Optimal. Leaf size=179 \[ -\frac {15 c^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 b^{13/4} \sqrt {b x^2+c x^4}}-\frac {30 c^2 \sqrt {b x^2+c x^4}}{77 b^3 x^{5/2}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2025, 2032, 329, 220} \[ -\frac {30 c^2 \sqrt {b x^2+c x^4}}{77 b^3 x^{5/2}}-\frac {15 c^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{77 b^{13/4} \sqrt {b x^2+c x^4}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 329
Rule 2025
Rule 2032
Rubi steps
\begin {align*} \int \frac {1}{x^{11/2} \sqrt {b x^2+c x^4}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}}-\frac {(9 c) \int \frac {1}{x^{7/2} \sqrt {b x^2+c x^4}} \, dx}{11 b}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}+\frac {\left (45 c^2\right ) \int \frac {1}{x^{3/2} \sqrt {b x^2+c x^4}} \, dx}{77 b^2}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac {30 c^2 \sqrt {b x^2+c x^4}}{77 b^3 x^{5/2}}-\frac {\left (15 c^3\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{77 b^3}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac {30 c^2 \sqrt {b x^2+c x^4}}{77 b^3 x^{5/2}}-\frac {\left (15 c^3 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{77 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac {30 c^2 \sqrt {b x^2+c x^4}}{77 b^3 x^{5/2}}-\frac {\left (30 c^3 x \sqrt {b+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{77 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 b x^{13/2}}+\frac {18 c \sqrt {b x^2+c x^4}}{77 b^2 x^{9/2}}-\frac {30 c^2 \sqrt {b x^2+c x^4}}{77 b^3 x^{5/2}}-\frac {15 c^{11/4} 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 )}{77 b^{13/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 57, normalized size = 0.32 \[ -\frac {2 \sqrt {\frac {c x^2}{b}+1} \, _2F_1\left (-\frac {11}{4},\frac {1}{2};-\frac {7}{4};-\frac {c x^2}{b}\right )}{11 x^{9/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {x}}{c x^{10} + b x^{8}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2}} x^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 147, normalized size = 0.82 \[ -\frac {30 c^{3} x^{6}+15 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \sqrt {-b c}\, c^{2} x^{5} \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+12 b \,c^{2} x^{4}-4 b^{2} c \,x^{2}+14 b^{3}}{77 \sqrt {c \,x^{4}+b \,x^{2}}\, b^{3} x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{4} + b x^{2}} x^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{11/2}\,\sqrt {c\,x^4+b\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {11}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________