Optimal. Leaf size=243 \[ \frac {4 b^2 (3 b B-5 A c) \sqrt {b x^2+c x^4}}{231 c^3 \sqrt {x}}-\frac {4 b (3 b B-5 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^2}-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {2 b^{11/4} (3 b B-5 A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 c^{13/4} \sqrt {b x^2+c x^4}} \]
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Rubi [A]
time = 0.25, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2064, 2046,
2049, 2057, 335, 226} \begin {gather*} -\frac {2 b^{11/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} (3 b B-5 A c) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{231 c^{13/4} \sqrt {b x^2+c x^4}}+\frac {4 b^2 \sqrt {b x^2+c x^4} (3 b B-5 A c)}{231 c^3 \sqrt {x}}-\frac {4 b x^{3/2} \sqrt {b x^2+c x^4} (3 b B-5 A c)}{385 c^2}-\frac {2 x^{7/2} \sqrt {b x^2+c x^4} (3 b B-5 A c)}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 335
Rule 2046
Rule 2049
Rule 2057
Rule 2064
Rubi steps
\begin {align*} \int x^{5/2} \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx &=\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {\left (2 \left (\frac {9 b B}{2}-\frac {15 A c}{2}\right )\right ) \int x^{5/2} \sqrt {b x^2+c x^4} \, dx}{15 c}\\ &=-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {(2 b (3 b B-5 A c)) \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx}{55 c}\\ &=-\frac {4 b (3 b B-5 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^2}-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}+\frac {\left (2 b^2 (3 b B-5 A c)\right ) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{77 c^2}\\ &=\frac {4 b^2 (3 b B-5 A c) \sqrt {b x^2+c x^4}}{231 c^3 \sqrt {x}}-\frac {4 b (3 b B-5 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^2}-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {\left (2 b^3 (3 b B-5 A c)\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{231 c^3}\\ &=\frac {4 b^2 (3 b B-5 A c) \sqrt {b x^2+c x^4}}{231 c^3 \sqrt {x}}-\frac {4 b (3 b B-5 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^2}-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {\left (2 b^3 (3 b B-5 A c) x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{231 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {4 b^2 (3 b B-5 A c) \sqrt {b x^2+c x^4}}{231 c^3 \sqrt {x}}-\frac {4 b (3 b B-5 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^2}-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {\left (4 b^3 (3 b B-5 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 )}{231 c^3 \sqrt {b x^2+c x^4}}\\ &=\frac {4 b^2 (3 b B-5 A c) \sqrt {b x^2+c x^4}}{231 c^3 \sqrt {x}}-\frac {4 b (3 b B-5 A c) x^{3/2} \sqrt {b x^2+c x^4}}{385 c^2}-\frac {2 (3 b B-5 A c) x^{7/2} \sqrt {b x^2+c x^4}}{55 c}+\frac {2 B x^{3/2} \left (b x^2+c x^4\right )^{3/2}}{15 c}-\frac {2 b^{11/4} (3 b B-5 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 )}{231 c^{13/4} \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 = 10.13, size = 136, normalized size = 0.56 \begin {gather*} \frac {2 \sqrt {x^2 \left (b+c x^2\right )} \left (\left (b+c x^2\right ) \sqrt {1+\frac {c x^2}{b}} \left (45 b^2 B+7 c^2 x^2 \left (15 A+11 B x^2\right )-3 b c \left (25 A+21 B x^2\right )\right )+15 b^2 (-3 b B+5 A c) \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {c x^2}{b}\right )\right )}{1155 c^3 \sqrt {x} \sqrt {1+\frac {c x^2}{b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 307, normalized size = 1.26
method | result | size |
risch | \(-\frac {2 \left (-77 B \,c^{3} x^{6}-105 A \,c^{3} x^{4}-14 B b \,c^{2} x^{4}-30 A b \,c^{2} x^{2}+18 B \,b^{2} c \,x^{2}+50 A \,b^{2} c -30 B \,b^{3}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{1155 \sqrt {x}\, c^{3}}+\frac {2 b^{3} \left (5 A c -3 B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {x \left (c \,x^{2}+b \right )}}{231 c^{4} \sqrt {c \,x^{3}+b x}\, x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}\) | \(241\) |
default | \(\frac {2 \sqrt {x^{4} c +b \,x^{2}}\, \left (77 B \,c^{5} x^{9}+105 A \,c^{5} x^{7}+91 B b \,c^{4} x^{7}+25 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, b^{3} c +135 A b \,c^{4} x^{5}-15 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, \EllipticF \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}\, b^{4}-4 B \,b^{2} c^{3} x^{5}-20 A \,b^{2} c^{3} x^{3}+12 B \,b^{3} c^{2} x^{3}-50 A \,b^{3} c^{2} x +30 B \,b^{4} c x \right )}{1155 x^{\frac {3}{2}} \left (c \,x^{2}+b \right ) c^{4}}\) | \(307\) |
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 = 0.33, size = 123, normalized size = 0.51 \begin {gather*} -\frac {2 \, {\left (10 \, {\left (3 \, B b^{4} - 5 \, A b^{3} c\right )} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) - {\left (77 \, B c^{4} x^{6} + 30 \, B b^{3} c - 50 \, A b^{2} c^{2} + 7 \, {\left (2 \, B b c^{3} + 15 \, A c^{4}\right )} x^{4} - 6 \, {\left (3 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}\right )}}{1155 \, c^{4} 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 x^{\frac {5}{2}} \sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B 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 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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