b^2}\\ &=\frac {2 x^3}{b \sqrt [4]{a-b x^2}}+\frac {12 x \left (a-b x^2\right )^{3/4}}{5 b^2}-\frac {\left (12 a \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{5 b^2 \sqrt [4]{a-b x^2}}\\ &=\frac {2 x^3}{b \sqrt [4]{a-b x^2}}+\frac {12 x \left (a-b x^2\right )^{3/4}}{5 b^2}-\frac {24 a^{3/2} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 b^{5/2} \sqrt [4]{a-b x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 66, normalized size = 0.65 \[ -\frac {2 \left (6 a x \sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )-6 a x+b x^3\right )}{5 b^2 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} x^{4}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (-b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (-b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\left (a-b\,x^2\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.95, size = 29, normalized size = 0.29 \[ \frac {x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5 a^{\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________