Optimal. Leaf size=163 \[ \frac {2 (b B+A c) \sqrt {b x^2+c x^4}}{3 b \sqrt {x}}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac {2 (b B+A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt [4]{c} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.17, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2063, 2046,
2057, 335, 226} \begin {gather*} \frac {2 x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (A c+b B) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt [4]{c} \sqrt {b x^2+c x^4}}+\frac {2 \sqrt {b x^2+c x^4} (A c+b B)}{3 b \sqrt {x}}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2046
Rule 2057
Rule 2063
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{7/2}} \, dx &=-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+-\frac {\left (2 \left (-\frac {3 b B}{2}-\frac {3 A c}{2}\right )\right ) \int \frac {\sqrt {b x^2+c x^4}}{x^{3/2}} \, dx}{3 b}\\ &=\frac {2 (b B+A c) \sqrt {b x^2+c x^4}}{3 b \sqrt {x}}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac {1}{3} (2 (b B+A c)) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {2 (b B+A c) \sqrt {b x^2+c x^4}}{3 b \sqrt {x}}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac {\left (2 (b B+A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{3 \sqrt {b x^2+c x^4}}\\ &=\frac {2 (b B+A c) \sqrt {b x^2+c x^4}}{3 b \sqrt {x}}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac {\left (4 (b B+A c) x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {b x^2+c x^4}}\\ &=\frac {2 (b B+A c) \sqrt {b x^2+c x^4}}{3 b \sqrt {x}}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{3 b x^{9/2}}+\frac {2 (b B+A c) 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 )}{3 \sqrt [4]{b} \sqrt [4]{c} \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 = 9.37, size = 97, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (-A \left (b+c x^2\right ) \sqrt {1+\frac {c x^2}{b}}+3 (b B+A c) x^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{b}\right )\right )}{3 b x^{5/2} \sqrt {1+\frac {c x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 239, normalized size = 1.47
method | result | size |
risch | \(-\frac {2 \left (-B \,x^{2}+A \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{3 x^{\frac {5}{2}}}+\frac {\left (\frac {2 A c}{3}+\frac {2 B b}{3}\right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{c \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(180\) |
default | \(\frac {2 \sqrt {x^{4} c +b \,x^{2}}\, \left (A \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) c x +B \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b x +B \,c^{2} x^{4}-A \,c^{2} x^{2}+b B \,x^{2} c -A b c \right )}{3 x^{\frac {5}{2}} \left (c \,x^{2}+b \right ) c}\) | \(239\) |
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 = 0.61, size = 61, normalized size = 0.37 \begin {gather*} \frac {2 \, {\left (2 \, {\left (B b + A c\right )} \sqrt {c} x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + \sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} - A c\right )} \sqrt {x}\right )}}{3 \, c x^{3}} \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 {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{\frac {7}{2}}}\, 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 = -0.01 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2}}{x^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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