Optimal. Leaf size=258 \[ -\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 b^{11/4} \sqrt {a+b x^4}}+\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^4}}-\frac {21 a x \sqrt {a+b x^4}}{10 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {x^7}{2 b \sqrt {a+b x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {288, 321, 305, 220, 1196} \[ -\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 b^{11/4} \sqrt {a+b x^4}}+\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {21 a x \sqrt {a+b x^4}}{10 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {x^7}{2 b \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 288
Rule 305
Rule 321
Rule 1196
Rubi steps
\begin {align*} \int \frac {x^{10}}{\left (a+b x^4\right )^{3/2}} \, dx &=-\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 \int \frac {x^6}{\sqrt {a+b x^4}} \, dx}{2 b}\\ &=-\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {(21 a) \int \frac {x^2}{\sqrt {a+b x^4}} \, dx}{10 b^2}\\ &=-\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {\left (21 a^{3/2}\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{10 b^{5/2}}+\frac {\left (21 a^{3/2}\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{10 b^{5/2}}\\ &=-\frac {x^7}{2 b \sqrt {a+b x^4}}+\frac {7 x^3 \sqrt {a+b x^4}}{10 b^2}-\frac {21 a x \sqrt {a+b x^4}}{10 b^{5/2} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{10 b^{11/4} \sqrt {a+b x^4}}-\frac {21 a^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{20 b^{11/4} \sqrt {a+b x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 66, normalized size = 0.26 \[ \frac {x^3 \left (7 a \sqrt {\frac {b x^4}{a}+1} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {b x^4}{a}\right )-7 a+b x^4\right )}{5 b^2 \sqrt {a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{4} + a} x^{10}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{10}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.01, size = 137, normalized size = 0.53 \[ \frac {a \,x^{3}}{2 \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}\, b^{2}}+\frac {\sqrt {b \,x^{4}+a}\, x^{3}}{5 b^{2}}-\frac {21 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \left (-\EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )+\EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )\right ) a^{\frac {3}{2}}}{10 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{10}}{{\left (b x^{4} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{10}}{{\left (b\,x^4+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.63, size = 37, normalized size = 0.14 \[ \frac {x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________