Optimal. Leaf size=167 \[ -\frac {2 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {b^{3/4} (5 b B-7 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 c^{9/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.17, antiderivative size = 167, 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 = {2064, 2049,
2057, 335, 226} \begin {gather*} \frac {b^{3/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (5 b B-7 A c) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 c^{9/4} \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4} (5 b B-7 A c)}{21 c^2 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2049
Rule 2057
Rule 2064
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {\left (2 \left (\frac {5 b B}{2}-\frac {7 A c}{2}\right )\right ) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{7 c}\\ &=-\frac {2 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {(b (5 b B-7 A c)) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{21 c^2}\\ &=-\frac {2 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {\left (b (5 b B-7 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{21 c^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {\left (2 b (5 b B-7 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 )}{21 c^2 \sqrt {b x^2+c x^4}}\\ &=-\frac {2 (5 b B-7 A c) \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 B x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {b^{3/4} (5 b B-7 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 )}{21 c^{9/4} \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 = 10.09, size = 97, normalized size = 0.58 \begin {gather*} \frac {2 x^{3/2} \left (-\left (\left (b+c x^2\right ) \left (5 b B-7 A c-3 B c x^2\right )\right )+b (5 b B-7 A c) \sqrt {1+\frac {c x^2}{b}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {c x^2}{b}\right )\right )}{21 c^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 248, normalized size = 1.49
method | result | size |
risch | \(\frac {2 \left (3 B c \,x^{2}+7 A c -5 B b \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{21 c^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {b \left (7 A c -5 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{21 c^{3} \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(191\) |
default | \(-\frac {\sqrt {x}\, \left (7 A \sqrt {-b c}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, b c -5 B \sqrt {-b c}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, b^{2}-6 B \,c^{3} x^{5}-14 A \,c^{3} x^{3}+4 B b \,c^{2} x^{3}-14 A b \,c^{2} x +10 B \,b^{2} c x \right )}{21 \sqrt {x^{4} c +b \,x^{2}}\, c^{3}}\) | \(248\) |
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 = 0.67, size = 73, normalized size = 0.44 \begin {gather*} \frac {2 \, {\left ({\left (5 \, B b^{2} - 7 \, A b c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + {\left (3 \, B c^{2} x^{2} - 5 \, B b c + 7 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{21 \, c^{3} x} \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 {x^{\frac {5}{2}} \left (A + B x^{2}\right )}{\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 = -0.01 \begin {gather*} \int \frac {x^{5/2}\,\left (B\,x^2+A\right )}{\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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