Optimal. Leaf size=176 \[ -\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}+\frac {20 c^2 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac {10 c^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {b x^2+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2045, 2050,
2057, 335, 226} \begin {gather*} \frac {10 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 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 b^{9/4} \sqrt {b x^2+c x^4}}+\frac {20 c^2 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 226
Rule 335
Rule 2045
Rule 2050
Rule 2057
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^{15/2}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}+\frac {1}{11} (2 c) \int \frac {1}{x^{7/2} \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}-\frac {\left (10 c^2\right ) \int \frac {1}{x^{3/2} \sqrt {b x^2+c x^4}} \, dx}{77 b}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}+\frac {20 c^2 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac {\left (10 c^3\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{231 b^2}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}+\frac {20 c^2 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac {\left (10 c^3 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{231 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}+\frac {20 c^2 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac {\left (20 c^3 x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{231 b^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{11 x^{13/2}}-\frac {4 c \sqrt {b x^2+c x^4}}{77 b x^{9/2}}+\frac {20 c^2 \sqrt {b x^2+c x^4}}{231 b^2 x^{5/2}}+\frac {10 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 )}{231 b^{9/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 57, normalized size = 0.32 \begin {gather*} -\frac {2 \sqrt {x^2 \left (b+c x^2\right )} \, _2F_1\left (-\frac {11}{4},-\frac {1}{2};-\frac {7}{4};-\frac {c x^2}{b}\right )}{11 x^{13/2} \sqrt {1+\frac {c x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 156, normalized size = 0.89
method | result | size |
default | \(\frac {2 \sqrt {c \,x^{4}+b \,x^{2}}\, \left (5 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, c^{2} x^{5}+10 c^{3} x^{6}+4 b \,c^{2} x^{4}-27 b^{2} c \,x^{2}-21 b^{3}\right )}{231 x^{\frac {13}{2}} \left (c \,x^{2}+b \right ) b^{2}}\) | \(156\) |
risch | \(-\frac {2 \left (-10 c^{2} x^{4}+6 b c \,x^{2}+21 b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{231 x^{\frac {13}{2}} b^{2}}+\frac {10 c^{2} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{231 b^{2} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.09, size = 64, normalized size = 0.36 \begin {gather*} \frac {2 \, {\left (10 \, c^{\frac {5}{2}} x^{7} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (10 \, c^{2} x^{4} - 6 \, b c x^{2} - 21 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{231 \, b^{2} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^4+b\,x^2}}{x^{15/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________