Optimal. Leaf size=326 \[ \frac {14 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}-\frac {14 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {b x^2+c x^4}}+\frac {7 c^{9/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.23, antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2050, 2057,
335, 311, 226, 1210} \begin {gather*} \frac {7 c^{9/4} 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 )}{15 b^{11/4} \sqrt {b x^2+c x^4}}-\frac {14 c^{9/4} 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 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{11/4} \sqrt {b x^2+c x^4}}+\frac {14 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2050
Rule 2057
Rubi steps
\begin {align*} \int \frac {1}{x^{9/2} \sqrt {b x^2+c x^4}} \, dx &=-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}-\frac {(7 c) \int \frac {1}{x^{5/2} \sqrt {b x^2+c x^4}} \, dx}{9 b}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}+\frac {\left (7 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx}{15 b^2}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac {\left (7 c^3\right ) \int \frac {x^{3/2}}{\sqrt {b x^2+c x^4}} \, dx}{15 b^3}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac {\left (7 c^3 x \sqrt {b+c x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {b+c x^2}} \, dx}{15 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac {\left (14 c^3 x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^3 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}+\frac {\left (14 c^{5/2} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^{5/2} \sqrt {b x^2+c x^4}}-\frac {\left (14 c^{5/2} x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {b}}}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{15 b^{5/2} \sqrt {b x^2+c x^4}}\\ &=\frac {14 c^{5/2} x^{3/2} \left (b+c x^2\right )}{15 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{9 b x^{11/2}}+\frac {14 c \sqrt {b x^2+c x^4}}{45 b^2 x^{7/2}}-\frac {14 c^2 \sqrt {b x^2+c x^4}}{15 b^3 x^{3/2}}-\frac {14 c^{9/4} 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 )}{15 b^{11/4} \sqrt {b x^2+c x^4}}+\frac {7 c^{9/4} 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 )}{15 b^{11/4} \sqrt {b x^2+c x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 57, normalized size = 0.17 \begin {gather*} -\frac {2 \sqrt {1+\frac {c x^2}{b}} \, _2F_1\left (-\frac {9}{4},\frac {1}{2};-\frac {5}{4};-\frac {c x^2}{b}\right )}{9 x^{7/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 230, normalized size = 0.71
method | result | size |
default | \(\frac {42 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticE \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} x^{4}-21 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b \,c^{2} x^{4}-42 c^{3} x^{6}-28 b \,c^{2} x^{4}+4 b^{2} c \,x^{2}-10 b^{3}}{45 \sqrt {c \,x^{4}+b \,x^{2}}\, x^{\frac {7}{2}} b^{3}}\) | \(230\) |
risch | \(-\frac {2 \left (c \,x^{2}+b \right ) \left (21 c^{2} x^{4}-7 b c \,x^{2}+5 b^{2}\right )}{45 b^{3} x^{\frac {7}{2}} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {7 c^{2} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{15 b^{3} \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(239\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.08, size = 72, normalized size = 0.22 \begin {gather*} -\frac {2 \, {\left (21 \, c^{\frac {5}{2}} x^{6} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (21 \, c^{2} x^{4} - 7 \, b c x^{2} + 5 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{45 \, b^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {9}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^{9/2}\,\sqrt {c\,x^4+b\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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