Optimal. Leaf size=296 \[ -\frac {3 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {b x^2+c x^4}}+\frac {6 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {b x^2+c x^4}}-\frac {6 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^2 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2025, 2032, 329, 305, 220, 1196} \[ -\frac {6 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^2 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {3 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {b x^2+c x^4}}+\frac {6 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {b x^2+c x^4}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 305
Rule 329
Rule 1196
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \sqrt {b x^2+c x^4}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}-\frac {(3 c) \int \frac {1}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx}{5 b}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac {\left (3 c^2\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{5 b^2}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac {\left (3 c^2 x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{5 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac {\left (6 c^2 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{5 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}-\frac {\left (6 c^{3/2} x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{5 b^{3/2} \sqrt {b x^2+c x^4}}+\frac {\left (6 c^{3/2} x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{5 b^{3/2} \sqrt {b x^2+c x^4}}\\ &=-\frac {6 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^2 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}+\frac {6 c \sqrt {b x^2+c x^4}}{5 b^2 x^{3/2}}+\frac {6 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {b x^2+c x^4}}-\frac {3 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{7/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 57, normalized size = 0.19 \[ -\frac {2 \sqrt {\frac {c x^2}{b}+1} \, _2F_1\left (-\frac {5}{4},\frac {1}{2};-\frac {1}{4};-\frac {c x^2}{b}\right )}{5 x^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c x^{7} + b x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2}} x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 215, normalized size = 0.73 \[ -\frac {-6 c^{2} x^{4}+6 \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, b c \,x^{2} \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, b c \,x^{2} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )-4 b c \,x^{2}+2 b^{2}}{5 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2}} x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^{5/2}\,\sqrt {c\,x^4+b\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________